Topic: Lehman's 'Bach' scale

19 scales

Thread (442 messages)

From: Carl Lumma (2005-08-18)
Subject: Lehman's 'Bach' scale

I don't want to get into an argument, but I thought I'd
make some observations.  I've read Brad's web site on a
number of occasions.

I'm not entirely clear if Brad is making a claim that his
scale is *the* scale of Bach, or just if Bach's music
sounds good in it.  He hasn't presented any objective way
of testing the latter, so the claim is essentially
meaningless.  It certainly can't be taken as evidence for
the former, as much of his text seems to do.

The other evidence for the tuning -- the squiggles on the
title page of the WTC -- am I the only one who thinks this
is a bit of a stretch?

Many of the other arguments on Brad's website focus on
attacking Barnes, Kellner, and other 'Bach scale'
scholars.  That certainly isn't hard to do, but again it
isn't evidence in favor of Brad's proposal.  As far as
iconography goes, Kellner's page (apparently down at the
moment) on Bach's signet ring was more convincing than
the squiggles.

As for evidence in the first place of there being one
'Bach scale' waiting for us to rediscovered, I haven't
seen much.

-Carl

From: Brad Lehman (2005-08-18)
Subject: Re: Lehman's 'Bach' scale

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> I don't want to get into an argument, but I thought I'd
> make some observations.  I've read Brad's web site on a
> number of occasions.
> 
> I'm not entirely clear if Brad is making a claim that his
> scale is *the* scale of Bach, or just if Bach's music
> sounds good in it.  He hasn't presented any objective way
> of testing the latter, so the claim is essentially
> meaningless.  It certainly can't be taken as evidence for
> the former, as much of his text seems to do.


Carl, the page of musical examples to "test the latter" empirically
is at
http://www-personal.umich.edu/~bpl/larips/testpieces.html
and a shorter version of that is in the second half of the article.  
Reproduce the musical experiment yourself, in practice, to hear what 
is being claimed!

Some practical instructions to set up the temperament for this test 
are at
http://www-personal.umich.edu/~bpl/larips/practical.html
and, of course, one should also set up any other comparative 
temperaments of one's own choice to hear the differences, in the same 
music.

The supplementary file for part 2 is available at
http://em.oxfordjournals.org/cgi/content/full/33/2/211/DC1
giving a closer analysis of BWV 622, 591, and 802-5.

Several Bach compositions are demonstrated directly in recorded 
examples at
http://www-personal.umich.edu/~bpl/larips/samples.html
with four full-length CDs to be released shortly.  Those are played
on organ and harpsichord.

An hour-long interview about this whole thing, including additional 
musical examples played by me and Richard Egarr, will be available 
within the next few weeks at
http://www.bbc.co.uk/radio3/earlymusicshow/
I have not heard Egarr's recording yet myself, but it is a new disc
of the Goldberg Variations.


Bradley Lehman

From: Michael Zapf (2005-08-18)
Subject: Re: [tuning] Lehman's 'Bach' scale

<Kellner's page (apparently down at the
moment)>

Kellner died several years ago, he can't defend
himself anymore.
Michael


	
	
		
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Yahoo! Messenger - NEW crystal clear PC to PC calling worldwide with voicemail http://uk.messenger.yahoo.com

From: rumsong (2005-08-18)
Subject: Re: Lehman's 'Bach' scale

--- In [email protected], Michael Zapf 
<zapfzapfzapf@y...> wrote:
> <Kellner's page (apparently down at the
> moment)>
> 
> Kellner died several years ago, he can't defend
> himself anymore.
> Michael

Greetings,

I had suspected this, but I am very sad to hear of it.  I owed him a 
letter, but life and work prevented my proper reply.

I studied his theory many years ago, and while I do not hold to it, I 
think he worked with honesty towards what he thought was a 
solution to a difficult problem.  Perhaps this is all we can do 
since Truth, Absolute Truth is such a slippery 'thing.'

All best wishes,

Gordon Rumson

From: Carl Lumma (2005-08-19)
Subject: Re: Lehman's 'Bach' scale

> > I don't want to get into an argument, but I thought I'd
> > make some observations.  I've read Brad's web site on a
> > number of occasions.
> > 
> > I'm not entirely clear if Brad is making a claim that his
> > scale is *the* scale of Bach, or just if Bach's music
> > sounds good in it.  He hasn't presented any objective way
> > of testing the latter, so the claim is essentially
> > meaningless.  It certainly can't be taken as evidence for
> > the former, as much of his text seems to do.
> 
> Carl, the page of musical examples to "test the latter"
> empirically is at
> http://www-personal.umich.edu/~bpl/larips/testpieces.html

That's just a list of pieces.  What is the procedure for
testing them?  Notice I used the word "objective".

>and a shorter version of that is in the second half of the
>article.  Reproduce the musical experiment yourself, in
>practice, to hear what is being claimed!

That sounds like going to a faith healer, and 'feeling the
results for myself'.

> Several Bach compositions are demonstrated directly in
> recorded examples at
> http://www-personal.umich.edu/~bpl/larips/samples.html
> with four full-length CDs to be released shortly.  Those are
> played on organ and harpsichord.

These aren't even comparisons -- only your temperament is
demonstrated.

By the way, it seems very unlikely that the Kleines Harmonisches
Labyrinth was written by Bach.

-Carl

From: monz (2005-08-19)
Subject: Re: Lehman's 'Bach' scale

Hi Brad and Michael,


--- In [email protected], Michael Zapf <zapfzapfzapf@y...> wrote:

> > [Brah Lehman:]
> > <Kellner's page (apparently down at the
> > moment)>
> 
> Kellner died several years ago, he can't defend
> himself anymore.



But he was a member of this group before his passing,
and he did indeed defend himself and his work on
"Bach's tuning".

... too bad that the Yahoo search feature sucks so much
that i'm not willing to invest the time to look for
his posts ... but hopefully someone else will.
They're in here.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Aaron Krister Johnson (2005-08-19)
Subject: Re: [tuning] Re: Lehman's 'Bach' scale

On Friday 19 August 2005 12:57 am, monz wrote:
> Hi Brad and Michael,
>
> --- In [email protected], Michael Zapf <zapfzapfzapf@y...> wrote:
> > > [Brah Lehman:]
> > > <Kellner's page (apparently down at the
> > > moment)>
> >
> > Kellner died several years ago, he can't defend
> > himself anymore.
>
> But he was a member of this group before his passing,
> and he did indeed defend himself and his work on
> "Bach's tuning".
>
> ... too bad that the Yahoo search feature sucks so much
> that i'm not willing to invest the time to look for
> his posts ... but hopefully someone else will.
> They're in here.

Where where you Monz, when I suggested we move over to Google groups for that 
very reason? ;)

-Aaron.

From: Brad Lehman (2005-08-19)
Subject: Re: Lehman's 'Bach' scale

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > http://www-personal.umich.edu/~bpl/larips/testpieces.html
> 
> That's just a list of pieces.  What is the procedure for
> testing them?  Notice I used the word "objective".

I have already outlined a suggested procedure, at
http://www-personal.umich.edu/~bpl/larips/faq2.html

- Set up this temperament on a good harpsichord: and preferably by
ear both to understand its structure and because Bach himself had no 
electronic aids. 

- Play through all the musical examples of the article, footnotes,
and supplementary files--listening carefully to the highlighted 
features in the sound. The article is about sound and listening, an 
attempt to hear things inside an aesthetic scheme written down by 
Bach, testing my hypothesis of its correct interpretation and its 
place in Bach's milieu. 

- Play Bach's repertoire, and especially the compositions that had 
been most problematic in other temperaments, listening closely for 
overall balance of musical elements. 

- Do similar listening tests with at least Werckmeister III,
Vallotti, regular 1/6 comma, and equal temperament to hear the 
differences; and additionally with any other proposed "Bach" 
temperaments for further comparison. A point here is to locate the 
Bach expectations within musical and historical context, which 
presupposes the reader's ability to hear and compare this with other 
contemporary layouts (including Neidhardt's and Sorge's, for the 
adventurous and thorough!).


Notice that my suggestion *does not* say listen to MIDI files run 
through a computer.  That in itself might be interesting, but Bach 
could not have done so.  It's up to hands-on playing on acoustic 
instruments, experiencing them by *playing* the repertoire directly 
and noting what it does to your own sense of melody/harmony/timing,
as to the tensions and resolutions in the music.  Bach wrote the WTC 
for musicians to work on in their own keyboard practice and lessons.  
If he somehow also envisioned computer-listening or recording-
listening at some point in his distant future, with people having the 
music played for them passively by a machine, that would really be 
something else!


> > Several Bach compositions are demonstrated directly in
> > recorded examples at
> > http://www-personal.umich.edu/~bpl/larips/samples.html
> > with four full-length CDs to be released shortly.  Those are
> > played on organ and harpsichord.
> 
> These aren't even comparisons -- only your temperament is
> demonstrated.

And those are for people who don't play keyboards at all, so they can 
at least hear the music on appropriate acoustic instruments with the 
proposed tuning.  Broader comparisons require actually doing some 
homework: finding other comparative recordings by other people, and 
(better yet) hands-on playing of the repertoire oneself.  I'm not 
going to have multiple additional recording sessions and production 
cost, just to excuse people from going and doing their own homework 
and investigations....!


> 
> By the way, it seems very unlikely that the Kleines Harmonisches
> Labyrinth was written by Bach.

I have addressed exactly that question of authenticity in the Oxford 
supplement to part 2 of the article, as part of discussing that
piece.  Some _Bach-Jahrbuch_ articles, and indeed a recent volume of 
the _Neue Bach-Ausgabe_ (2003) as well, do include this piece in the 
Bach repertoire after all.  The books/articles I've cited are from 
2000, 2001, and 2003 on this.  And yes, I know that Ledbetter in 2002 
and Williams in 2003 continue to vote it out, and that the 1998 BWV 
still has it in the "questionable" index.  

Whether some people would allow this composition as a testing example 
or not, based *only* on the question whether Bach himself wrote it
(to everybody's unanimous satisfaction), is to miss the broader
point.  The piece, as it stands, is a useful one with which to
explore temperament issues hands-on.  And it's a fairly easy one to 
play, too, on any keyboard: with only one optional pedal-point in the 
last section, or just restrike it a couple of times if there's no 
pedalboard available.

As I mentioned here a couple of days ago, the four Duetti BWV 802-5 
are even better in that regard as hands-on test pieces, and they are 
unquestionably by JSB.  There's nowhere for any less-than-worthy 
temperament to hide in their brutally exposed texture of two voices, 
either melodically or with two-note intervals harmonically.


Bradley Lehman

From: monz (2005-08-19)
Subject: migration of the tuning list (was: Lehman's 'Bach' scale)

--- In [email protected], Aaron Krister Johnson <aaron@a...> 
wrote:

> On Friday 19 August 2005 12:57 am, monz wrote:
> >
> > ... too bad that the Yahoo search feature sucks so much
> > that i'm not willing to invest the time to look for
> > his posts ... but hopefully someone else will.
> > They're in here.
> 
> Where where you Monz, when I suggested we move over
> to Google groups for that very reason? ;)



I was busy working on the new Tonalsoft website ...
where *we* plan to eventually host the tuning list!


-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Carl Lumma (2005-08-19)
Subject: Re: Lehman's 'Bach' scale

> > > http://www-personal.umich.edu/~bpl/larips/testpieces.html
> > 
> > That's just a list of pieces.  What is the procedure for
> > testing them?  Notice I used the word "objective".
> 
> I have already outlined a suggested procedure, at
> http://www-personal.umich.edu/~bpl/larips/faq2.html
> 
> - Set up this temperament on a good harpsichord: and preferably by
> ear both to understand its structure and because Bach himself had
> no electronic aids. 
> 
> - Play through all the musical examples of the article, footnotes,
> and supplementary files--listening carefully to the highlighted 
> features in the sound. The article is about sound and listening, an 
> attempt to hear things inside an aesthetic scheme written down by 
> Bach, testing my hypothesis of its correct interpretation and its 
> place in Bach's milieu. 
> 
> - Play Bach's repertoire, and especially the compositions that had 
> been most problematic in other temperaments, listening closely for 
> overall balance of musical elements. 
> 
> - Do similar listening tests with at least Werckmeister III,
> Vallotti, regular 1/6 comma, and equal temperament to hear the 
> differences; and additionally with any other proposed "Bach" 
> temperaments for further comparison. A point here is to locate the 
> Bach expectations within musical and historical context, which 
> presupposes the reader's ability to hear and compare this with
> other contemporary layouts (including Neidhardt's and Sorge's,
> for the  adventurous and thorough!).
> 
> Notice that my suggestion *does not* say listen to MIDI files run 
> through a computer.

This isn't an objective test, it's a subjective one.  Who cares
if I do this and think your temperament sounds 'better' in some
way?  Who cares if I think Werckmeister III sounds better?

-Carl

From: Brad Lehman (2005-08-20)
Subject: Re: Lehman's 'Bach' scale

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > > > http://www-personal.umich.edu/~bpl/larips/testpieces.html
> > > 
> > > That's just a list of pieces.  What is the procedure for
> > > testing them?  Notice I used the word "objective".
> > 
> > I have already outlined a suggested procedure, at
> > http://www-personal.umich.edu/~bpl/larips/faq2.html
> > 
> > - Set up this temperament on a good harpsichord: and preferably by
> > ear both to understand its structure and because Bach himself had
> > no electronic aids. 
> > 
> > - Play through all the musical examples of the article, footnotes,
> > and supplementary files--listening carefully to the highlighted 
> > features in the sound. The article is about sound and listening, 
an 
> > attempt to hear things inside an aesthetic scheme written down by 
> > Bach, testing my hypothesis of its correct interpretation and its 
> > place in Bach's milieu. 
> > 
> > - Play Bach's repertoire, and especially the compositions that
had 
> > been most problematic in other temperaments, listening closely
for 
> > overall balance of musical elements. 
> > 
> > - Do similar listening tests with at least Werckmeister III,
> > Vallotti, regular 1/6 comma, and equal temperament to hear the 
> > differences; and additionally with any other proposed "Bach" 
> > temperaments for further comparison. A point here is to locate
the 
> > Bach expectations within musical and historical context, which 
> > presupposes the reader's ability to hear and compare this with
> > other contemporary layouts (including Neidhardt's and Sorge's,
> > for the  adventurous and thorough!).
> > 
> > Notice that my suggestion *does not* say listen to MIDI files run 
> > through a computer.
> 
> This isn't an objective test, it's a subjective one.  Who cares
> if I do this and think your temperament sounds 'better' in some
> way?  Who cares if I think Werckmeister III sounds better?


The point is not to evaluate things as better/worse/whatever.  The 
point is to experience them directly in the first place, in the same 
way that Bach would have done: sitting and working directly at an 
acoustic instrument, hands-on, playing through the music carefully in 
different ways to hear what is there, and to experience any effect it 
might have on one's own musicality or conception of the piece.

How might one explain the concept of "orange" to a person who has 
never troubled to view an orange object directly?  Orange isn't 
qualitatively better or worse than other colors; it's just something 
to experience directly rather than speculating about/against.  It's 
some measurable range of a wave pattern, objectively.  One might 
quibble what the boundaries of orange are, on either side toward the 
neighboring colors of different frequencies, sure.  But the simplest 
way to distinguish among orange, green, and violet is to look at them 
side by side with a fair and controlled test....  Whether someone 
personally prefers the green or the violet ahead of the orange is 
beside the point.

Your question, in the first place, was how to try my results 
empirically.  I recommended some ways to do so, and in Bach
repertoire where differences are especially noticeable (i.e. where it 
might most affect someone's choice of a temperament for a Bach 
concert).  A refusal to go do it, then, is just backpedaling against 
an attempt to be empirical, isn't it?....  Empiricism only when it's 
not too inconvenient, and won't take time/effort?


Brad Lehman

From: Carl Lumma (2005-08-21)
Subject: Re: Lehman's 'Bach' scale

> Your question, in the first place, was how to try my results
> empirically.  I recommended some ways to do so, and in Bach
> repertoire where differences are especially noticeable (i.e.
> where it might most affect someone's choice of a temperament
> for a Bach concert).  A refusal to go do it, then, is just
> backpedaling against an attempt to be empirical, isn't it?....
> Empiricism only when it's not too inconvenient, and won't take
> time/effort?

My point is that you've made some very strong claims for
something that you derived through what can only be called
far-fetched means.  I don't doubt that your scale is a
fine well temperament, distinguishable (under very careful
listening) from equal or even other well temperaments.  I
do doubt that Bach intended a particular scale for the WTC.
I believe he, like most keyboardists, had a favored method
he used to tune his intruments, which he probably taught to
his students.  But I am not convinced that your method
corresponds to it.  If your point is, "Hey, well temperaments
are great!" then great.  But there are lots of people saying
that.  You distinguish yourself by making a particular
argument, and I happen to think it's unconvincing.  That's
all.

-Carl

From: George D. Secor (2005-08-24)
Subject: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> --- In [email protected], "Brad Lehman" <bpl@u...> wrote:
> > Your question, in the first place, was how to try my results
> > empirically.  I recommended some ways to do so, and in Bach
> > repertoire where differences are especially noticeable (i.e.
> > where it might most affect someone's choice of a temperament
> > for a Bach concert).  A refusal to go do it, then, is just
> > backpedaling against an attempt to be empirical, isn't it?....
> > Empiricism only when it's not too inconvenient, and won't take
> > time/effort?
> 
> My point is that you've made some very strong claims for
> something that you derived through what can only be called
> far-fetched means.  I don't doubt that your scale is a
> fine well temperament, 

Not so fast, Carl!  You'll find the specifics of the Lehman-"Bach" 
temperament here:
http://www-personal.umich.edu/~bpl/larips/math.html

After looking at the numbers, I must say that I have a few doubts.  
Here's a table showing the total error in cents of the intervals in 
each major triad:

Db Ab Eb Bb F. C. G. D. A. E. B. F#
39 35 31 23 20 20 27 35 43 39 35 31

For comparison, following is the total error of the intervals of a 
major triad in each of the "big three" historical regular 
temperaments:

Meantone: 11 cents
12-ET: 31 cents
Pythagorean: 43 cents

In the Lehman-"Bach" temperament the numbers progressively decrease 
going from Db to F and then level off at C -- so far so good.  They 
then increase going from C to A; the intonation favors the keys in 
the flat direction (a minor flaw IMO, characteristic of many well-
temperaments, including Werckmeister III), so there's nothing 
unexpected there.  The numbers then *decrease* going from A to F#, 
which is exactly the *opposite* of what is expected (and desired) in 
a well-temperament -- a very serious flaw indeed.  This leads me to 
think that either Brad has not properly deciphered some of 
Bach's "squiggles", or if he has, then Bach's temperament is not a 
very good one.  While you might get some interesting results playing 
some of Bach's music in it, I would not expect anything in the key of 
A major (its worst triad) to fare particularly "well" (pun intended.

Besides this, there's a wide fifth (Bb-F) that, strictly speaking, 
would by itself exclude this temperament from the "well-temperament" 
category, inasmuch as this causes the total error of the intervals in 
the 12 major triads (379.3c) to exceed the minimum theoretically 
possible (387.5c).  My reason for bringing this point up is not to 
nitpick about a categorical technicality, but rather to make the 
point that the inclusion of a wide fifth is *totally unnecessary* in 
a temperament in which the narrowest fifths are tempered only 1/6 of 
a pythagorean comma (see following paragraph).

I found on this website:
http://www.rollingball.com/TemperamentsFrames.htm
a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799) that's 
a perfect (even if "unauthentic") alternative to Lehman-"Bach".  It 
has 1/6-p-comma fifths in a chain from C to F# and just fifths from 
F#=Gb to C.  If you check the numbers in the following table, you'll 
see that none of the major triads from Eb to E (corresponding to the 
8 usable ones in a standard meantone tuning) has an error 
significantly exceeding that of 12-ET:

Major triad:.. Db Ab Eb Bb F. C. G. D. A. E. B. F#
Lehman-"Bach": 39 35 31 23 20 20 27 35 43 39 35 31
Valotti-Young: 43 43 35 26 20 20 20 20 27 35 43 43
Werckmst. III: 43 43 31 20 _8 20 31 31 31 31 43 43

I've also included Werckmeister III for comparison; although it has 
room for improvement, at least its numbers demonstrate a reasonable 
progression of intonation from the best to worst keys, and we *do 
know* that Bach actually played organs that were so tuned.  Maybe he 
tuned his harpsichord this way (at least sometimes), or maybe he 
tuned it differently -- I'll leave that for others to debate.  But I 
would have a difficult time accepting the notion that Bach intended 
his _Well-tempered Clavier_ to be played in a temperament unworthy of 
the name "well-tempered".

Brad, it was not my intention to be unkind in my criticism of your 
work, and I hope that you'll be open to reviewing your methodology in 
light of my observations.

Best,

--George

From: George D. Secor (2005-08-24)
Subject: Re: Further doubts about Lehman's 'Bach' scale

Correction:

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> ... inasmuch as this causes the total error of the intervals in 
> the 12 major triads (379.3c) to exceed the minimum theoretically 
> possible (387.5c).  ...

Sorry!  This should have read:

... inasmuch as this causes the total error of the intervals in 
the 12 major triads (379.3c) to exceed the minimum theoretically 
possible (375.4c).  ...

--George

From: wallyesterpaulrus (2005-08-24)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

[...]

Hiya George,

Why not show the total absolute errors in the intervals in each of the 
minor triads as well? I mean, as long as you're showing it in each of 
the major triads already . . .

Cheers,
Paul

From: Ozan Yarman (2005-08-24)
Subject: Re: [tuning] Further doubts about Lehman's 'Bach' scale

George, I'm still quite in the dark as to why you think Bach would not have approved MY well-temperament. It IS very easy to tune by ear too. 8-)

Cordially,
Ozan


12-tone Well Temperament from practically 159tET
|
    0:          1/1                 C    Dbb   unison, perfect prime
  1:         90.225 cents   C#   Db
  2:        196.090 cents   D    Ebb
  3:        294.135 cents   D#   Eb
  4:        392.180 cents   E    Fb
  5:        498.045 cents   F    Gbb
  6:        588.270 cents   F#   Gb
  7:        701.955 cents   G    Abb
  8:        792.180 cents   G#   Ab
  9:        898.045 cents   A    Bbb
 10:        996.090 cents   A#   Bb
 11:       1086.315 cents   B    Cb
 12:          2/1                     C    Dbb   octave


  0:         0.000 cents   0.000    0     0 commas                               C
  7:       701.955 cents  -0.000    0     0 commas                             G
  2:       694.135 cents  -7.820   -240  -1/3 Pyth. commas              D
  9:       701.955 cents  -7.820   -240  -1/3 Pyth. commas              A
  4:       694.135 cents  -15.640  -480  -2/3 Pyth. commas              E
 11:      694.135 cents  -23.460  -720  -1 Pyth. commas                B
  6:       701.955 cents  -23.460  -720  -1 Pyth. commas                F#
  1:       701.955 cents  -23.460  -720  -1 Pyth. commas                C#
  8:       701.955 cents  -23.460  -720  -1 Pyth. commas                G#
  3:       701.955 cents  -23.460  -720  -1 Pyth. commas                Eb
 10:      701.955 cents  -23.460  -720  -1 Pyth. commas                Bb
  5:       701.955 cents  -23.460  -720  -1 Pyth. commas                F
 12:      701.955 cents  -23.460  -720  -Pythagorean comma, ditonic co C
Average absolute difference:     18.2467 cents
Root mean square difference:     20.8237 cents
Maximum absolute difference:     23.4600 cents
Maximum formal fifth difference: 7.8200 cents


 0-4-7: Major Triad  diff.  5.866, 0.000
  C E G
 1-5-8: Major Triad  diff.  21.506, 0.000
  C# F G#
  2-6-9: Major Triad  diff.  5.866, 0.000
  D F# A
 3-7-10: Major Triad  diff.  21.506, 0.000
  Eb G Bb
 4-8-11: Major Triad  diff.  13.686,-7.820
  E G# B
 5-9-12: Major Triad  diff.  13.686, 0.000
  F A C
6-10-13: Major Triad  diff.  21.506, 0.000
  F# Bb C#
 7-11-14: Major Triad  diff. -1.954,-7.820
  G B D
 8-12-15: Major Triad  diff.  21.506, 0.000
  G# C Eb
  9-13-16: Major Triad  diff.  5.866,-7.820
  A C# E
  10-14-17: Major Triad  diff.  13.686, 0.000
  Bb D F

  ----- Original Message ----- 
  From: George D. Secor 
  To: [email protected] 
  Sent: 24 Ağustos 2005 Çarşamba 22:45 
  Subject: [tuning] Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
  > --- In [email protected], "Brad Lehman" <bpl@u...> wrote:
  > > Your question, in the first place, was how to try my results
  > > empirically.  I recommended some ways to do so, and in Bach
  > > repertoire where differences are especially noticeable (i.e.
  > > where it might most affect someone's choice of a temperament
  > > for a Bach concert).  A refusal to go do it, then, is just
  > > backpedaling against an attempt to be empirical, isn't it?....
  > > Empiricism only when it's not too inconvenient, and won't take
  > > time/effort?
  > 
  > My point is that you've made some very strong claims for
  > something that you derived through what can only be called
  > far-fetched means.  I don't doubt that your scale is a
  > fine well temperament, 

  Not so fast, Carl!  You'll find the specifics of the Lehman-"Bach" 
  temperament here:
  http://www-personal.umich.edu/~bpl/larips/math.html

  After looking at the numbers, I must say that I have a few doubts.  
  Here's a table showing the total error in cents of the intervals in 
  each major triad:

  Db Ab Eb Bb F. C. G. D. A. E. B. F#
  39 35 31 23 20 20 27 35 43 39 35 31

  For comparison, following is the total error of the intervals of a 
  major triad in each of the "big three" historical regular 
  temperaments:

  Meantone: 11 cents
  12-ET: 31 cents
  Pythagorean: 43 cents

  In the Lehman-"Bach" temperament the numbers progressively decrease 
  going from Db to F and then level off at C -- so far so good.  They 
  then increase going from C to A; the intonation favors the keys in 
  the flat direction (a minor flaw IMO, characteristic of many well-
  temperaments, including Werckmeister III), so there's nothing 
  unexpected there.  The numbers then *decrease* going from A to F#, 
  which is exactly the *opposite* of what is expected (and desired) in 
  a well-temperament -- a very serious flaw indeed.  This leads me to 
  think that either Brad has not properly deciphered some of 
  Bach's "squiggles", or if he has, then Bach's temperament is not a 
  very good one.  While you might get some interesting results playing 
  some of Bach's music in it, I would not expect anything in the key of 
  A major (its worst triad) to fare particularly "well" (pun intended.

  Besides this, there's a wide fifth (Bb-F) that, strictly speaking, 
  would by itself exclude this temperament from the "well-temperament" 
  category, inasmuch as this causes the total error of the intervals in 
  the 12 major triads (379.3c) to exceed the minimum theoretically 
  possible (387.5c).  My reason for bringing this point up is not to 
  nitpick about a categorical technicality, but rather to make the 
  point that the inclusion of a wide fifth is *totally unnecessary* in 
  a temperament in which the narrowest fifths are tempered only 1/6 of 
  a pythagorean comma (see following paragraph).

  I found on this website:
  http://www.rollingball.com/TemperamentsFrames.htm
  a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799) that's 
  a perfect (even if "unauthentic") alternative to Lehman-"Bach".  It 
  has 1/6-p-comma fifths in a chain from C to F# and just fifths from 
  F#=Gb to C.  If you check the numbers in the following table, you'll 
  see that none of the major triads from Eb to E (corresponding to the 
  8 usable ones in a standard meantone tuning) has an error 
  significantly exceeding that of 12-ET:

  Major triad:.. Db Ab Eb Bb F. C. G. D. A. E. B. F#
  Lehman-"Bach": 39 35 31 23 20 20 27 35 43 39 35 31
  Valotti-Young: 43 43 35 26 20 20 20 20 27 35 43 43
  Werckmst. III: 43 43 31 20 _8 20 31 31 31 31 43 43

  I've also included Werckmeister III for comparison; although it has 
  room for improvement, at least its numbers demonstrate a reasonable 
  progression of intonation from the best to worst keys, and we *do 
  know* that Bach actually played organs that were so tuned.  Maybe he 
  tuned his harpsichord this way (at least sometimes), or maybe he 
  tuned it differently -- I'll leave that for others to debate.  But I 
  would have a difficult time accepting the notion that Bach intended 
  his _Well-tempered Clavier_ to be played in a temperament unworthy of 
  the name "well-tempered".

  Brad, it was not my intention to be unkind in my criticism of your 
  work, and I hope that you'll be open to reviewing your methodology in 
  light of my observations.

  Best,

  --George

From: Carl Lumma (2005-08-24)
Subject: Re: Further doubts about Lehman's 'Bach' scale

> > My point is that you've made some very strong claims for
> > something that you derived through what can only be called
> > far-fetched means.  I don't doubt that your scale is a
> > fine well temperament, 
> 
> Not so fast, Carl!  You'll find the specifics of the
> Lehman-"Bach" temperament here:
> http://www-personal.umich.edu/~bpl/larips/math.html
> 
> After looking at the numbers, I must say that I have a few
> doubts.  Here's a table showing the total error in cents of
> the intervals in each major triad:
> 
> Db Ab Eb Bb F. C. G. D. A. E. B. F#
> 39 35 31 23 20 20 27 35 43 39 35 31
> 
> For comparison, following is the total error of the intervals
> of a major triad in each of the "big three" historical regular 
> temperaments:
> 
> Meantone: 11 cents
> 12-ET: 31 cents
> Pythagorean: 43 cents

This makes it look like the average figure for 12-tone scales
can be less than 31.  I think a more apt comparison would be to
show all 12 keys of some popular well temperaments.

Personally I'd use only the errors of the major third and fifth.
I don't hear nearly as much of a difference in the consonance of
minor thirds in the range of 290-320 cents as of major thirds in
the 380-410 cents range.

> In the Lehman-"Bach" temperament the numbers progressively
> decrease going from Db to F and then level off at C -- so far
> so good.  They then increase going from C to A; the intonation
> favors the keys in the flat direction (a minor flaw IMO,
> characteristic of many well-temperaments, including Werckmeister
> III), so there's nothing unexpected there.  The numbers then
> *decrease* going from A to F#, which is exactly the *opposite*
> of what is expected (and desired) in  a well-temperament -- a
> very serious flaw indeed.  This leads me to think that either
> Brad has not properly deciphered some of Bach's "squiggles", or
> if he has, then Bach's temperament is not a very good one.
> While you might get some interesting results playing some of
> Bach's music in it, I would not expect anything in the key of 
> A major (its worst triad) to fare particularly "well" (pun
> intended.

In fact I hadn't noticed this.  Good point.
 
> the total error of the intervals in the 12 major
> triads (379.3c) to exceed the minimum theoretically
> possible (387.5c).

There are maybe some rounding errors going on... above you
say the value for a 12-tET triad is 31 cents; here the minimum
possible is over 32.

-Carl

From: wallyesterpaulrus (2005-08-24)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> Personally I'd use only the errors of the major third and fifth.
> I don't hear nearly as much of a difference in the consonance of
> minor thirds in the range of 290-320 cents as of major thirds in
> the 380-410 cents range.

But in the context of a major triad, which is what George was talking 
about, I find that the tuning of the minor third matters just as much 
(particularly if you allow for inversions). A triad like 0-400-720 
sounds markedly better to me than 0-380-720, even though though the 
latter has a better major third and the same perfect fifth. Similar 
conclusions held for me when I compared different triads with narrow 
perfect fifths.

From: wallyesterpaulrus (2005-08-24)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> 
> > Personally I'd use only the errors of the major third and fifth.
> > I don't hear nearly as much of a difference in the consonance of
> > minor thirds in the range of 290-320 cents as of major thirds in
> > the 380-410 cents range.
> 
> But in the context of a major triad, which is what George was talking 
> about, I find that the tuning of the minor third matters just as much 
> (particularly if you allow for inversions). A triad like 0-400-720 
> sounds markedly better to me than 0-380-720,

Oops -- that was supposed to be 390, not 380.

> even though though the 
> latter has a better major third and the same perfect fifth. Similar 
> conclusions held for me when I compared different triads with narrow 
> perfect fifths.

Perhaps a better example given your ranges would be to follow George 
Secor and compare the 22-equal (0-382-709) and 19-equal (0-379-695) 
major triads. The 19-equal one is a good deal more concordant, and the 
explanation clearly can't lie with either the 'fifth' or the 'major 
third'. Try this and some other examples out for yourself and see if 
you don't agree.

From: Ozan Yarman (2005-08-24)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Paul, could the "proportional stretch" factor explain the triadic consonances you mentioned? I noticed similar things myself during my investigations.

Cordially,
Ozan

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 25 Ağustos 2005 Perşembe 0:32 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "wallyesterpaulrus" 
  <wallyesterpaulrus@y...> wrote:
  > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
  > 
  > > Personally I'd use only the errors of the major third and fifth.
  > > I don't hear nearly as much of a difference in the consonance of
  > > minor thirds in the range of 290-320 cents as of major thirds in
  > > the 380-410 cents range.
  > 
  > But in the context of a major triad, which is what George was talking 
  > about, I find that the tuning of the minor third matters just as much 
  > (particularly if you allow for inversions). A triad like 0-400-720 
  > sounds markedly better to me than 0-380-720,

  Oops -- that was supposed to be 390, not 380.

  > even though though the 
  > latter has a better major third and the same perfect fifth. Similar 
  > conclusions held for me when I compared different triads with narrow 
  > perfect fifths.

  Perhaps a better example given your ranges would be to follow George 
  Secor and compare the 22-equal (0-382-709) and 19-equal (0-379-695) 
  major triads. The 19-equal one is a good deal more concordant, and the 
  explanation clearly can't lie with either the 'fifth' or the 'major 
  third'. Try this and some other examples out for yourself and see if 
  you don't agree.

From: Carl Lumma (2005-08-25)
Subject: Re: Further doubts about Lehman's 'Bach' scale

> > A triad like 0-400-720 
> > sounds markedly better to me than 0-380-720,
> 
> Oops -- that was supposed to be 390, not 380.

720 is out of well temperament range for a 5th.

> Perhaps a better example given your ranges would be to follow
> George Secor and compare the 22-equal (0-382-709) and
> 19-equal (0-379-695) major triads. The 19-equal one is a good
> deal more concordant, and the explanation clearly can't lie
> with either the 'fifth' or the 'major third'. Try this and some
> other examples out for yourself and see if you don't agree.

That's not in my ranges, with a 327 minor third in the first case.

I only hear 19's triads (in root position) as a little bit more
concordant than 22's.

This does bear further investigation...

-Carl

From: Bradley P Lehman (2005-08-25)
Subject: re: Further doubts about Lehman's 'Bach' scale

> Subject: Further doubts about Lehman's 'Bach' scale
> (...)  While you might get some interesting results playing
> some of Bach's music in it, I would not expect anything in the key of
> A major (its worst triad) to fare particularly "well" (pun intended.
>

Rather than "expecting" some, or speculating, how about tuning a 
harpsichord and an organ this way and playing some?  From more than a year 
of doing so, A major has become one of my favorite keys to play in and 
listen to, with its melodic smoothness and the brightness of those sharps. 
The note C#, as it turns out, is exactly midway between A and F.  One 
could turn your observations right around and say them the other way: in 
other temperaments, the interval Db-F is so much worse than A-C# that the 
music played using it (such as quite a bit of Bach's in F minor) "doesn't 
fare particularly well".


> Besides this, there's a wide fifth (Bb-F) that, strictly speaking,
> would by itself exclude this temperament from the "well-temperament"
> category, inasmuch as this causes the total error of the intervals in
> the 12 major triads (379.3c) to exceed the minimum theoretically
> possible (387.5c).

No, look again: *strictly speaking* that interval is the diminished 6th 
A#-F.  I've explained this thoroughly in my FAQ pages; please take a look.
http://www-personal.umich.edu/~bpl/larips/faq3.html

Better yet, read the whole paper too, and try its musical points hands-on 
at a harpsichord, to hear the effects in real music.  Believe me, I was as 
skeptical/dubious as anybody BEFORE setting it up on harpsichord and 
playing it for a bit; it looks ludicrous on paper IN SPECULATION but music 
isn't speculation.

Also keep in mind that your argument from "the well-temperament category" 
here is based on a *modern* definition made up by Owen Jorgensen et al; 
there's nothing historically that would constrain Bach to conform to 
categories made up a couple hundred years after he was gone.


> My reason for bringing this point up is not to
> nitpick about a categorical technicality, but rather to make the
> point that the inclusion of a wide fifth is *totally unnecessary* in
> a temperament in which the narrowest fifths are tempered only 1/6 of
> a pythagorean comma (see following paragraph).

I see that Jorgensen's speculative notions about the undesirability of 
"harmonic waste" are alive and well.  But within such a paradigm, where 
the expectations *have been made up in the 20th century*, of course my 
solution is going to be judged and found wanting.  The key here is to come 
to it with 17th century expectations and habits, and then to see/hear how 
new it is against that background.  Against *that* relief it sounds so 
smoothly equal and so unremarkable that it doesn't call a huge amount of 
attention *to itself*, but rather it calls attention to the *the music* 
and the way tonal progressions behave.

Keep in mind that Bach was as master of harmony, both simple and complex. 
A couple of days ago I played through the "Schmucke dich" BWV 654 and the 
"Nun komm der heiden Heiland" BWV 659, marveling at the blend of 
harmonic/melodic motion in these pieces, in this tuning (there is a pipe 
organ tuned this way in Indiana, and a second one being completed now in 
Helsinki).  In that same session I played through the "Liebster Jesu" 
settings BWV 633-634 [in A major!] with their canons, suspended 6ths, D#s, 
etc...*musically* it's wonderful.  And obviously there's no way I can 
convince anybody who would prefer his/her own speculations on paper ahead 
of listening to and actually *playing* the music hands-on.

For any who are willing to listen, there is a BBC radio 3 program about 
this on Sunday, and available on the web all next week:
http://www.bbc.co.uk/radio3/earlymusicshow/pip/z9s45/


>
> I found on this website:
> http://www.rollingball.com/TemperamentsFrames.htm
> a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799) that's
> a perfect (even if "unauthentic") alternative to Lehman-"Bach".

What do you mean "unauthentic"?  Werckmeister knew the simple layout of 
six chained tempered 5ths, contrasted against the other six chained pure 
5ths, in the 1680s as a typical Venetian style.  And his own publications 
were an attempt to do something different.  I've addressed this point in 
my article.  The "Vallotti" style of tuning is *older than* the 
Neidhardts, Werckmeisters, and Sorges!


> But I
> would have a difficult time accepting the notion that Bach intended
> his _Well-tempered Clavier_ to be played in a temperament unworthy of
> the name "well-tempered".

Again: "unworthy of the name 'well-tempered'" is going by definitions that 
were made up in the 20th century.  Why would Bach care about that?  My 
hypothesis is that Bach used the term _wohltemperirt_ to mean this 
specific layout that he wrote down himself; and if some people *today* 
can't accept that it might have a wide "5th" (really a diminished 6th) in 
it, or the little zigzag where B major is better in tune than E major is, 
that's a problem of modern expectations, not a problem of Bach's practice. 
And to see what's in this hypothesis, please read the actual article and 
try the music as it recommends, rather than relying solely on internet 
chatter about it.


> Brad, it was not my intention to be unkind in my criticism of your
> work, and I hope that you'll be open to reviewing your methodology in
> light of my observations.

I haven't taken offense at your remarks, George; it's only that I've 
already covered all that ground myself in the months *before* writing the 
article, and indeed in the 21 years of working as a harpsichord tuner 
before that.  Your remarks here haven't told me anything yet that I didn't 
already consider *before* putting my boldly anti-establishment stuff into 
the article.  I know that this article calls for a paradigm shift in the 
way people think about and listen to temperaments, and it took me some 
months myself to realize what was necessary to say in it.  I had to drop 
the encrusted habits of 20+ years out the window myself before taking this 
different harmonic balance seriously myself.  And then, having heard it 
and played in it for a while, I had to figure out the theory as to why it 
works.  The layout of intervals is different enough that it calls for its 
own methods of analysis, arising from the sounds produced.  Its handling 
of tonics/dominants is simple and logical, *after* one is able to dump old 
habits and expectations out the window and play directly in the music, to 
hear what's there.


Bradley Lehman
http://www.larips.com

From: George D. Secor (2005-08-25)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], Bradley P Lehman <bpl@u...> wrote:
> > Subject: Further doubts about Lehman's 'Bach' scale
> > (...)  While you might get some interesting results playing
> > some of Bach's music in it, I would not expect anything in the 
key of
> > A major (its worst triad) to fare particularly "well" (pun 
intended.
> >
> 
> Rather than "expecting" some, or speculating, how about tuning a 
> harpsichord and an organ this way and playing some?  From more than 
a year 
> of doing so, A major has become one of my favorite keys to play in 
and 
> listen to, with its melodic smoothness and the brightness of those 
sharps. 
> The note C#, as it turns out, is exactly midway between A and F.  
One 
> could turn your observations right around and say them the other 
way: in 
> other temperaments, the interval Db-F is so much worse than A-C# 
that the 
> music played using it (such as quite a bit of Bach's in F 
minor) "doesn't 
> fare particularly well".

I agree with much of what you have to say.  I had my own piano tuned 
to my own #2 well-temperament for about 5 years (did it myself, by 
ear) and, as I observed in a previous message (#59697):

<< After a very short time I found that, as I became accustomed to 
the differences in intonation in the various keys, I tended to hear 
the "remote" pythagorean triads as having a different "mood" rather 
than being unacceptably out of tune.  I might compare my first 
reaction to swimming in cold water: initially shocking, but 
ultimately invigorating. >>

Obviously *some* of the triads in a circulating unequal temperament 
will be "worse" than others, so the question comes down to 
determining which ones.  As long as we agree that the temperament 
you've presented is at variance with conventional expectations (that 
an F# major triad should sound more dissonant than A), then I suppose 
I should drop my objections (which were on acoustic grounds) and 
leave it to others to debate the other issues.

> > Besides this, there's a wide fifth (Bb-F) that, strictly speaking,
> > would by itself exclude this temperament from the "well-
temperament"
> > category, inasmuch as this causes the total error of the 
intervals in
> > the 12 major triads (379.3c) to exceed the minimum theoretically
> > possible (387.5c).
> 
> No, look again: *strictly speaking* that interval is the diminished 
6th 
> A#-F.  I've explained this thoroughly in my FAQ pages; please take 
a look.
> http://www-personal.umich.edu/~bpl/larips/faq3.html

Yes, it can be a diminished 6th in certain contexts (such as your 
tuning routine), but musically those two tones occur much more often 
as a 5th, Bb-F.

> Better yet, read the whole paper too, and try its musical points 
hands-on 
> at a harpsichord, to hear the effects in real music.  

If only I had the time.  :-(

> Believe me, I was as 
> skeptical/dubious as anybody BEFORE setting it up on harpsichord 
and 
> playing it for a bit; it looks ludicrous on paper IN SPECULATION 
but music 
> isn't speculation.
> 
> Also keep in mind that your argument from "the well-temperament 
category" 
> here is based on a *modern* definition made up by Owen Jorgensen et 
al; 
> there's nothing historically that would constrain Bach to conform 
to 
> categories made up a couple hundred years after he was gone.

Yes, I was aware of that, hence my following disclaimer:

> > My reason for bringing this point up is not to
> > nitpick about a categorical technicality, but rather to make the
> > point that the inclusion of a wide fifth is *totally unnecessary* 
in
> > a temperament in which the narrowest fifths are tempered only 1/6 
of
> > a pythagorean comma (see following paragraph).
> 
> I see that Jorgensen's speculative notions about the undesirability 
of 
> "harmonic waste" are alive and well.  

I aquired that notion independently.  I devised my #2 (well-
temperament) and #3 (temperament ordinaire) tunings in 1975, several 
years before I learned about Jorgensen's work (from a copy of the 
limited-edition _Tuning the Historical Temperaments by Ear_ that I 
purchased around 1978), by which I discovered that my #2 tuning met 
all of Jorgensen's (rather redundant) conditions for a well-
temperament.  My #3 temperament has considerable "harmonic waste", 
but that's unavoidable (and therefore acceptable), given the purpose 
for which it's intended.

> But within such a paradigm, where 
> the expectations *have been made up in the 20th century*, of course 
my 
> solution is going to be judged and found wanting.  The key here is 
to come 
> to it with 17th century expectations and habits, and then to 
see/hear how 
> new it is against that background.  Against *that* relief it sounds 
so 
> smoothly equal and so unremarkable that it doesn't call a huge 
amount of 
> attention *to itself*, but rather it calls attention to the *the 
music* 
> and the way tonal progressions behave.
> 
> Keep in mind that Bach was as master of harmony, both simple and 
complex. 
> A couple of days ago I played through the "Schmucke dich" BWV 654 
and the 
> "Nun komm der heiden Heiland" BWV 659, marveling at the blend of 
> harmonic/melodic motion in these pieces, in this tuning (there is a 
pipe 
> organ tuned this way in Indiana, and a second one being completed 
now in 
> Helsinki).  In that same session I played through the "Liebster 
Jesu" 
> settings BWV 633-634 [in A major!] with their canons, suspended 
6ths, D#s, 
> etc...*musically* it's wonderful.  And obviously there's no way I 
can 
> convince anybody who would prefer his/her own speculations on paper 
ahead 
> of listening to and actually *playing* the music hands-on.

It may be argued that if you listen to practically any unfamiliar 
tuning enough times, you'll eventually get accustomed to it and might 
end up preferring it for that (and no other) reason.

Brad, if you really want to make a convincing demonstration that 
you've discovered Bach's keyboard temperament, then I think that you 
should also make parallel recordings of some of these selections in 
the (theoretically "perfect" alternative) Valotti-Young 1/6-
pythagorean-comma well-temperament (on the same instrument) so that 
anyone can make a head-to-head comparison.  I realize that it may not 
be very practical to retune an entire organ, but I think a 
harpsichord would suffice.  The listener would be allowed repeated 
hearings in order to become accustomed to each tuning and could then 
draw conclusions.

> For any who are willing to listen, there is a BBC radio 3 program 
about 
> this on Sunday, and available on the web all next week:
> http://www.bbc.co.uk/radio3/earlymusicshow/pip/z9s45/
> 
> > I found on this website:
> > http://www.rollingball.com/TemperamentsFrames.htm
> > a 1/6-pythagorean-comma well-temperament (Valotti-Young, 1799) 
that's
> > a perfect (even if "unauthentic") alternative to Lehman-"Bach".
> 
> What do you mean "unauthentic"?  Werckmeister knew the simple 
layout of 
> six chained tempered 5ths, contrasted against the other six chained 
pure 
> 5ths, in the 1680s as a typical Venetian style.  And his own 
publications 
> were an attempt to do something different.  I've addressed this 
point in 
> my article.  The "Vallotti" style of tuning is *older than* the 
> Neidhardts, Werckmeisters, and Sorges!

Okay, I stand corrected.  My chief interest in these temperaments is 
in arriving at theoretically optimal or elegant solutions, rather 
than involvement with historical details.  I first encountered the 
tuning as "Young's No. 2" in Barbour's _Tuning and Temperament_ and 
had the impression that it was post-Bach.  The head-to-head 
comparison I suggested above would then be between two 
arguably "authentic" tunings.

> > But I
> > would have a difficult time accepting the notion that Bach 
intended
> > his _Well-tempered Clavier_ to be played in a temperament 
unworthy of
> > the name "well-tempered".
> 
> Again: "unworthy of the name 'well-tempered'" is going by 
definitions that 
> were made up in the 20th century.  Why would Bach care about that?  
My 
> hypothesis is that Bach used the term _wohltemperirt_ to mean this 
> specific layout that he wrote down himself; and if some people 
*today* 
> can't accept that it might have a wide "5th" (really a diminished 
6th) in 
> it, or the little zigzag where B major is better in tune than E 
major is, 
> that's a problem of modern expectations, not a problem of Bach's 
practice. 
> And to see what's in this hypothesis, please read the actual 
article and 
> try the music as it recommends, rather than relying solely on 
internet 
> chatter about it.

Point taken (IF your interpretation of Bach's notations are correct).

> > Brad, it was not my intention to be unkind in my criticism of your
> > work, and I hope that you'll be open to reviewing your 
methodology in
> > light of my observations.
> 
> I haven't taken offense at your remarks, George; it's only that 
I've 
> already covered all that ground myself in the months *before* 
writing the 
> article, and indeed in the 21 years of working as a harpsichord 
tuner 
> before that.  Your remarks here haven't told me anything yet that I 
didn't 
> already consider *before* putting my boldly anti-establishment 
stuff into 
> the article.  I know that this article calls for a paradigm shift 
in the 
> way people think about and listen to temperaments, and it took me 
some 
> months myself to realize what was necessary to say in it.  I had to 
drop 
> the encrusted habits of 20+ years out the window myself before 
taking this 
> different harmonic balance seriously myself.  

Okay, I can appreciate that.

> And then, having heard it 
> and played in it for a while, I had to figure out the theory as to 
why it 
> works.  The layout of intervals is different enough that it calls 
for its 
> own methods of analysis, arising from the sounds produced.  Its 
handling 
> of tonics/dominants is simple and logical, *after* one is able to 
dump old 
> habits and expectations out the window and play directly in the 
music, to 
> hear what's there.

And even if the rest of us can get past that, it will still take 
quite a bit more convincing.  You're facing a uphill battle, my 
friend, but I wish you the best.

--George

From: George D. Secor (2005-08-25)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> George, I'm still quite in the dark as to why you think Bach would 
not have approved MY well-temperament. It IS very easy to tune by ear 
too. 8-)
> 
> Cordially,
> Ozan
> 
> 12-tone Well Temperament from practically 159tET
> |
>   0:          1/1           C    Dbb   unison, perfect prime
>   1:         90.225 cents   C#   Db
>   2:        196.090 cents   D    Ebb
>   3:        294.135 cents   D#   Eb
>   4:        392.180 cents   E    Fb
>   5:        498.045 cents   F    Gbb
>   6:        588.270 cents   F#   Gb
>   7:        701.955 cents   G    Abb
>   8:        792.180 cents   G#   Ab
>   9:        898.045 cents   A    Bbb
>  10:        996.090 cents   A#   Bb
>  11:       1086.315 cents   B    Cb
>  12:          2/1                     C    Dbb   octave

The alternation of narrow and just 5ths that Gene noted produces a 
zig-zag progression in intonation going (by 5ths) from F to A.  You 
also have 6 triads with total error (43) equal to pythagorean:

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Ozan Yarman: 43 43 43 27 27 12 16 12 27 43 43 43
Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

As the numbers for my #2 temperament indicate, it's possible to have 
3 triads with error in the lower 'teens (1/4-comma meantone error = 
11) in combination with 5 ~pythagorean triads.  If you want meantone-
like intonation in your 3 best triads in combination with only 4 
triads having significantly greater error than 12-ET, you'd have to 
have some wide fifths (as in my #3 temperament).

As for whether Bach would approve of your temperament, after hearing 
what Brad had to say, perhaps I should not be so prejudicial.  ;-)

Best,

--George

From: George D. Secor (2005-08-25)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> --- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> 
> [...]
> 
> Hiya George,
> 
> Why not show the total absolute errors in the intervals in each of 
the 
> minor triads as well? I mean, as long as you're showing it in each of 
> the major triads already . . .

Because I don't have the figures handy.  I set up a spreadsheet that 
only calculates the individual and total errors in the major triads, 
but there's an easy way to determine these for the minor triads.  As 
long as there are no 5ths wider than just or major 3rds narrower than 
just, the major and minor minor triads contained in a major-7th chord 
will have the same total error (i.e., twice the error of the minor 3rd 
of the triad).  Thus the F major and A minor triads in a well-
temperament will have the same total error.

--George

From: Ozan Yarman (2005-08-25)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

So, did I finally made it into the "temperament league"? Does this well temperament of mine (12 out of practically 159-tET) deserve to be called "Yarman #5" ?

Then the others would be:

Yarman #1 : 12-tone 79 MOS 159-tET
Yarman #2:  12-tone 67 MOS 135-tET
Yarman #3:  12-tone 55 MOS 277-tET
Yarman #4:  12-tone pure fourths (inverse of `Cordier pure fifths`)

Cordially,
Ozan



  ----- Original Message ----- 
  From: George D. Secor 
  To: [email protected] 
  Sent: 25 Ağustos 2005 Perşembe 20:35 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  > George, I'm still quite in the dark as to why you think Bach would 
  not have approved MY well-temperament. It IS very easy to tune by ear 
  too. 8-)
  > 
  > Cordially,
  > Ozan
  > 
  > 12-tone Well Temperament from practically 159tET
  > |
  >   0:          1/1           C    Dbb   unison, perfect prime
  >   1:         90.225 cents   C#   Db
  >   2:        196.090 cents   D    Ebb
  >   3:        294.135 cents   D#   Eb
  >   4:        392.180 cents   E    Fb
  >   5:        498.045 cents   F    Gbb
  >   6:        588.270 cents   F#   Gb
  >   7:        701.955 cents   G    Abb
  >   8:        792.180 cents   G#   Ab
  >   9:        898.045 cents   A    Bbb
  >  10:        996.090 cents   A#   Bb
  >  11:       1086.315 cents   B    Cb
  >  12:          2/1                     C    Dbb   octave

  The alternation of narrow and just 5ths that Gene noted produces a 
  zig-zag progression in intonation going (by 5ths) from F to A.  You 
  also have 6 triads with total error (43) equal to pythagorean:

  Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
  Ozan Yarman: 43 43 43 27 27 12 16 12 27 43 43 43
  Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
  Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

  As the numbers for my #2 temperament indicate, it's possible to have 
  3 triads with error in the lower 'teens (1/4-comma meantone error = 
  11) in combination with 5 ~pythagorean triads.  If you want meantone-
  like intonation in your 3 best triads in combination with only 4 
  triads having significantly greater error than 12-ET, you'd have to 
  have some wide fifths (as in my #3 temperament).

  As for whether Bach would approve of your temperament, after hearing 
  what Brad had to say, perhaps I should not be so prejudicial.  ;-)

  Best,

  --George

From: George D. Secor (2005-08-25)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> So, did I finally made it into the "temperament league"? 

Yep!

> Does this well temperament of mine (12 out of practically 159-tET) 
deserve to be called "Yarman #5" ?

It's your privilege to number them as you wish.

--George

From: Ozan Yarman (2005-08-26)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Great!! but dear George, can you show me how to practically tune the cent values in a modified 1/4 comma meantone:


  0:          1/1           C    Dbb   unison, perfect prime
  1:         76.049 cents   C#   Db
  2:        193.157 cents   D    Ebb
  3:        279.471 cents   D#   Eb
  4:        386.314 cents   E    Fb
  5:        493.157 cents   F    Gbb
  6:        579.471 cents   F#   Gb
  7:        696.578 cents   G    Abb
  8:        772.627 cents   G#   Ab
  9:        889.735 cents   A    Bbb
 10:        986.314 cents   A#   Bb
 11:       1082.892 cents   B    Cb
 12:       1200.000 cents   C    Dbb


  0:         0.000 cents   0.000    0     0 commas                      C
  7:       696.578 cents  -5.377   -165  -1/4 synt. commas              G
  2:       696.578 cents  -10.753  -330  -1/2 synt. commas              D
  9:       696.578 cents  -16.130  -495  -3/4 synt. commas              A
  4:       696.578 cents  -21.506  -660  -1 synt. commas                E
 11:       696.578 cents  -26.883  -825  -5/4 synt. commas              B
  6:       696.578 cents  -32.259  -990  -3/2 synt. commas              F#
  1:       696.578 cents  -37.636  -1155 -7/4 synt. commas              C#
  8:       696.578 cents  -43.013  -1320 -2 synt. commas                G#
  3:       706.843 cents  -38.124  -1170 -39/22 synt. commas            Eb
 10:       706.843 cents  -33.236  -1020 -17/11 synt. commas            Bb
  5:       706.843 cents  -28.348  -870  -29/24 Pyth. commas            F
 12:       706.843 cents  -23.460  -720  -1 Pyth. commas                C
Average absolute difference:     26.3938 cents
Root mean square difference:     29.8622 cents
Maximum absolute difference:     43.0126 cents
Maximum formal fifth difference: 5.3766 cents



so that I can achieve equal beatings for the major triads? What is the most feasible formula that calibrates the beats in a tempered 4:5:6 to equal? Specifically, how can I distrubute the `wolf-fifth remnant` to the tones in an optimal manner so as to do minimal damage in consonant interval approximations?

Cordially,
Ozan

  ----- Original Message ----- 
  From: George D. Secor 
  To: [email protected] 
  Sent: 26 Ağustos 2005 Cuma 0:38 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  > So, did I finally made it into the "temperament league"? 

  Yep!

  > Does this well temperament of mine (12 out of practically 159-tET) 
  deserve to be called "Yarman #5" ?

  It's your privilege to number them as you wish.

  --George





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From: wallyesterpaulrus (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

Hi Ozan,

I have no idea what you mean by 'the "proportional stretch" factor'. 
Would you explain?

Thanks,
Paul


--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Paul, could the "proportional stretch" factor explain the triadic 
consonances you mentioned? I noticed similar things myself during my 
investigations.
> 
> Cordially,
> Ozan
> 
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 25 Aðustos 2005 Perþembe 0:32 
>   Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
> 
> 
>   --- In [email protected], "wallyesterpaulrus" 
>   <wallyesterpaulrus@y...> wrote:
>   > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>   > 
>   > > Personally I'd use only the errors of the major third and 
fifth.
>   > > I don't hear nearly as much of a difference in the consonance 
of
>   > > minor thirds in the range of 290-320 cents as of major thirds 
in
>   > > the 380-410 cents range.
>   > 
>   > But in the context of a major triad, which is what George was 
talking 
>   > about, I find that the tuning of the minor third matters just 
as much 
>   > (particularly if you allow for inversions). A triad like 0-400-
720 
>   > sounds markedly better to me than 0-380-720,
> 
>   Oops -- that was supposed to be 390, not 380.
> 
>   > even though though the 
>   > latter has a better major third and the same perfect fifth. 
Similar 
>   > conclusions held for me when I compared different triads with 
narrow 
>   > perfect fifths.
> 
>   Perhaps a better example given your ranges would be to follow 
George 
>   Secor and compare the 22-equal (0-382-709) and 19-equal (0-379-
695) 
>   major triads. The 19-equal one is a good deal more concordant, 
and the 
>   explanation clearly can't lie with either the 'fifth' or 
the 'major 
>   third'. Try this and some other examples out for yourself and see 
if 
>   you don't agree.

From: wallyesterpaulrus (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> > Perhaps a better example given your ranges would be to follow
> > George Secor and compare the 22-equal (0-382-709) and
> > 19-equal (0-379-695) major triads. The 19-equal one is a good
> > deal more concordant, and the explanation clearly can't lie
> > with either the 'fifth' or the 'major third'. Try this and some
> > other examples out for yourself and see if you don't agree.
> 
> That's not in my ranges, with a 327 minor third in the first case.

Did you try other examples? How about 0-385-705 vs. 0-382-698?

From: wallyesterpaulrus (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> > --- In [email protected], "George D. Secor" <gdsecor@y...> 
wrote:
> > 
> > [...]
> > 
> > Hiya George,
> > 
> > Why not show the total absolute errors in the intervals in each 
of 
> the 
> > minor triads as well? I mean, as long as you're showing it in 
each of 
> > the major triads already . . .
> 
> Because I don't have the figures handy.  I set up a spreadsheet 
that 
> only calculates the individual and total errors in the major 
triads, 
> but there's an easy way to determine these for the minor triads.  
As 
> long as there are no 5ths wider than just or major 3rds narrower 
than 
> just, the major and minor minor triads contained in a major-7th 
chord 
> will have the same total error (i.e., twice the error of the minor 
3rd 
> of the triad).  Thus the F major and A minor triads in a well-
> temperament will have the same total error.
> 
> --George

Egads, you're right! So for some of the tunings, you really didn't 
need the figures to be 'handy' -- just rotate the figures for the 
major triads four positions and you get the figures for the minor 
triads. But that wouldn't have worked for the other tunings, because 
some of the fifths were wider than pure?

P.S. I can't see what sense it makes to talk about A#-F as a 
diminished sixth, while not also talking about B-Gb, etc., as 
diminished sixths, in the context of a circulating 12-note fixed 
tuning system.

From: George D. Secor (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Great!! but dear George, can you show me how to practically tune 
the cent values in a modified 1/4 comma meantone:
> 
> 
>   0:          1/1           C    Dbb   unison, perfect prime
>   1:         76.049 cents   C#   Db
>   2:        193.157 cents   D    Ebb
>   3:        279.471 cents   D#   Eb
>   4:        386.314 cents   E    Fb
>   5:        493.157 cents   F    Gbb
>   6:        579.471 cents   F#   Gb
>   7:        696.578 cents   G    Abb
>   8:        772.627 cents   G#   Ab
>   9:        889.735 cents   A    Bbb
>  10:        986.314 cents   A#   Bb
>  11:       1082.892 cents   B    Cb
>  12:       1200.000 cents   C    Dbb

Sorry, Ozan.  I know how to calculate beat rates, but tuning routines 
aren't one of my areas of expertise.

Could someone else please help?

--George

From: George D. Secor (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Great!! but dear George, can you show me how to practically tune 
the cent values in a modified 1/4 comma meantone:
> ...
> so that I can achieve equal beatings for the major triads? What is 
the most feasible formula that calibrates the beats in a tempered 
4:5:6 to equal? Specifically, how can I distrubute the `wolf-fifth 
remnant` to the tones in an optimal manner so as to do minimal damage 
in consonant interval approximations?

Wait a minute -- I think I may have partially misread your first 
question (which had a couple of tables inserted), but in any case, 
there needs to be a little clarification.

By "equal beatings", do you mean:
1) Equal beat-rates for intervals *within* a single triad, or
2) Equal beat rates for the same interval in two different triads?

Whatever the case, this is something that I haven't thought about 
until quite recently, so I'm not the best person to answer this.

> What is the most feasible formula that calibrates the beats in a 
tempered 4:5:6 to equal?

Do you mean "calculates the beats ... "?  If I needed to do that, I'd 
set it up in a spreadsheet, but there must be a resource somewhere 
out on the Internet that will do it for you.

> Specifically, how can I distrubute the `wolf-fifth remnant` to the 
tones in an optimal manner so as to do minimal damage in consonant 
interval approximations?

I can't find any wolf fifth here:

>   0:          1/1           C    Dbb   unison, perfect prime
>   1:         76.049 cents   C#   Db
>   2:        193.157 cents   D    Ebb
>   3:        279.471 cents   D#   Eb
>   4:        386.314 cents   E    Fb
>   5:        493.157 cents   F    Gbb
>   6:        579.471 cents   F#   Gb
>   7:        696.578 cents   G    Abb
>   8:        772.627 cents   G#   Ab
>   9:        889.735 cents   A    Bbb
>  10:        986.314 cents   A#   Bb
>  11:       1082.892 cents   B    Cb
>  12:       1200.000 cents   C    Dbb

Are you referring to the wolf fifth in a strict 12-tone 1/4-comma 
meantone temperament?  It's not too hard to eliminate the wolf fifth, 
but the answer to your question depends on how much you're willing to 
temper the thirds.  For some examples of modified meantone 
temperaments, check out:

http://www.rollingball.com/TemperamentsFrames.htm

--George

From: George D. Secor (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> --- In [email protected], "George D. Secor" <gdsecor@y...> 
wrote:
> > ... there's an easy way to determine these for the minor triads.  
As 
> > long as there are no 5ths wider than just or major 3rds narrower 
than 
> > just, the major and minor minor triads contained in a major-7th 
chord 
> > will have the same total error (i.e., twice the error of the 
minor 3rd 
> > of the triad).  Thus the F major and A minor triads in a well-
> > temperament will have the same total error.
> > 
> > --George
> 
> Egads, you're right! So for some of the tunings, you really didn't 
> need the figures to be 'handy' -- just rotate the figures for the 
> major triads four positions and you get the figures for the minor 
> triads. 

Yep!

> But that wouldn't have worked for the other tunings, because 
> some of the fifths were wider than pure?

Yes, in triads with wide fifths the total error is twice the absolute 
error of the third having the larger absolute error (generally the 
major 3rd).  In the Lehman-Bach temperament Bb-F is a wide fifth, and 
Bb-D has 2 cents greater absolute error than D-F.

--George

> P.S. I can't see what sense it makes to talk about A#-F as a 
> diminished sixth, while not also talking about B-Gb, etc., as 
> diminished sixths, in the context of a circulating 12-note fixed 
> tuning system.

Yeah, I thought the same thing, even after reading the link to Brad's 
FAQ.

--George

From: Ozan Yarman (2005-08-26)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Simple! If you stretch the fifth by 20 cents, you should stretch the major third as much as 1 over 702/386=0.54985754985754985754985754985766, which makes it 10.1 cents to achieve harmonic consonance.

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 26 Ağustos 2005 Cuma 21:56 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  Hi Ozan,

  I have no idea what you mean by 'the "proportional stretch" factor'. 
  Would you explain?

  Thanks,
  Paul

From: Carl Lumma (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

> > > Perhaps a better example given your ranges would be to follow
> > > George Secor and compare the 22-equal (0-382-709) and
> > > 19-equal (0-379-695) major triads. The 19-equal one is a good
> > > deal more concordant, and the explanation clearly can't lie
> > > with either the 'fifth' or the 'major third'. Try this and some
> > > other examples out for yourself and see if you don't agree.
> > 
> > That's not in my ranges, with a 327 minor third in the first case.
> 
> Did you try other examples? How about 0-385-705 vs. 0-382-698?

I clearly prefer the Maj3rd and 5th of the first chord, and
the min3rd of the second, in two different timbres.  With
a clarinet, both triads sound the same except the first one
beats more.  With an english horn, both sound the same except
the second one beats more.

What do you hear?

Anybody else?

-Carl

From: wallyesterpaulrus (2005-08-26)
Subject: Re: Further doubts about Lehman's 'Bach' scale

I get 11 cents, not 10.1 cents, according to your own calculation. 
Sure, this is one way of looking at it, but it seems less mysterious 
simply (or firstly) to acknowledge that the mistuning of the minor 
third matters too (as well as the mistunings of the fifth and major 
third), especially if we are looking for an 'explanation'. You wrote:

'Paul, could the "proportional stretch" factor explain the triadic 
consonances you mentioned? I noticed similar things myself during my 
investigations.'

Although trying to stretch things proportionally may make sense, and 
one may be able to derive it as a solution in certain circumstances 
under certain assumptions, it doesn't seem to serve as 
an 'explanation' for anything. Instead, it's something that might 
present itself in someone's investigations and brainstormings and 
then cry out for an explanation itself. In the case of a triad with 
given fifth, the explanation seems to be that the mistuning of the 
minor third matters too.



--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Simple! If you stretch the fifth by 20 cents, you should stretch 
the major third as much as 1 over 
702/386=0.54985754985754985754985754985766, which makes it 10.1 cents 
to achieve harmonic consonance.
> 
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 26 Aðustos 2005 Cuma 21:56 
>   Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
> 
> 
>   Hi Ozan,
> 
>   I have no idea what you mean by 'the "proportional stretch" 
factor'. 
>   Would you explain?
> 
>   Thanks,
>   Paul

From: Ozan Yarman (2005-08-27)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Yes! sorry, you are right, it was 11 cents. My mistake there. But Paul, the mistuning of the minor third is - obviously - directly correlated to the mistuning of the major third in a consonant triad.

I have merely proposed a model to explain the phenomenon observed. By no means do I make any ontological claims in that respect. If the model works, then a triad indeed should require the "proportional stretch" formula to sound consonant. Do you have any relevant evidence to the contrary?

Cordially,
Ozan

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 27 Ağustos 2005 Cumartesi 1:39 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  I get 11 cents, not 10.1 cents, according to your own calculation. 
  Sure, this is one way of looking at it, but it seems less mysterious 
  simply (or firstly) to acknowledge that the mistuning of the minor 
  third matters too (as well as the mistunings of the fifth and major 
  third), especially if we are looking for an 'explanation'. You wrote:

  'Paul, could the "proportional stretch" factor explain the triadic 
  consonances you mentioned? I noticed similar things myself during my 
  investigations.'

  Although trying to stretch things proportionally may make sense, and 
  one may be able to derive it as a solution in certain circumstances 
  under certain assumptions, it doesn't seem to serve as 
  an 'explanation' for anything. Instead, it's something that might 
  present itself in someone's investigations and brainstormings and 
  then cry out for an explanation itself. In the case of a triad with 
  given fifth, the explanation seems to be that the mistuning of the 
  minor third matters too.



  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  > Simple! If you stretch the fifth by 20 cents, you should stretch 
  the major third as much as 1 over 
  702/386=0.54985754985754985754985754985766, which makes it 10.1 cents 
  to achieve harmonic consonance.
  > 
  >   ----- Original Message ----- 
  >   From: wallyesterpaulrus 
  >   To: [email protected] 
  >   Sent: 26 Ağustos 2005 Cuma 21:56 
  >   Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
  > 
  > 
  >   Hi Ozan,
  > 
  >   I have no idea what you mean by 'the "proportional stretch" 
  factor'. 
  >   Would you explain?
  > 
  >   Thanks,
  >   Paul




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From: Ozan Yarman (2005-08-27)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Dear George, by "equal beating" I mean firstly:

1) Equal beat-rates for intervals *within* a single triad,

and secondly:


2) Equal beat rates for the same interval in two different triads.

if you are in the know as to how to convert pitch information to vibrations per second, I may perhaps demonstrate whether or not I understand how to tune to `equal (or rather, even) beating`.

I am indeed referring to the wolf of 1/4 comma meantone which I `tempered` - so to speak - in such a way as to make sure the fifths do not howl anymore. But I am not sure how to distribute the remnant optimally so as to preserve the `equal-beating` property.

Does your modified meantone contain any `equal-beating` intervals?

Cordially,
Ozan

  ----- Original Message ----- 
  From: George D. Secor 
  To: [email protected] 
  Sent: 27 Ağustos 2005 Cumartesi 0:19 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  > Great!! but dear George, can you show me how to practically tune 
  the cent values in a modified 1/4 comma meantone:
  > ...
  > so that I can achieve equal beatings for the major triads? What is 
  the most feasible formula that calibrates the beats in a tempered 
  4:5:6 to equal? Specifically, how can I distrubute the `wolf-fifth 
  remnant` to the tones in an optimal manner so as to do minimal damage 
  in consonant interval approximations?

  Wait a minute -- I think I may have partially misread your first 
  question (which had a couple of tables inserted), but in any case, 
  there needs to be a little clarification.

  By "equal beatings", do you mean:
  1) Equal beat-rates for intervals *within* a single triad, or
  2) Equal beat rates for the same interval in two different triads?

  Whatever the case, this is something that I haven't thought about 
  until quite recently, so I'm not the best person to answer this.

  > What is the most feasible formula that calibrates the beats in a 
  tempered 4:5:6 to equal?

  Do you mean "calculates the beats ... "?  If I needed to do that, I'd 
  set it up in a spreadsheet, but there must be a resource somewhere 
  out on the Internet that will do it for you.

  > Specifically, how can I distrubute the `wolf-fifth remnant` to the 
  tones in an optimal manner so as to do minimal damage in consonant 
  interval approximations?

  I can't find any wolf fifth here:

  >   0:          1/1           C    Dbb   unison, perfect prime
  >   1:         76.049 cents   C#   Db
  >   2:        193.157 cents   D    Ebb
  >   3:        279.471 cents   D#   Eb
  >   4:        386.314 cents   E    Fb
  >   5:        493.157 cents   F    Gbb
  >   6:        579.471 cents   F#   Gb
  >   7:        696.578 cents   G    Abb
  >   8:        772.627 cents   G#   Ab
  >   9:        889.735 cents   A    Bbb
  >  10:        986.314 cents   A#   Bb
  >  11:       1082.892 cents   B    Cb
  >  12:       1200.000 cents   C    Dbb

  Are you referring to the wolf fifth in a strict 12-tone 1/4-comma 
  meantone temperament?  It's not too hard to eliminate the wolf fifth, 
  but the answer to your question depends on how much you're willing to 
  temper the thirds.  For some examples of modified meantone 
  temperaments, check out:

  http://www.rollingball.com/TemperamentsFrames.htm

  --George

From: wallyesterpaulrus (2005-08-29)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

> Yes! sorry, you are right, it was 11 cents. My mistake there. But 
>Paul, the mistuning of the minor third is - obviously - directly 
>correlated to the mistuning of the major third in a consonant triad.

Sometimes it's independent, for example if you're only specifying 
absolute deviations from JI.

> I have merely proposed a model to explain the phenomenon observed. 
>By no means do I make any ontological claims in that respect. If the 
>model works, then a triad indeed should require the "proportional 
>stretch" formula to sound consonant. Do you have any relevant 
>evidence to the contrary?

Not necessarily evidence, but I can propose a test. I believe that if 
we're talking about a root-position close-voiced major triad, 
approximating 4:5:6, and the 4:6 (or 2:3) is stretched by 20 cents, 
then the triad is likely to sound more concordant when the 4:5 is 
stretched by only 9 1/3 cents (and hence the 5:6 is stretched by 10 
2/3 cents). The reasoning here is that 4:5 is a slightly stronger 
concordance than 5:6; stronger concordances are slightly more 
sensitive to mistuning than weaker ones; thus the 4:5 should be 
mistuned just a bit less than the 5:6 in order to prevent one or the 
other interval from being too damaged by mistuning. So the test would 
be to compare your proposal,

0 - 397 1/3 - 720

against mine,

0 - 395 2/3 - 720

with various timbres and in various registers, and try to determine 
which one sounds concordant. Probably a tough test to get significant 
results with, due to the closeness of the two triads to one another.

> Cordially,
> Ozan

Best,
Paul

From: George D. Secor (2005-08-29)
Subject: Temperament (Extra)ordinare! (Was: Further doubts ...)

HEY EVERYONE!

Having been unable to ignore the discussion about equal-beating 
temperaments lately, I have some startling news about a "temperament 
(extra)ordinaire!"

I say "startling" because I never expected to be thinking about 
dumping a couple of "optimal" tunings I devised 30 years ago in favor 
of something that it took me less than a couple of hours to arrive at 
this past Saturday.

But first I need to answer Ozan's message:

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George, by "equal beating" I mean firstly:
> 
> 1) Equal beat-rates for intervals *within* a single triad,
> 
> and secondly:
> 
> 2) Equal beat rates for the same interval in two different triads.
> 
> if you are in the know as to how to convert pitch information to 
vibrations per second, I may perhaps demonstrate whether or not I 
understand how to tune to `equal (or rather, even) beating`.

Well, whichever you mean, why don't I provide you with a spreadsheet 
I was already working on to calculate frequencies (cols. W-Y), beat 
rates (cols. S-U), beat ratios (cols. O-Q), and errors from just 
(cols. G,I,K,M) from cents (cols.E-F):

http://groups.yahoo.com/group/tuning-math/files/secor/WT-wksht.xls

You can input the cents directly into cells E8-E19, or you can 
specify the departure of each tone from 12-ET, expressed as a 
fraction of a comma (either didymus, pythagorean, archytas/septimal, 
or skhisma) and let the spreadsheet calculate the cents.  Put your 
base frequency for A in K2, the type of comma (d/p/a/s) in G3, the 
fraction of that comma to use as your unit of departure in G4, and 
the numerator of the fraction in B15.  I presently have figures 
entered for a regular 5/17-didymus-comma temperament with A=220, for 
which the intervals in a root-position major triad are all equal-
beating.  If you change G4 to 7 and b15 to -1 (negative number for a 
*narrow* 5th), you'll see that the 1/7th-comma temperament is very 
close to having proportional-beating triads.

Now change G4 to 23 and b15 to -5, and you'll see that the 5/23rd-
comma temperament is very close to having equal-beating 5ths and 
major 3rds (with minor 3rds beating 4X -- more about this below).  
I'm a bit perplexed as to why the ratios between the beat-rates are 
*almost* exact integers (imprecision in the calculations perhaps, but 
I'm not so sure).

Observe in all of the above regular temperaments that F# and Gb are 
different pitches.  The numbers to the right of column F are all 
calculated using Gb (not F#).

If you want to see the Valotti-Young 1/6-pythagorean-comma well-
temperament, put "p" in G3, 6 in G4, -1 in B15, and 0 in B8 through 
B13.  (Note that cents for F# and Gb are now the same.)

You can construct your own temperament either by changing the numbers 
in the light-cyan-colored cells, or you can enter cents directly in 
cells E8-E19.  If your temperament does not conform to Owen 
Jorgensen's strict definition of a well-temperament, this will be 
indicated by one or more asterisks appearing in columns H, J, L, or N.

> I am indeed referring to the wolf of 1/4 comma meantone which I 
`tempered` - so to speak - in such a way as to make sure the fifths 
do not howl anymore. But I am not sure how to distribute the remnant 
optimally so as to preserve the `equal-beating` property.

Now you can experiment using the spreadsheet.  Besides the fractions 
of a didymus comma I already mentioned above, you may find 5/14, 
5/19, and 5/21 to be of interest.

> Does your modified meantone contain any `equal-beating` intervals?

Nope, but my brand-new temperament (extra)ordinare does:

! secor12_1.scl
!
George Secor's 12-tone temperament ordinaire #1, proportional beating
 12
!
 86.53330
 194.55680
 294.12876
 389.11361
 499.91792
 585.54105
 697.27840
 789.37483
 891.83521
 997.96292
 1086.39201

 2/1

It's in the 'temperament ordinaire' category, but after trying it out 
on my Scalatron, I'd have to consider it a bit 'extradordinaire'.

I've made available a temperament worksheet with all sorts of 
figures, should anyone be interested:

http://groups.yahoo.com/group/tuning-math/files/secor/GS_1.xls

Observe that all of the major triads from Eb to E are proportional-
beating (i.e., the beat rates are related by rational integers or 
fractions -- I had to tweak the numbers in column B slightly to get 
the beat-ratios to come out exactly proportional).  I also inserted 
an extra column (G) so that Kraig Grady (and myself) could try this 
on a Scalatron.  (I'd be interested in your opinion, Kraig!)

Here's a summary of how the triad errors compare to my other two 
temperaments:

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
Secor #1:... 54 49 34 21 15 15 15 19 26 37 48 54
Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

Again, for comparison, following is the total error of the intervals 
of a major triad in each of the "big three" historical regular 
temperaments:

1/4-comma meantone: 11 cents
12-ET: 31 cents
Pythagorean: 43 cents

However, the numbers don't tell the whole story (see below).

--- In [email protected], "monz" <monz@t...> wrote: (#59901)
> IMO, when a chord is played in which two or more of
> the intervals beat at the same rate, it sets up an
> unwanted pulse which will probably interfere with the
> rhythmic structure of the piece.

I've observed that this is also occurs to some extent in a triad with 
a just interval, such as 1/3 or 1/4-comma meantone temperament, in 
which the beat rates are proportional.  However, I don't find it 
particularly objectionable -- rather like a good vibrato.

Anyway, I've had a chance to play around with this on my Scalatron 
the past couple of days, and I've decided to call it "#1", because I 
think it tops both my #2 (well-temperament) and (in all respects) my 
#3 (temperament ordinaire) tunings.  (I devised my old #1 temperament 
in 1964, and #2 and #3 eventually replaced it.  Now, IMO, this new #1 
could replace all of them.)

After listening and comparing, I've concluded that the 8 equal- and 
proportional-beating major triads in this temperament sound as if 
they have less total error than the numbers would indicate, with the 
faster beats tending to be masked by the slower ones.  In conducting 
some head-to-head comparisons of the C and G major triads with those 
in 31-ET (12c total error), I found that they sound at least as good; 
likewise the E and Eb major triads as compared with 12-ET.  (The 
Scalatron permits me to jump back and forth instantly from one tuning 
to another, which easily allows me to make comparisons using the same 
timbres.)  I also found, for some reason, that the B and Ab major 
triads sound only about as dissonant as pythagorean triads, even 
though they have about 5 cents greater total error.

I was frankly a bit surprised at how quickly I was able to come up 
with proportional beat rates for so many triads, thinking that it 
should have been a lot harder to do this and beginning to experience 
some feelings of doubt that there must be some room for improvement --
 at this point I don't know.  (But the more I play around with it, 
the less my doubt.)

Anyway, I hope that some others will try this tuning and give me your 
reactions -- at least go through the triads in Scala using the chord 
playing function in the keyboard clavier.  Convince me that this 
isn't all a dream!

--George

From: Ozan Yarman (2005-08-29)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Paul, absolute deviations from JI is not independent of consonance in my opinion. However, I understand and approve of your harmonic entropy model, and agree that 5/4 is more sensitive to mistuning as compared to 6/5.

Therefore, I am eager to hear of the results of the test you proposed. If the good folk of tuning list is willing to participate in it, we can load the MIDI or WAV files up on my website for auditory comparison.

Cordially,
Ozan



  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 0:03 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

  > Yes! sorry, you are right, it was 11 cents. My mistake there. But 
  >Paul, the mistuning of the minor third is - obviously - directly 
  >correlated to the mistuning of the major third in a consonant triad.

  Sometimes it's independent, for example if you're only specifying 
  absolute deviations from JI.

  > I have merely proposed a model to explain the phenomenon observed. 
  >By no means do I make any ontological claims in that respect. If the 
  >model works, then a triad indeed should require the "proportional 
  >stretch" formula to sound consonant. Do you have any relevant 
  >evidence to the contrary?

  Not necessarily evidence, but I can propose a test. I believe that if 
  we're talking about a root-position close-voiced major triad, 
  approximating 4:5:6, and the 4:6 (or 2:3) is stretched by 20 cents, 
  then the triad is likely to sound more concordant when the 4:5 is 
  stretched by only 9 1/3 cents (and hence the 5:6 is stretched by 10 
  2/3 cents). The reasoning here is that 4:5 is a slightly stronger 
  concordance than 5:6; stronger concordances are slightly more 
  sensitive to mistuning than weaker ones; thus the 4:5 should be 
  mistuned just a bit less than the 5:6 in order to prevent one or the 
  other interval from being too damaged by mistuning. So the test would 
  be to compare your proposal,

  0 - 397 1/3 - 720

  against mine,

  0 - 395 2/3 - 720

  with various timbres and in various registers, and try to determine 
  which one sounds concordant. Probably a tough test to get significant 
  results with, due to the closeness of the two triads to one another.

  > Cordially,
  > Ozan

  Best,
  Paul




  You can configure your subscription by sending an empty email to one
  of these addresses (from the address at which you receive the list):
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From: Carl Lumma (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

> ! secor12_1.scl
> !
> George Secor's 12-tone temperament ordinaire #1
>  12
> !
>  86.53330
>  194.55680
>  294.12876
>  389.11361
>  499.91792
>  585.54105
>  697.27840
>  789.37483
>  891.83521
>  997.96292
>  1086.39201
>  2/1
> 
> It's in the 'temperament ordinaire' category, but after trying
> it out on my Scalatron, I'd have to consider it a
> bit 'extradordinaire'.
> 
> I've made available a temperament worksheet with all sorts of 
> figures, should anyone be interested:
> 
> http://groups.yahoo.com/group/tuning-math/files/secor/GS_1.xls

Hiya George (and Ozan),

Do you find you like the triads where all the intervals beat
at the same rate?  I would think it would be better to stagger
the beats, so that they do not result in large amplitude
changes.  Or does this not happen?

-Carl

From: Jon Szanto (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> HEY EVERYONE!

Look, George is shouting with GLEE! :)

> Nope, but my brand-new temperament (extra)ordinare does:
> 
> ! secor12_1.scl

Copied to a file...

> It's in the 'temperament ordinaire' category, but after trying 
> it out on my Scalatron...

George, you *do* realize just how extraordinary the last part of that
sentence is, at least to 99.99999% of the people out here? (the one's
without Scalatrons, that is)

> Anyway, I hope that some others will try this tuning and give me your 
> reactions -- at least go through the triads in Scala using the chord 
> playing function in the keyboard clavier.  Convince me that this 
> isn't all a dream!

Some of the best stuff in the world is in dreams. But I've put it on
my "must check out" pile.

Cheers,
Jon

From: wallyesterpaulrus (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
 
> I'm a bit perplexed as to why the ratios between the beat-rates are 
> *almost* exact integers (imprecision in the calculations perhaps, 
but 
> I'm not so sure).

No, your calculations are quite correct. This observation of *almost* 
exact-integer beat rates has come up a few times here and 
particularly on the tuning-math list. Gene in particular has posted a 
great deal about it -- I hope he'll point you to some relevant 
postings and/or articles.

> I was frankly a bit surprised at how quickly I was able to come up 
> with proportional beat rates for so many triads, thinking that it 
> should have been a lot harder to do this

Bob Wendell's posts on the subject might lead one to believe that 
this is extremely difficult, almost an intractable problem, while 
Gene's paint a different picture.

> and beginning to experience 
> some feelings of doubt that there must be some room for 
improvement --
>  at this point I don't know.  (But the more I play around with it, 
> the less my doubt.)

Again, since Gene has worked a lot on this subject, I'm hoping to see 
him provide you with an illuminating, well-rounded reply and 
assessment of your new tuning systems alongside relevant alternatives.

Gene?

From: Carl Lumma (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

> > I was frankly a bit surprised at how quickly I was able to
> > come up with proportional beat rates for so many triads,
> > thinking that it should have been a lot harder to do this
> 
> Bob Wendell's posts on the subject might lead one to believe that 
> this is extremely difficult, almost an intractable problem, while 
> Gene's paint a different picture.

However, didn't Gene agree that his main tuning, the "Natural
Synchronous Well" was indeed a unique point in the space of
such tunings?

-Carl

From: wallyesterpaulrus (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > > I was frankly a bit surprised at how quickly I was able to
> > > come up with proportional beat rates for so many triads,
> > > thinking that it should have been a lot harder to do this
> > 
> > Bob Wendell's posts on the subject might lead one to believe that 
> > this is extremely difficult, almost an intractable problem, while 
> > Gene's paint a different picture.
> 
> However, didn't Gene agree that his main tuning, the "Natural
> Synchronous Well" was indeed a unique point in the space of
> such tunings?

To the best of my recollection, yes, but really Gene should answer 
this. And I'd love to see a comparison of George's new proposal against 
this and other relevant alternatives.

From: wallyesterpaulrus (2005-08-29)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

> Paul, absolute deviations from JI is not independent of consonance 
>in my opinion.

If you thought I stated something to the contrary, we must have 
misunderstood one another.

> Therefore, I am eager to hear of the results of the test you 
>proposed. If the good folk of tuning list is willing to participate 
>in it, we can load the MIDI or WAV files up on my website for 
>auditory comparison.

I would never trust MIDI files for this purpose, especially as the 
difference between the two triads is of the same order of magnitude as 
the resolution of most sound cards! But I suppose I could prepare a 
series of .wav files for this purpose -- we need a variety of timbres 
and registers to test -- any suggestions? Also, we would need to 
instruct the listener to try several different volume levels for each 
example, or something similar . . .

From: Carl Lumma (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

> > > > I was frankly a bit surprised at how quickly I was able to
> > > > come up with proportional beat rates for so many triads,
> > > > thinking that it should have been a lot harder to do this
> > > 
> > > Bob Wendell's posts on the subject might lead one to believe
> > > that this is extremely difficult, almost an intractable
> > > problem, while Gene's paint a different picture.
> > 
> > However, didn't Gene agree that his main tuning, the "Natural
> > Synchronous Well" was indeed a unique point in the space of
> > such tunings?
> 
> To the best of my recollection, yes, but really Gene should answer 
> this. And I'd love to see a comparison of George's new proposal
> against this and other relevant alternatives.

Bob's scale has brats of 2 and 3:2 on all triads.  George's has
brats of 1 on most triads, 6:5 on one or two, and some more
complex ones (if memory (from 20 min ago) serves).

-Carl

From: Ozan Yarman (2005-08-29)
Subject: Re: [tuning] Temperament (Extra)ordinare! (Was: Further doubts ...)

Hi again George!
  ----- Original Message ----- 
  From: George D. Secor 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 0:43 
  Subject: [tuning] Temperament (Extra)ordinare! (Was: Further doubts ...)


  HEY EVERYONE!

  Having been unable to ignore the discussion about equal-beating 
  temperaments lately, I have some startling news about a "temperament 
  (extra)ordinaire!"


Wowsers!




  I say "startling" because I never expected to be thinking about 
  dumping a couple of "optimal" tunings I devised 30 years ago in favor 
  of something that it took me less than a couple of hours to arrive at 
  this past Saturday.

  But first I need to answer Ozan's message:


Let's get to it!


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  > Dear George, by "equal beating" I mean firstly:
  > 
  > 1) Equal beat-rates for intervals *within* a single triad,
  > 
  > and secondly:
  > 
  > 2) Equal beat rates for the same interval in two different triads.
  > 
  > if you are in the know as to how to convert pitch information to 
  vibrations per second, I may perhaps demonstrate whether or not I 
  understand how to tune to `equal (or rather, even) beating`.

  Well, whichever you mean, why don't I provide you with a spreadsheet 
  I was already working on to calculate frequencies (cols. W-Y), beat 
  rates (cols. S-U), beat ratios (cols. O-Q), and errors from just 
  (cols. G,I,K,M) from cents (cols.E-F):

  http://groups.yahoo.com/group/tuning-math/files/secor/WT-wksht.xls


  SNIP!


  You can construct your own temperament either by changing the numbers 
  in the light-cyan-colored cells, or you can enter cents directly in 
  cells E8-E19.  If your temperament does not conform to Owen 
  Jorgensen's strict definition of a well-temperament, this will be 
  indicated by one or more asterisks appearing in columns H, J, L, or N.



Goodness gracious! It was just the kind of thing I was thinking about yesterday! Dear George, you are truly blessed. :) I congratulate you for providing such a resourceful little gadget. It's the dream of any tuning enthusiast!


  > I am indeed referring to the wolf of 1/4 comma meantone which I 
  `tempered` - so to speak - in such a way as to make sure the fifths 
  do not howl anymore. But I am not sure how to distribute the remnant 
  optimally so as to preserve the `equal-beating` property.

  Now you can experiment using the spreadsheet.  Besides the fractions 
  of a didymus comma I already mentioned above, you may find 5/14, 
  5/19, and 5/21 to be of interest.



I am eager to start my experiments, but how do I input these fractions you mentioned?


  > Does your modified meantone contain any `equal-beating` intervals?

  Nope, but my brand-new temperament (extra)ordinare does:

  ! secor12_1.scl
  !
  George Secor's 12-tone temperament ordinaire #1, proportional beating
  12
  !
  86.53330
  194.55680
  294.12876
  389.11361
  499.91792
  585.54105
  697.27840
  789.37483
  891.83521
  997.96292
  1086.39201

  2/1


It sounds terrific on my organ! Why, this is the best temperament I have ever heard so far and it should obviously qualify to replace the horrible 12-tET default of all the synths currently under production.




  It's in the 'temperament ordinaire' category, but after trying it out 
  on my Scalatron, I'd have to consider it a bit 'extradordinaire'.

  I've made available a temperament worksheet with all sorts of 
  figures, should anyone be interested:

  http://groups.yahoo.com/group/tuning-math/files/secor/GS_1.xls

  Observe that all of the major triads from Eb to E are proportional-
  beating (i.e., the beat rates are related by rational integers or 
  fractions -- I had to tweak the numbers in column B slightly to get 
  the beat-ratios to come out exactly proportional).  




Uh, slow down please! And tell me what beat-rates, or BRATS are and what their significance is in your own words.



  I also inserted 
  an extra column (G) so that Kraig Grady (and myself) could try this 
  on a Scalatron.  (I'd be interested in your opinion, Kraig!)

  Here's a summary of how the triad errors compare to my other two 
  temperaments:

  Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
  Secor #2:... 43 43 36 25 14 11 14 25 36 43 43 43
  Secor #1:... 54 49 34 21 15 15 15 19 26 37 48 54
  Secor #3:... 65 57 36 14 11 11 11 14 25 36 54 65

  Again, for comparison, following is the total error of the intervals 
  of a major triad in each of the "big three" historical regular 
  temperaments:

  1/4-comma meantone: 11 cents
  12-ET: 31 cents
  Pythagorean: 43 cents



And what is the total error for this extra-ordinary temperament of yours?



  However, the numbers don't tell the whole story (see below).

  --- In [email protected], "monz" <monz@t...> wrote: (#59901)
  > IMO, when a chord is played in which two or more of
  > the intervals beat at the same rate, it sets up an
  > unwanted pulse which will probably interfere with the
  > rhythmic structure of the piece.

  I've observed that this is also occurs to some extent in a triad with 
  a just interval, such as 1/3 or 1/4-comma meantone temperament, in 
  which the beat rates are proportional.  However, I don't find it 
  particularly objectionable -- rather like a good vibrato.


Playing with dozens of 12-tone systems just showed me that beats should be calibrated in only a special way, or otherwise the temperament in question becomes intolerable.



  Anyway, I've had a chance to play around with this on my Scalatron 
  the past couple of days, and I've decided to call it "#1", because I 
  think it tops both my #2 (well-temperament) and (in all respects) my 
  #3 (temperament ordinaire) tunings.  (I devised my old #1 temperament 
  in 1964, and #2 and #3 eventually replaced it.  Now, IMO, this new #1 
  could replace all of them.)

  After listening and comparing, I've concluded that the 8 equal- and 
  proportional-beating major triads in this temperament sound as if 
  they have less total error than the numbers would indicate, with the 
  faster beats tending to be masked by the slower ones.  In conducting 
  some head-to-head comparisons of the C and G major triads with those 
  in 31-ET (12c total error), I found that they sound at least as good; 
  likewise the E and Eb major triads as compared with 12-ET.  (The 
  Scalatron permits me to jump back and forth instantly from one tuning 
  to another, which easily allows me to make comparisons using the same 
  timbres.)  I also found, for some reason, that the B and Ab major 
  triads sound only about as dissonant as pythagorean triads, even 
  though they have about 5 cents greater total error.



I can personally say that it is by far the most optimal solution in my opinion that deserves to replace 12-tET utilized in contemporary music education and practice worldwide.


I was frankly a bit surprised at how quickly I was able to come up 
with proportional beat rates for so many triads, thinking that it 
should have been a lot harder to do this and beginning to experience 
some feelings of doubt that there must be some room for improvement --
at this point I don't know.  (But the more I play around with it, 
the less my doubt.)

Anyway, I hope that some others will try this tuning and give me your 
reactions -- at least go through the triads in Scala using the chord 
playing function in the keyboard clavier.  Convince me that this 
isn't all a dream!

--George






It is a dream come true!

Cordially,
Ozan

From: Ozan Yarman (2005-08-29)
Subject: Re: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Carl, do not underestimate the power of the beat! Or in other words, may the force beat with you!  8=)

Cordially,
Ozan



  ----- Original Message ----- 
  From: Carl Lumma 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 1:01 
  Subject: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)


  > ! secor12_1.scl
  > !
  > George Secor's 12-tone temperament ordinaire #1
  >  12
  > !
  >  86.53330
  >  194.55680
  >  294.12876
  >  389.11361
  >  499.91792
  >  585.54105
  >  697.27840
  >  789.37483
  >  891.83521
  >  997.96292
  >  1086.39201
  >  2/1
  > 
  > It's in the 'temperament ordinaire' category, but after trying
  > it out on my Scalatron, I'd have to consider it a
  > bit 'extradordinaire'.
  > 
  > I've made available a temperament worksheet with all sorts of 
  > figures, should anyone be interested:
  > 
  > http://groups.yahoo.com/group/tuning-math/files/secor/GS_1.xls

  Hiya George (and Ozan),

  Do you find you like the triads where all the intervals beat
  at the same rate?  I would think it would be better to stagger
  the beats, so that they do not result in large amplitude
  changes.  Or does this not happen?

  -Carl

From: Ozan Yarman (2005-08-29)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Dear Paul,
  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 2:02 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

  > Paul, absolute deviations from JI is not independent of consonance 
  >in my opinion.

  If you thought I stated something to the contrary, we must have 
  misunderstood one another.


Yes, once more it seems! Hopefully there will be no end to it. ;)


> Therefore, I am eager to hear of the results of the test you 
>proposed. If the good folk of tuning list is willing to participate 
>in it, we can load the MIDI or WAV files up on my website for 
>auditory comparison.

I would never trust MIDI files for this purpose, especially as the 
difference between the two triads is of the same order of magnitude as 
the resolution of most sound cards! But I suppose I could prepare a 
series of .wav files for this purpose -- we need a variety of timbres 
and registers to test -- any suggestions? Also, we would need to 
instruct the listener to try several different volume levels for each 
example, or something similar . . .





Timbres:
Piano, Trumpet, Clarinet, Celesta and Flutes

Common Register:
G3 to C5

We may safely disregard volume at this time I believe.

Cordially,
Ozan

From: Gene Ward Smith (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

> ! secor12_1.scl
> !
> George Secor's 12-tone temperament ordinaire #1, proportional beating

I get 4, ~640/441, 3, 4/3, 2, 2, 14/9, 4, 4/3, 7/3, 3/2, ~795/469
for the brats. It does look interesting.

From: Gene Ward Smith (2005-08-29)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> No, your calculations are quite correct. This observation of *almost* 
> exact-integer beat rates has come up a few times here and 
> particularly on the tuning-math list. Gene in particular has posted a 
> great deal about it -- I hope he'll point you to some relevant 
> postings and/or articles.

I gave a table of meantone fractional comma tunings which had near
low-integer-ratio brats to go with them a few days back.

> Again, since Gene has worked a lot on this subject, I'm hoping to see 
> him provide you with an illuminating, well-rounded reply and 
> assessment of your new tuning systems alongside relevant alternatives.
> 
> Gene?

So far I haven't had a chance to test it, but I wonder what sort of
reply you are looking for. Cooking up alternatives already?

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

> It sounds terrific on my organ! Why, this is the best temperament I 
>have ever heard so far and it should obviously qualify to replace >the 
horrible 12-tET default of all the synths currently under >production.

Unless you plan to force the musical world to suddenly revert to 
spending most of its time in the key of C major and nearby keys, and/or 
to playing mostly early 18th century music, or foresee that such 
regressive changes are for some reason in our future, such a 
replacement would seem extremely inappropriate. As an additional 
transposable tuning *option* (or preset) for synths, along with Robert 
Wendell's and other equal-beating schemes, any of which could be 
employed at the discretion of the performer, I could certainly endorse 
its adoption, though! Perhaps I just took your enthusiasm too 
literally, my dear Ozan.

From: Gene Ward Smith (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> > However, didn't Gene agree that his main tuning, the "Natural
> > Synchronous Well" was indeed a unique point in the space of
> > such tunings?
> 
> To the best of my recollection, yes, but really Gene should answer 
> this. And I'd love to see a comparison of George's new proposal against 
> this and other relevant alternatives.

I'm not sure what you mean by a "unique point", but certainly it
didn;t have the barqoue complexity of George's system, and stuck with
brats that clearly seemed to be worthwhile.

One possibility would be to start with a circulating temperament, and
see how well one could mutate it into something with more of the
interesting beat ratios. Starting with bifrost, for instance, we have
brats of infinity, ~9/7, ~3, 3/2, ~17/9, infinity, 3/2, ~59/50, ~9/4,
3/2, 3/2. If we kept the infinites and 3/2s, and made the 3 an exact
3, how close can we get the others to good ratios?

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

> Timbres:
> Piano, Trumpet, Clarinet, Celesta and Flutes

Interesting. Piano of course gets you into issues of inharmonic 
timbres, and the Celesta into other issues (I really doubt I could hear 
the difference between those two chords on the Celesta). I can 
create .wav files with Matlab but I certainly doubt my ability to 
emulate these instruments with mathematical functions! Maybe someone 
else can come to the rescue here . . .

> Common Register:
> G3 to C5

This would be a range of pitch for the lowest note? Highest note?

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 
> > No, your calculations are quite correct. This observation of 
*almost* 
> > exact-integer beat rates has come up a few times here and 
> > particularly on the tuning-math list. Gene in particular has 
posted a 
> > great deal about it -- I hope he'll point you to some relevant 
> > postings and/or articles.
> 
> I gave a table of meantone fractional comma tunings which had near
> low-integer-ratio brats to go with them a few days back.

Interestingly, I just referenced that table in a post here, but I was 
thinking more about your well-temperament stuff.

> > Again, since Gene has worked a lot on this subject, I'm hoping to 
see 
> > him provide you with an illuminating, well-rounded reply and 
> > assessment of your new tuning systems alongside relevant 
alternatives.
> > 
> > Gene?
> 
> So far I haven't had a chance to test it, but I wonder what sort of
> reply you are looking for. Cooking up alternatives already?

I'm under the impression that George is unfamiliar with a lot of the 
work on equal-beating well-temperaments that Robert Wendell, you, and 
others have done. So someone should start introducing him to it. Of 
course, there are two sometimes different meanings of "equal-beating 
well-temperament" -- well-temperaments in which some of the different 
occurences of like intervals have synchronized beatings, making 
tuning by ear easier (Jorgensen is big on these); and well-
temperaments in which unlike intervals cohabitating in a given 
consonant chord (for all or many occurences of that chord) have 
synchronized beatings, giving the chord a sort of "second-order 
coherence" that may somewhat mitigate its impurity. We seem to be 
discussing the second meaning here.

From: Ozan Yarman (2005-08-30)
Subject: Re: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Huh, I hate enforcing ideas on people. In fact, that is one of the reasons why I stick to this forum given its open-mindedness and all that.

Still, you do a great deal of injustice to George's calculations. One can very well play any kind of 12-tone music in his temperament on any key one likes. The tone-color is a bonus and it is only a matter of which frequency you want to take as basis in order to start the cycle of fifths. I played it, did you?

Maybe it escaped you, but I simply imagined 12-tET to be too crude to be accepted as default. I didn't mean to dump the availabilities out of the window! Simply put, non-microtonal synths can perform much better with George's tuning as default instead of the borin' ol' 12-tET in my opinion.

I for one wish to tune my piano to this temperament straight away.

Great work George!

Cordially,
Ozan

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 3:00 
  Subject: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

  > It sounds terrific on my organ! Why, this is the best temperament I 
  >have ever heard so far and it should obviously qualify to replace >the 
  horrible 12-tET default of all the synths currently under >production.

  Unless you plan to force the musical world to suddenly revert to 
  spending most of its time in the key of C major and nearby keys, and/or 
  to playing mostly early 18th century music, or foresee that such 
  regressive changes are for some reason in our future, such a 
  replacement would seem extremely inappropriate. As an additional 
  transposable tuning *option* (or preset) for synths, along with Robert 
  Wendell's and other equal-beating schemes, any of which could be 
  employed at the discretion of the performer, I could certainly endorse 
  its adoption, though! Perhaps I just took your enthusiasm too 
  literally, my dear Ozan.

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> 
> > > However, didn't Gene agree that his main tuning, the "Natural
> > > Synchronous Well" was indeed a unique point in the space of
> > > such tunings?
> > 
> > To the best of my recollection, yes, but really Gene should 
answer 
> > this. And I'd love to see a comparison of George's new proposal 
against 
> > this and other relevant alternatives.
> 
> I'm not sure what you mean by a "unique point", but certainly it
> didn;t have the barqoue complexity of George's system, and stuck 
with
> brats that clearly seemed to be worthwhile.

Now wouldn't this be a worthwhile fact, if accompanied with plenty of 
supporting data on "Natural Synchronous Well" and other systems of 
Bob's and your devising, and other helpful explanatory writing, to 
point out to George in this thread?

I'm just trying to get some *real* communication going here.

From: Gene Ward Smith (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I'm under the impression that George is unfamiliar with a lot of the 
> work on equal-beating well-temperaments that Robert Wendell, you, and 
> others have done. So someone should start introducing him to it.

Wendell came up with some well-temperaments, not ordinaire, which had
some very nice beat ratios. I pointed out that you can set up systems
of equations and solve for specific arrangements of beat ratios as a
way to construct these sorts of temperaments, and that therefore the
problem may not be as intractible as suggested. One question that you
need to answer for yourself at the outset is which brats are
interesting enough to be worth bothering with, and therefore of
solving for. I'd say 3/2, 2, 3, 4, infinity and -1 are all worthwhile,
but I'm not sure about some of the stuff George seems to be using.

From: Ozan Yarman (2005-08-30)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Paul,
  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 3:23 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

  > Timbres:
  > Piano, Trumpet, Clarinet, Celesta and Flutes

  Interesting. Piano of course gets you into issues of inharmonic 
  timbres, and the Celesta into other issues (I really doubt I could hear 
  the difference between those two chords on the Celesta). 


I trust you will.


  I can 
  create .wav files with Matlab but I certainly doubt my ability to 
  emulate these instruments with mathematical functions! Maybe someone 
  else can come to the rescue here . . .



Gene perhaps? Or Charles?



> Common Register:
> G3 to C5

This would be a range of pitch for the lowest note? Highest note?


If you are familiar with the note-naming conventions, you will recall that C4 is the middle C of the piano. I am thinking of a common range for all of the timbres above. 1.5 octaves is plenty room for chordal modulation. But the last note should have been C#5.

Cordially,
Ozan

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

>> > Common Register:
>> > G3 to C5
> 
>> This would be a range of pitch for the lowest note? Highest note?
> 
> 
> If you are familiar with the note-naming conventions, you will recall 
>that C4 is the middle C of the piano.

Of course.

>I am thinking of a common range for all of the timbres above. 1.5 
>octaves is plenty room for chordal modulation. But the last note 
>should have been C#5.

You don't seem to have answered my question. Whether you meant the 
lowest note or the highest note, you're still talking about a range of 
1.5 octave within which to transpose the chord.

From: Aaron Krister Johnson (2005-08-30)
Subject: Suggestion for Ozan....1/7-comma meantone

On Monday 29 August 2005 7:34 pm, Ozan Yarman wrote:
> Huh, I hate enforcing ideas on people. In fact, that is one of the reasons
> why I stick to this forum given its open-mindedness and all that.
>
> Still, you do a great deal of injustice to George's calculations. One can
> very well play any kind of 12-tone music in his temperament on any key one
> likes. The tone-color is a bonus and it is only a matter of which frequency
> you want to take as basis in order to start the cycle of fifths. I played
> it, did you?
>
> Maybe it escaped you, but I simply imagined 12-tET to be too crude to be
> accepted as default. I didn't mean to dump the availabilities out of the
> window! Simply put, non-microtonal synths can perform much better with
> George's tuning as default instead of the borin' ol' 12-tET in my opinion.
>
> I for one wish to tune my piano to this temperament straight away.

Ozan,

It sounds to me like you would also rather like 1/7-comma meantone. It has 
some very sweet builtin beat ratios, and is at a kind of 'sweet spot' in the 
space of meantones: the fifths are large enough to make it a circulating 
temperament, the largest fifth is a tiny bit larger than Pythagorean, and 
very close to being 19/15, and it's just subversive enough in sound to be not 
the same old 12-equal, but you can play music of any Western era in it 
without fundamentally 'breaking it'!

What I do is also distribute the wolf a bit between Ab-Eb and Eb-Bb. The wolf 
in the case of 1/7-comma is really samll enough that putting it between 2 
fifths makes it sound like a very very mild distortion of a well-temperament, 
or rather, a very very mild temperament ordinaire.

Margo and I were talking about 1/7-comma for about an hour on the phone a few 
days ago....check it out!

In the meantime, George's tuning does seem intriguing...

How is Istanbul these days?

Warmly,
Aaron.

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 
> > I'm under the impression that George is unfamiliar with a lot of 
the 
> > work on equal-beating well-temperaments that Robert Wendell, you, 
and 
> > others have done. So someone should start introducing him to it.
> 
> Wendell came up with some well-temperaments, not ordinaire, which had
> some very nice beat ratios.

A well-temperament, not ordinaire, would seem to be a better candidate 
for the type of replacement of 12-equal Ozan is talking about. I hope 
Ozan will give Wendell's tunings the kind of hearing he gave one of 
Secor's.

From: Herman Miller (2005-08-30)
Subject: Re: [tuning] Temperament (Extra)ordinare! (Was: Further doubts ...)

George D. Secor wrote:

> Anyway, I hope that some others will try this tuning and give me your 
> reactions -- at least go through the triads in Scala using the chord 
> playing function in the keyboard clavier.  Convince me that this 
> isn't all a dream!

I tried it a little with the keyboard hooked up to Scala, and retuned 
some MIDI files, and I have to say that it's a very colorful tuning. The 
synchronized beating creates an interesting effect in the remote keys, 
and the keys around C are very pleasant. I'm not sure how it compares 
with similar tunings; it's been a while since I was looking at tunings 
like this, but it seems to hold up pretty well with a range of different 
styles.

From: Gene Ward Smith (2005-08-30)
Subject: Re: Suggestion for Ozan....1/7-comma meantone

--- In [email protected], Aaron Krister Johnson <aaron@a...> wrote:

> It sounds to me like you would also rather like 1/7-comma meantone. 

Has anyone ever sung the praises of 91-equal meantone, I wonder? It's
got a brat of 1.99, which ought to do for it in the beat department.
It's wolf fifth is 712 cents, which as Aaron notes you can pretty well
get away with.

From: Gene Ward Smith (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> A well-temperament, not ordinaire, would seem to be a better candidate 
> for the type of replacement of 12-equal Ozan is talking about

Especially since I meant to say "not extraordinaire". Mixing pure
fifths with the 1/7 comma meantone fifths Aaron is advocating leads to
a mild tempering, with beat ratios to smooth out the pain of the
mistuning for you if you feel they help.

From: Carl Lumma (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

> > > However, didn't Gene agree that his main tuning, the "Natural
> > > Synchronous Well" was indeed a unique point in the space of
> > > such tunings?
> > 
> > To the best of my recollection, yes, but really Gene should
> > answer this. And I'd love to see a comparison of George's new
> > proposal against this and other relevant alternatives.
> 
> I'm not sure what you mean by a "unique point", but certainly it
> didn;t have the barqoue complexity of George's system, and stuck
> with brats that clearly seemed to be worthwhile.

I seem to remember the claim being that it is the only well
temperament with all triads having brats of 3:2 or 2, and that
you agreed.  Now let's check that...

...crud.  That must have been when I was reading the list on the
web.
 
-Carl

From: Carl Lumma (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

> Wendell came up with some well-temperaments, not ordinaire, which had
> some very nice beat ratios. I pointed out that you can set up systems
> of equations and solve for specific arrangements of beat ratios as a
> way to construct these sorts of temperaments, and that therefore the
> problem may not be as intractible as suggested. One question that you
> need to answer for yourself at the outset is which brats are
> interesting enough to be worth bothering with, and therefore of
> solving for. I'd say 3/2, 2, 3, 4, infinity and -1 are all
> worthwhile, but I'm not sure about some of the stuff George seems
> to be using.

Hi Gene,

Wondering if you could help clear up the importance of sign with
brats.  With a brat of 1, the major and minor thirds beat together,
right?  How is -1 different?

-Carl

From: Gene Ward Smith (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> Wondering if you could help clear up the importance of sign with
> brats.  With a brat of 1, the major and minor thirds beat together,
> right?  How is -1 different?

It would differ in intonation, in ways that depend on the tuning. For
instance, a meantone fifth with a brat of +1 is a useless 679.56
cents, whereas -1 gives 695.63 cents. In general, if we stick to pure
meantone (no circulating temperaments) a brat of b goes with a fifth
which is a root of (3-2b)x^4 - 10x + 10b = 0. Other linear
temperaments, or nonregular temperaments, would be quite different.

From: Kraig Grady (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

BTW i also find George's tuning is very impressive and sounds good in 
all keys

>
>  
>

-- 
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

From: Ozan Yarman (2005-08-30)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

>I am thinking of a common range for all of the timbres above. 1.5 
>octaves is plenty room for chordal modulation. But the last note 
>should have been C#5.

You don't seem to have answered my question. Whether you meant the 
lowest note or the highest note, you're still talking about a range of 1.5 octave within which to transpose the chord.

-------

Paul, if you mean a base frequency to commence the test, that would be a 260Hz C4.

From: Ozan Yarman (2005-08-30)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

>I am thinking of a common range for all of the timbres above. 1.5
>octaves is plenty room for chordal modulation. But the last note
>should have been C#5.

You don't seem to have answered my question. Whether you meant the
lowest note or the highest note, you're still talking about a range of 1.5
octave within which to transpose the chord.

-------

Paul, if you mean a base frequency to commence the test, that would be a
260Hz C4.

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 
> > A well-temperament, not ordinaire, would seem to be a better 
candidate 
> > for the type of replacement of 12-equal Ozan is talking about
> 
> Especially since I meant to say "not extraordinaire". Mixing pure
> fifths with the 1/7 comma meantone fifths Aaron is advocating leads to
> a mild tempering, with beat ratios to smooth out the pain of the
> mistuning for you if you feel they help.

And Wendell's well-temperaments?

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > > > However, didn't Gene agree that his main tuning, the "Natural
> > > > Synchronous Well" was indeed a unique point in the space of
> > > > such tunings?
> > > 
> > > To the best of my recollection, yes, but really Gene should
> > > answer this. And I'd love to see a comparison of George's new
> > > proposal against this and other relevant alternatives.
> > 
> > I'm not sure what you mean by a "unique point", but certainly it
> > didn;t have the barqoue complexity of George's system, and stuck
> > with brats that clearly seemed to be worthwhile.
> 
> I seem to remember the claim being that it is the only well
> temperament with all triads having brats of 3:2 or 2, and that
> you agreed.  Now let's check that...
> 
> ...crud.  That must have been when I was reading the list on the
> web.
>  
> -Carl

I'd be interested to see Gene's reply.

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> 
> > Wondering if you could help clear up the importance of sign with
> > brats.  With a brat of 1, the major and minor thirds beat 
together,
> > right?  How is -1 different?
> 
> It would differ in intonation, in ways that depend on the tuning. 
For
> instance, a meantone fifth with a brat of +1 is a useless 679.56
> cents,

Far from useless, I've been extolling the virtues of such Mavila 
tunings here and on MMM, but mostly to deaf ears. Another Mavila 
tuning with synchronous beat rates is Wilson's meta-mavila:

http://www.anaphoria.com/meantone-mavila.PDF

whose fifth is 1200*(1-.436385705396) = 676.34 cents.

> whereas -1 gives 695.63 cents.

That's Wilson's meta-meantone (see the article above);

1200*.579692031034 = 695.63 cents.

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Further doubts about Lehman's 'Bach' scale

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> 
> >I am thinking of a common range for all of the timbres above. 1.5 
> >octaves is plenty room for chordal modulation. But the last note 
> >should have been C#5.
> 
> You don't seem to have answered my question. Whether you meant the 
> lowest note or the highest note, you're still talking about a range 
of 1.5 octave within which to transpose the chord.
> 
> -------
> 
> Paul, if you mean a base frequency to commence the test, that would 
be a 260Hz C4.

Oddly, the text below the dashes doesn't show up until I click 
on "Reply"! This is crazy! What on earth is Yahoo doing here?

But you still didn't answer my question. You gave the range G3 to 
C#5, did you not? I still don't know whether that refers to the range 
for lowest note or the highest note (or something else).

From: Ozan Yarman (2005-08-30)
Subject: Re: [tuning] Re: Further doubts about Lehman's 'Bach' scale

Oh I see what you mean! C4 should be the lowest note of the triad in root position.

Oz.
  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 19:36 
  Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  > 
  > >I am thinking of a common range for all of the timbres above. 1.5 
  > >octaves is plenty room for chordal modulation. But the last note 
  > >should have been C#5.
  > 
  > You don't seem to have answered my question. Whether you meant the 
  > lowest note or the highest note, you're still talking about a range 
  of 1.5 octave within which to transpose the chord.
  > 
  > -------
  > 
  > Paul, if you mean a base frequency to commence the test, that would 
  be a 260Hz C4.

  Oddly, the text below the dashes doesn't show up until I click 
  on "Reply"! This is crazy! What on earth is Yahoo doing here?

  But you still didn't answer my question. You gave the range G3 to 
  C#5, did you not? I still don't know whether that refers to the range 
  for lowest note or the highest note (or something else).

From: wallyesterpaulrus (2005-08-30)
Subject: Re: Further doubts about Lehman's 'Bach' scale

Ok, I hope someone can prepare some examples.

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Oh I see what you mean! C4 should be the lowest note of the triad 
in root position.
> 
> Oz.
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 30 Aðustos 2005 Salý 19:36 
>   Subject: [tuning] Re: Further doubts about Lehman's 'Bach' scale
> 
> 
>   --- In [email protected], "Ozan Yarman" <ozanyarman@s...> 
wrote:
>   > 
>   > >I am thinking of a common range for all of the timbres above. 
1.5 
>   > >octaves is plenty room for chordal modulation. But the last 
note 
>   > >should have been C#5.
>   > 
>   > You don't seem to have answered my question. Whether you meant 
the 
>   > lowest note or the highest note, you're still talking about a 
range 
>   of 1.5 octave within which to transpose the chord.
>   > 
>   > -------
>   > 
>   > Paul, if you mean a base frequency to commence the test, that 
would 
>   be a 260Hz C4.
> 
>   Oddly, the text below the dashes doesn't show up until I click 
>   on "Reply"! This is crazy! What on earth is Yahoo doing here?
> 
>   But you still didn't answer my question. You gave the range G3 to 
>   C#5, did you not? I still don't know whether that refers to the 
range 
>   for lowest note or the highest note (or something else).

From: Gene Ward Smith (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> And Wendell's well-temperaments?

That's what I was describing--his mixture of pure fifths (brat 3/2)
with 1/7 comma fifths (brat 2.)

From: Gene Ward Smith (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> I'd be interested to see Gene's reply.

I don't recall proving such a thing, and one of the brats is going to
be only almost exactly a pure 2 or 3/2, instead of exactly. However,
if you arrange the fifths in a non-waste way, there's really only one
way to do this, modulo differences which don't actually matter.

From: George D. Secor (2005-08-30)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Jon Szanto" <jszanto@c...> wrote:
> --- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> > HEY EVERYONE!
> 
> Look, George is shouting with GLEE! :)

And I'm (gleefully) overwhelmed just trying to read the replies!!!  
(Still haven't gotten thru all of them -- please be patient -- I'll 
answer soon!

--George

From: Carl Lumma (2005-08-31)
Subject: Re: Further doubts about Lehman's 'Bach' scale

> Oddly, the text below the dashes doesn't show up until I click 
> on "Reply"! This is crazy! What on earth is Yahoo doing here?

They think it's a signature, probably.  Google started filtering
those out in gmail, and yahoo's probably trying to copy them.
The whole idea is stupid.

-Carl

From: terry0051 (2005-08-31)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> HEY EVERYONE!
> 
> Having been unable to ignore the discussion about equal-beating 
> temperaments lately, I have some startling news about a
"temperament 
> (extra)ordinaire!"

That's really interesting:  and it put me in mind of
a temp.ordinaire of interestingly similar type that was 
discussed in one of the piano tuning groups about four 
years ago -- (apologies for the repetition if you are 
already familiar with this one).

According to the reports of the time, it was tried out and 
much liked, and there, too, some of its benefits were being 
tentatively attributed to "close attention to internal 
agreements of beat speeds and not just purity of intervals". 

The starting message of the older thread is at
http://www.ptg.org/pipermail/caut/2001-May/004075.html

and the temperament used was defined by cent deviations from
unstretched equal temperament 100 cents/semitone:
A,  Bb, B, C ,C#, D, Eb, E,  F,  F#,  G,  G#, A
0, +7, -6,+9,-2, +3, +2,-3, +12, -4, +6, -1, 0
[possibly "G +6" may have been a typo for "G +5"].

The source was cited as "The Well Tempered Organ", Charles A
Padgham (Oxford 1986).

I confess that I haven't yet managed to have a listen-in 
(I haven't yet succeeded in getting my computer to do 
this scale stuff, even with 'Scala', but maybe I will 
soon crack this thing!), but if I may add a question:- 

I notice that your temperament is defined with 
precision down to 10^-5 cents, while the discussion four 
years ago was based on trial of a temperament defined only 
to the nearest cent.  How much precision of implementation 
do you think is needed in practice to realise these 
internal agreements of beat-rate?

Kind regards
Terry Stancliffe
Cambridge

From: terry0051 (2005-08-31)
Subject: [correction] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Correction to previous message: 
Sorry -- the possible typo seemed to be in g# (-2?), not in g.
(would remove a seeming +/- anomaly in tempering of fifths).
Terry

--- In [email protected], "terry0051" <terry0051@y...> wrote:
> --- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

From: [email protected] (2005-08-31)
Subject: MOS at folding points other than the 2:1

I was pushing around numbers last night and decided to see what would be 
generated when I used 5^(1/2) as a generator and 3:1 folding point (see 
attachment) -- as was expected, MOSs happened at various steps of the 
procedure -- but when reduced back into the 2:1 space, the MOS evaporated. 
 

Has anyone toyed around with this notion of folding around other points 
besides the 2:1?  If so, anything curious come out of it?

Paul

From: George D. Secor (2005-09-01)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Hi again George!

Hello, again!  Sorry for my delay in replying -- I had to take some 
time off yesterday due to an illness in my family.  (And I have time 
today to reply to only a few things.)

>   ----- Original Message ----- 
>   From: George D. Secor 
>   --- In [email protected], "Ozan Yarman" <ozanyarman@s...> 
wrote:
>   > ...
>   > > if you are in the know as to how to convert pitch information 
to 
>   vibrations per second, I may perhaps demonstrate whether or not I 
>   understand how to tune to `equal (or rather, even) beating`.
> 
>   Well, whichever you mean, why don't I provide you with a 
spreadsheet 
>   I was already working on to calculate frequencies (cols. W-Y), 
beat 
>   rates (cols. S-U), beat ratios (cols. O-Q), and errors from just 
>   (cols. G,I,K,M) from cents (cols.E-F):
> 
>   http://groups.yahoo.com/group/tuning-math/files/secor/WT-wksht.xls
> ...
> >   Now you can experiment using the spreadsheet.  Besides the 
fractions 
> >   of a didymus comma I already mentioned above, you may find 
5/14, 
> >   5/19, and 5/21 to be of interest.
> 
> I am eager to start my experiments, but how do I input these 
fractions you mentioned?

For n/d-comma, put the number d in cell G4 and -n in B15.  Observe 
that all of the major triads from Gb thru G have the same beat ratios.

>   > Does your modified meantone contain any `equal-beating` 
intervals?
> 
>   Nope, but my brand-new temperament (extra)ordinare does:
> ...
>   I've made available a temperament worksheet with all sorts of 
>   figures, should anyone be interested:
> 
>   http://groups.yahoo.com/group/tuning-math/files/secor/GS_1.xls
> 
>   Observe that all of the major triads from Eb to E are 
proportional-
>   beating (i.e., the beat rates are related by rational integers or 
>   fractions -- I had to tweak the numbers in column B slightly to 
get 
>   the beat-ratios to come out exactly proportional).  
> 
> Uh, slow down please! And tell me what beat-rates, or BRATS are and 
what their significance is in your own words.

A beat-rate is simply the number of beats per second that occur in a 
tempered consonance.  A triad has synchronous beating when the ratios 
between the beat rates for its constituent intervals can be expressed 
with small integers.

> ...
> And what is the total error for this extra-ordinary temperament of 
yours?

For the 12 major triads: ~386.5 cents (in cell H21); since this isn't 
a well-temperament, that figure exceeds the theoretical minimum of 
~375.4 cents for a curculating 12-tone temperament.

> Anyway, I hope that some others will try this tuning and give me 
your 
> reactions -- at least go through the triads in Scala using the 
chord 
> playing function in the keyboard clavier.  Convince me that this 
> isn't all a dream!

And for those others who did: thank you for your feedback -- much 
appreciated.

Best,

--George

From: George D. Secor (2005-09-01)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > ! secor12_1.scl
> > ...
> > http://groups.yahoo.com/group/tuning-math/files/secor/GS_1.xls
> 
> Hiya George (and Ozan),
> 
> Do you find you like the triads where all the intervals beat
> at the same rate?  I would think it would be better to stagger
> the beats, so that they do not result in large amplitude
> changes.  Or does this not happen?
> 
> -Carl

It seems to give the consonant chords a more "unified" 
or "integrated" sound, as if the instrument is "singing" with a 
vibrato.

Only the 5th & major 3rd of the C and G major triads beat at a 1:1 
ratio in this temperament, but the overwhelming majority of the other 
beat ratios in the best triads (from Eb to E major) are also 
integers, so those triads also have a "singing" quality.

I don't think that the amplitude changes are objectionably large -- 
even in the 5/17-comma temperament with its 1:1:1 beat ratio (which I 
previously incorporated in the best keys of my 19-tone well-
temperament -- a tuning I currently have stored in my Scalatron).  I 
expect that one would have to deliberately adjust the phase 
relationships between the tones in order for the beats to reinforce 
one another so as to produce an amplitude large enough to be judged 
objectionable.

--George

From: Carl Lumma (2005-09-01)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

> > Hiya George (and Ozan),
> > 
> > Do you find you like the triads where all the intervals beat
> > at the same rate?  I would think it would be better to stagger
> > the beats, so that they do not result in large amplitude
> > changes.  Or does this not happen?
> > 
> > -Carl
> 
> It seems to give the consonant chords a more "unified" 
> or "integrated" sound, as if the instrument is "singing" with a 
> vibrato.
> 
> Only the 5th & major 3rd of the C and G major triads beat at a 1:1 
> ratio in this temperament, but the overwhelming majority of the
> other beat ratios in the best triads (from Eb to E major) are also 
> integers, so those triads also have a "singing" quality.
> 
> I don't think that the amplitude changes are objectionably large -- 
> even in the 5/17-comma temperament with its 1:1:1 beat ratio (which
> I previously incorporated in the best keys of my 19-tone well-
> temperament -- a tuning I currently have stored in my Scalatron).
> I  expect that one would have to deliberately adjust the phase 
> relationships between the tones in order for the beats to reinforce 
> one another so as to produce an amplitude large enough to be judged 
> objectionable.

Interesting.

-Carl

From: wallyesterpaulrus (2005-09-02)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "terry0051" <terry0051@y...> wrote:
> --- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> How much precision of implementation 
> do you think is needed in practice to realise these 
> internal agreements of beat-rate?
> 
> Kind regards
> Terry Stancliffe
> Cambridge

I don't see that George replied; but one often needs far better than 1 
cent precision to get the beating ratios to sound audibly right. So 
most electronic synths are out of the question for hearing the effects.

From: wallyesterpaulrus (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], pgreenhaw@n... wrote:

> I was pushing around numbers last night and decided to see what would 
be 
> generated when I used 5^(1/2) as a generator and 3:1 folding point 
(see 
> attachment) -- as was expected, MOSs happened at various steps of the 
> procedure -- but when reduced back into the 2:1 space, the MOS 
evaporated. 
>  
> 
> Has anyone toyed around with this notion of folding around other 
points 
> besides the 2:1?

Yes.

> If so, anything curious come out of it?

Are you familiar with the Bohlen-Pierce scale?

I'd like to snail-mail you my 'Middle Path' paper, if I may. The new 
version is over a year old now, but I'd love your comments. May I have 
your address?

> 
> Paul

-Another Paul

From: monz (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

Hi Paul and Paul,


--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:

> --- In [email protected], pgreenhaw@n... wrote:
> 
> > Has anyone toyed around with this notion of folding
> > around other points besides the 2:1?
> 
> Yes.
> 
> > If so, anything curious come out of it?
> 
> Are you familiar with the Bohlen-Pierce scale?



Bohlen-Pierce uses 3:1 as the identity interval.

Tonescape allows you to use any interval as the
identity interval, and also allows the creation of
tunings which have no identity interval.

In addition, the generators of the tuning can be
anything.

Both the identity interval and generators can be
specified as either ratios or cents values (to
3 decimal places).



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Ozan Yarman (2005-09-02)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

Identity interval??? What happened to generator and period?

  ----- Original Message ----- 
  From: monz 
  To: [email protected] 
  Sent: 02 Eylül 2005 Cuma 9:47 
  Subject: [tuning] Re: MOS at folding points other than the 2:1


  Hi Paul and Paul,

  SNIP!

  Both the identity interval and generators can be
  specified as either ratios or cents values (to
  3 decimal places).



  -monz
  http://tonalsoft.com
  Tonescape microtonal music software

From: [email protected] (2005-09-02)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

What about other MOSs at places other than the 2:1?  Does finding MOSs at 
a 3:1 folding-point even matter in the grand scheme of things?  Is there 
anything "coherent" with that?  Or any integrity to be gained?

P

___________________________________________
Paul Greenhaw
Music Specialist II
The New York Public Library for the Performing Arts
40 Lincoln Center Plaza
New York, NY 10023
(212) 870-1892
__________________________________________




"monz" <[email protected]>
Sent by: [email protected]
09/02/2005 02:47 AM
Please respond to tuning
 
        To:     [email protected]
        cc: 
        Subject:        [tuning] Re: MOS at folding points other than the 
2:1


Hi Paul and Paul,


--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:

> --- In [email protected], pgreenhaw@n... wrote:
> 
> > Has anyone toyed around with this notion of folding
> > around other points besides the 2:1?
> 
> Yes.
> 
> > If so, anything curious come out of it?
> 
> Are you familiar with the Bohlen-Pierce scale?



Bohlen-Pierce uses 3:1 as the identity interval.

Tonescape allows you to use any interval as the
identity interval, and also allows the creation of
tunings which have no identity interval.

In addition, the generators of the tuning can be
anything.

Both the identity interval and generators can be
specified as either ratios or cents values (to
3 decimal places).



-monz
http://tonalsoft.com
Tonescape microtonal music software
















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From: Carl Lumma (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Identity interval??? What happened to generator and period?

Period = identity interval.

-Carl

From: Carl Lumma (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], pgreenhaw@n... wrote:
> What about other MOSs at places other than the 2:1?  Does finding
> MOSs at a 3:1 folding-point even matter in the grand scheme of
> things?  Is there anything "coherent" with that?  Or any
> integrity to be gained?

I don't think there are universal answers to these questions at
this point.  Listen, make some music, and if you go platinum
go back and say it was coherent.  :)

-Carl


PS- Charles Carpenter's band made 2 CDs of Bohlen-Pierce music
in a rather aggressive fusion style.  If you're interested in
scales like this, you might want to get these recordings.  But
even if you don't like them, it isn't conclusive evidence against
the 3:1-equivalence idea.

From: Graham Breed (2005-09-02)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

Ozan Yarman wrote:
> Identity interval??? What happened to generator and period?

The period is an equal division of the identity interval.  Usually the 
identity interval is an octave.  The period is the interval by which the 
tuning repeats, and the identity interval is a higher level concept.


                    Graham

From: Carl Lumma (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

> > Identity interval??? What happened to generator and period?
> 
> The period is an equal division of the identity interval.  Usually
> the identity interval is an octave.  The period is the interval by
> which the tuning repeats, and the identity interval is a higher
> level concept.
>                     Graham

That's an alternate usage.  But in this usage, the identity
interval has nothing to do with the pitches -- it's just a
statement that you do not intend the period to sound like an
equivalence interval.

-Carl

From: wallyesterpaulrus (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> > Identity interval??? What happened to generator and period?
> 
> Period = identity interval.

No, it isn't necessarily. For example, most composers who use the 
octatonic (diminished) scale in 12-equal still consider the identity 
interval to be the octave, even though the period of the scale is 1/4 
octave.

From: wallyesterpaulrus (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

The identity interval, or interval of equivalence, is usually taken 
to be the octave (2:1 or thereabouts), while a scale may repeat more 
than once per octave and thus have a period that is 1/2, 1/3, 1/4, 
1/5, . . . octave. If you're talking about 2D tuning systems, then 
you just need to then specify a generator, and you can make that 
specification unique if you insist that it's less than 1/2 the size 
of the period.


--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Identity interval??? What happened to generator and period?
> 
>   ----- Original Message ----- 
>   From: monz 
>   To: [email protected] 
>   Sent: 02 Eylül 2005 Cuma 9:47 
>   Subject: [tuning] Re: MOS at folding points other than the 2:1
> 
> 
>   Hi Paul and Paul,
> 
>   SNIP!
> 
>   Both the identity interval and generators can be
>   specified as either ratios or cents values (to
>   3 decimal places).
> 
> 
> 
>   -monz
>   http://tonalsoft.com
>   Tonescape microtonal music software

From: wallyesterpaulrus (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], pgreenhaw@n... wrote:

> What about other MOSs at places other than the 2:1?  Does finding 
MOSs at 
> a 3:1 folding-point even matter in the grand scheme of things?

Well, what does matter?

> Is there 
> anything "coherent" with that?  Or any integrity to be gained?

The Bohlen-Pierce scale is quite unique and interesting. Some great 
music has been made in it, for example by Charles Carpenter. Can we 
hear 3:1 as an interval of equivalence? I'm still skeptical. But 
sometimes the music is *right* even if the theory behind it is wrong.

From: wallyesterpaulrus (2005-09-02)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > > Identity interval??? What happened to generator and period?
> > 
> > The period is an equal division of the identity interval.  Usually
> > the identity interval is an octave.  The period is the interval by
> > which the tuning repeats, and the identity interval is a higher
> > level concept.
> >                     Graham
> 
> That's an alternate usage.

It seems to be the more meaningful one, by far. Why do you say 
it's "alternate"?

> But in this usage, the identity
> interval has nothing to do with the pitches -- it's just a
> statement that you do not intend the period to sound like an
> equivalence interval.

When you start naming notes, notating things, and even simply saying 
how many notes there are in the scale, knowing the identity interval 
(or interval of equivalence) is necessary.

From: monz (2005-09-03)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:

> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> 
> > --- In [email protected], "Ozan Yarman" <ozanyarman@s...> 
wrote:
> > >
> > > Identity interval??? What happened to generator and period?
> > 
> > Period = identity interval.
> 
> No, it isn't necessarily. For example, most composers
> who use the octatonic (diminished) scale in 12-equal
> still consider the identity interval to be the octave,
> even though the period of the scale is 1/4 octave.


It should also be noted that, at least concerning the
way Tonescape works, the period and identity-interval
are also generators. 

They're just considered differently from what's called
"generators", because of the equivalence aspect.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: monz (2005-09-03)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:

> When you start naming notes, notating things, and even
> simply saying how many notes there are in the scale,
> knowing the identity interval (or interval of equivalence)
> is necessary.


*If* there is an identity-interval.

As i pointed out (and as you well know, of course, Paul),
scales don't necessarily have to have an identity-interval.

The ancient Greek tunings, based on similar tetrachords,
had a "sort-of" identity-interval of 4/3 ratio ...
however, because of the "tone of disjunction" which
occurs above _mese_ in the Greater Perfect System and
above _proslambanomenos_, the identity aspect isn't
totally consistent. Thus, the Greek tunings are best
designed in Tonescape without an identity-interval.

(BTW, word is that John Chalmers is currently composing
some sample pieces in these Greek scales with Tonescape.)



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Ozan Yarman (2005-09-06)
Subject: Re: [tuning] Suggestion for Ozan....1/7-comma meantone

Equal beating 6/5 = 8/5 same. Almost 1/7-comma
|
  0:          1/1           C          unison, perfect prime
  1:         92.146 cents
  2:        197.756 cents   D
  3:        303.366 cents   Dx   Eb
  4:        395.512 cents   E
  5:        501.122 cents   F
  6:        593.268 cents
  7:        698.878 cents   G
  8:        804.488 cents   Gx   Ab
  9:        896.634 cents   A
 10:       1002.244 cents   Ax   Bb
 11:       1094.390 cents   B
 12:          2/1           C          octave


> I for one wish to tune my piano to this temperament straight away.

Ozan,

It sounds to me like you would also rather like 1/7-comma meantone. It has some very sweet builtin beat ratios, and is at a kind of 'sweet spot' in the space of meantones: the fifths are large enough to make it a circulating temperament, the largest fifth is a tiny bit larger than Pythagorean, and very close to being 19/15, and it's just subversive enough in sound to be not the same old 12-equal, but you can play music of any Western era in it without fundamentally 'breaking it'!

---------

Aaron, 1/7-comma meantone does indeed sound sweet to my ears when I play a first inversion dominant seventh. However, when one modulates to major or minor from there, the triad becomes pretty ugly. Moreover, I hardly think that 712 cents to be `tiny bit larger than Pythagorean`. This is a good temperament, but it can be improved in my opinion.

Now, George has formulated an excellent temperament which I have just yesterday implemented on my Bechstein Imperial Grand, and it sounds absolutely terrific.

-----------


What I do is also distribute the wolf a bit between Ab-Eb and Eb-Bb. The wolf in the case of 1/7-comma is really samll enough that putting it between 2 fifths makes it sound like a very very mild distortion of a well-temperament, or rather, a very very mild temperament ordinaire.

Margo and I were talking about 1/7-comma for about an hour on the phone a few days ago....check it out!


-------------

Your modification would require both Ab-Eb and Eb-Bb to be about 706 cents wide. I'm not sure if that would bode well with the equal-beating nature of 1/7-comma meantone.

-------------


In the meantime, George's tuning does seem intriguing...



Yes, indeed! I'm the honored musician who first implemented George's brand new tuning to an acoustic grand piano. I'm delighted with the results!


How is Istanbul these days?


------------


Mildly warm, windy and humid due to the Marmara climate, but not intolerably so. It's good though for my hypoglycaemic diet. 


Warmly,
Aaron.


Cordially,
Ozan

From: Ozan Yarman (2005-09-06)
Subject: Re: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Paul, Wendell's temperaments are also very elegant, however, my ears prefer Secor temperament ordinaire #1. It seems to be more balanced on the white keys and flat tonalities in comparison.

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 30 Ağustos 2005 Salı 4:47 
  Subject: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)




  A well-temperament, not ordinaire, would seem to be a better candidate for the type of replacement of 12-equal Ozan is talking about. I hope Ozan will give Wendell's tunings the kind of hearing he gave one of Secor's.

From: Ozan Yarman (2005-09-06)
Subject: Re: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Hello, again!  Sorry for my delay in replying -- I had to take some 
time off yesterday due to an illness in my family.  (And I have time today to reply to only a few things.)



It's quite alright dear George. I have delayed to reply myself.


For n/d-comma, put the number d in cell G4 and -n in B15.  Observe 
that all of the major triads from Gb thru G have the same beat ratios.


Now that is fascinating! But how does one go to a temperament where the octave closes with equal-beating triads after, say, 30-40 tones?


> 
> Uh, slow down please! And tell me what beat-rates, or BRATS are and what their significance is in your own words.

A beat-rate is simply the number of beats per second that occur in a tempered consonance.  A triad has synchronous beating when the ratios between the beat rates for its constituent intervals can be expressed with small integers.


Okay, ratios between the beat-rates... as in the ratio of the beat rate of 5/4 to the beat rate of 3/2? In the case of 5/17-comma temperament, the major third beats 4.14 times per second and the fifth the same, so the ratio is 1/1. In the case of the 5/14-comma temperament, the BRAT of the consonant major triad is 10/5=2. I think I got it! The comma-fractions you gave yield multiples of beat-ratios. Right?


> ...
> And what is the total error for this extra-ordinary temperament of yours?

For the 12 major triads: ~386.5 cents (in cell H21); since this isn't a well-temperament, that figure exceeds the theoretical minimum of ~375.4 cents for a curculating 12-tone temperament.



That is because you have chosen to increase the error in the remote keys in favor of the more accustomed keys. Correct?


> Anyway, I hope that some others will try this tuning and give me 
your reactions -- at least go through the triads in Scala using the 
chord playing function in the keyboard clavier.  Convince me that this isn't all a dream!

And for those others who did: thank you for your feedback -- much 
appreciated.

Best,

--George



I have implemented your tuning on my Bechstein Imperial Grand just yesterday. Being the first to tune an acoustic instrument to your new temperament (my tuners almost demanded double the price for the extra effort!), let me also be the first to say that I simply loved the results. Good Work dear George!

Cordially,
Ozan

From: George D. Secor (2005-09-06)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "terry0051" <terry0051@y...> wrote:
> --- In [email protected], "George D. Secor" <gdsecor@y...> 
wrote:
> > HEY EVERYONE!
> > 
> > Having been unable to ignore the discussion about equal-beating 
> > temperaments lately, I have some startling news about 
a "temperament 
> > (extra)ordinaire!"
> 
> That's really interesting:  and it put me in mind of
> a temp.ordinaire of interestingly similar type that was 
> discussed in one of the piano tuning groups about four 
> years ago -- (apologies for the repetition if you are 
> already familiar with this one).

No, I wasn't aware of it.  My main interest is in non-12 tunings, so 
I haven't made it a point to keep fully informed about the more 
recent developments concerning well temperaments or _temperaments 
ordinaires_.

> According to the reports of the time, it was tried out and 
> much liked, and there, too, some of its benefits were being 
> tentatively attributed to "close attention to internal 
> agreements of beat speeds and not just purity of intervals". 
> 
> The starting message of the older thread is at
> http://www.ptg.org/pipermail/caut/2001-May/004075.html
> 
> and the temperament used was defined by cent deviations from
> unstretched equal temperament 100 cents/semitone:
> A,  Bb, B, C ,C#, D, Eb, E,  F,  F#,  G,  G#, A
> 0, +7, -6,+9,-2, +3, +2,-3, +12, -4, +6, -1, 0
> [possibly "G +6" may have been a typo for "G +5"].
> 
> [Correction:
> Sorry -- the possible typo seemed to be in g# (-2?), not in g.
> (would remove a seeming +/- anomaly in tempering of fifths).
> Terry]

I agree that this is probably a typo and have taken this into account 
in my attempts to reconstruct the temperament below.

> The source was cited as "The Well Tempered Organ", Charles A
> Padgham (Oxford 1986).

Not having that at hand, it's not clear to me whether the table of 
offsets defines a temperament that was devised in the 20th century or 
is an attempt to enumerate (in cents) a description of a temperment 
dating from a much earlier time.  (Should the latter be the case, 
then I might question whether the figures are even correct to the 
nearest cent.)

> I confess that I haven't yet managed to have a listen-in 
> (I haven't yet succeeded in getting my computer to do 
> this scale stuff, even with 'Scala', but maybe I will 
> soon crack this thing!), but if I may add a question:- 
> 
> I notice that your temperament is defined with 
> precision down to 10^-5 cents, while the discussion four 
> years ago was based on trial of a temperament defined only 
> to the nearest cent.  How much precision of implementation 
> do you think is needed in practice to realise these 
> internal agreements of beat-rate?
> 
> Kind regards
> Terry Stancliffe
> Cambridge

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> I don't see that George replied; but one often needs far better 
than 1 
> cent precision to get the beating ratios to sound audibly right. So 
> most electronic synths are out of the question for hearing the 
effects.

I'm inclined to agree with Paul about this.  Since the pitch 
increments on my Scalatron vary (from 0.8 to 1.6 cents) according to 
the interval the pitch makes with "A", I would also have to 
investigate how many of the triads have the pitches close enough to 
exhibit the desired effect on my instrument.  (I do remember as I 
consulted a lookup table to set the Scalatron switches for each note, 
that many of the pitches were within 0.2 cents or so -- I'll have to 
listen closely to the beating of the individual intervals in each 
triad and do a qualitative comparison.  In any case, I'm *extremely* 
happy with the way my temperament sounds.)

In attempting to analyze the temperament described by whole-cent 
offsets, I found that this precision is barely adequate for 
*determining* the intended beat rates.  For example, C-G is 3 cents 
narrower than in 12-ET, which would be 5 cents narrower than just, 
but this could apply to a 5th ranging anywhere from a 1/4-comma (5.4 
cents) narrow to the 5/23-comma (4.7 cents narrow) fifth that I have 
in my #1 temperament.  If one (reasonably) assumes that the chain of 
fifths from F to B are the same size and that all of the pitches in 
the table are indeed to the nearest cent, then the fifth must be 
rather closer to 5.0 cents.  But it's difficult to say for sure, 
since the offsets may be, at best, a second-hand description.

The text in the above link provides another clue:

<< In this tempering in the central keys of 1 or 2 sharps and flats, 
for example, beats in thirds or fifths agree with beats in sixths 
which creates a wonderful clarity of texture to chords and harmonies. 
>>

If "agree" is interpreted as a 1:1 beat-ratio, this description could 
apply to the F, C, and G major triads in a temperament with 5ths 5/23-
comma narrow from F to B (1:1 beat ratio between 5th & M3), as long 
as we allow a little leeway with the cents.  Using this as a starting 
point, I've constructed a temperament here:

http://groups.yahoo.com/group/tuning-math/files/secor/TO-523.xls

! TO-523.scl
!
Possible 5/23-comma temperament ordinaire
 12
!
86.53508
194.55680
294.12920
389.11361
502.72160
585.53449
697.27840
789.38184
891.83521
997.01621
1086.39201
2/1

The description could also apply to a temperament with 10/43-comma 
(5.002 cents narrow) fifths from F to B.  Although the M3:5th ratio 
(for F, C, and G major triads) is 1:2, the m6:5th ratio (in a 
tempered 4:5:6:8) would be 1:1.  If Bb-F is then made ~6.5/43-comma 
wide (which gives the correct number of cents for Bb), then the m3:M3 
ratio in the Bb major triad (root position) will be 1:1.  If B-F# is 
made 1/43-comma narrow, the M3:5th ratio in the D major triad (root 
position) would be 2:1, which could still be interpreted 
as "agreeable"; this gives F# an offset of -4.7 cents instead of -4, 
which is close.  Following this, I've attempted to construct a 
temperament reasonably close to the offsets given in the table:

http://groups.yahoo.com/group/tuning-math/files/secor/TO-1043.xls

! TO-1043.scl
!
Possible 10/43-comma temperament ordinaire
 12
!
87.62999
193.90577
292.61883
387.81153
503.04712
586.20299
696.95288
789.04548
890.85865
997.83297
1084.76441
2/1

Both of the above constructions conform fairly well to figures 
derived from the cents-offset table, particularly when the total 
absolute error of the major triads are calculated (and then compared 
with my #1 temperament):

Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
From table:. 59 49 35 19 13 13 13 23 33 39 43 49
TO-1043.xls: 59 49 36 20 13 13 13 22 31 40 44 52
TO-523.xls:. 60 49 34 22 15 15 15 19 26 37 48 52
Secor #1:... 54 49 34 21 15 15 15 19 26 37 48 54

The most important difference between those three and my #1 
temperament (extra)ordinaire is that mine has the narrowest 5ths in a 
chain only from C-B rather than F-B, which allows Db-F to have 
significantly less error (without increasing the total absolute error 
in the F major triad).  This, in turn, significantly reduces the 
total absolute error of the Db major triad -- the characteristic 
essential to making it "extraordinaire" (and still #1 in my book, 
BTW!).

I've tightened up the calculations for #1 and have updated the 
spreadsheet accordingly:

http://groups.yahoo.com/group/tuning-math/files/secor/GS_1.xls

This is the updated .scl file:

! secor12_1.scl
!
George Secor's 12-tone temperament ordinaire #1, proportional beating
 12
!
 86.53508
 194.55680
 294.12834
 389.11361
 499.91715
 585.53449
 697.27840
 789.38184
 891.83521
 997.96215
 1086.39201
 2/1

Best,

--George

From: wallyesterpaulrus (2005-09-06)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "monz" <monz@t...> wrote:
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> 
> > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > 
> > > --- In [email protected], "Ozan Yarman" <ozanyarman@s...> 
> wrote:
> > > >
> > > > Identity interval??? What happened to generator and period?
> > > 
> > > Period = identity interval.
> > 
> > No, it isn't necessarily. For example, most composers
> > who use the octatonic (diminished) scale in 12-equal
> > still consider the identity interval to be the octave,
> > even though the period of the scale is 1/4 octave.
> 
> 
> It should also be noted that, at least concerning the
> way Tonescape works, the period and identity-interval
> are also generators. 

That's unfortunate, because for plenty of important scales and 
tunings, such as diminished, the identity-interval (the octave) is 
*not* one of the generators.

> They're just considered differently from what's called
> "generators", because of the equivalence aspect.

For the diminished example, one of the two generators is 1/4-octave. 
Neither of the generators is the octave or identity-interval.

From: wallyesterpaulrus (2005-09-06)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "monz" <monz@t...> wrote:
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> 
> > When you start naming notes, notating things, and even
> > simply saying how many notes there are in the scale,
> > knowing the identity interval (or interval of equivalence)
> > is necessary.
> 
> 
> *If* there is an identity-interval.
> 
> As i pointed out (and as you well know, of course, Paul),
> scales don't necessarily have to have an identity-interval.

The pitches of scales are unaffected whether there is an identity 
interval or not. It's simply how we *name* and *notate* the scales 
that is affected.

> The ancient Greek tunings, based on similar tetrachords,
> had a "sort-of" identity-interval of 4/3 ratio ...
> however, because of the "tone of disjunction" which
> occurs above _mese_ in the Greater Perfect System and
> above _proslambanomenos_, the identity aspect isn't
> totally consistent.

Monz, you're confusing "identity" with "periodicity".

> Thus, the Greek tunings are best
> designed in Tonescape without an identity-interval.

I think this is just silly because the Greeks used similar 
nomenclature for notes an octave apart, just as we do today.

From: Gene Ward Smith (2005-09-06)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

> Okay, ratios between the beat-rates... as in the ratio of the beat
rate of 5/4 to the beat rate of 3/2? In the case of 5/17-comma
temperament, the major third beats 4.14 times per second and the fifth
the same, so the ratio is 1/1. In the case of the 5/14-comma
temperament, the BRAT of the consonant major triad is 10/5=2. I think
I got it! The comma-fractions you gave yield multiples of beat-ratios.
Right?

A brat of 2 goes with 1/7 comma. 5/14 comma is a brat of 1/4, and is
for a fifth (a root of f^4-4f+1) even flatter than 19-et. However,
there *is* a 2 in there; in fact, two 2s, if we look at the associated
beat ratios for all three chord consituants taken in pairs, we get two 
2 beat ratios.

If someone were looking for a 19-tone circulating temperament, this
fifth would be a good one to keep in mind.

From: George D. Secor (2005-09-06)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> [gs:]
> Hello, again!  Sorry for my delay in replying -- I had to take some 
> time off yesterday due to an illness in my family.  (And I have 
time today to reply to only a few things.)
> [oy:]
> It's quite alright dear George. I have delayed to reply myself.

This has turned into even more of a delay -- I learned last Friday 
afternoon that my mother-in-law (87 years old) died the previous 
night, and I have had almost no time until today to catch up with the 
discussion.

Tom Dent has contributed some valuable observations (which will take 
me some time to analyze and/or digest), to which I hope to make a 
detailed reply in 2 or 3 days (hang in there, Tom!).

> [gs:]
> For n/d-comma, put the number d in cell G4 and -n in B15.  Observe 
> that all of the major triads from Gb thru G have the same beat 
ratios.
> 
> [oy:]
> Now that is fascinating! But how does one go to a temperament where 
the octave closes with equal-beating triads after, say, 30-40 tones?

That's something I've never tried.  The first thing I would do is add 
more rows to my spreadsheet, then take it from there.  You would have 
to change some of the formulas for a 41-tone circulating system, 
since you'd no longer be dealing mainly with narrow 5ths.

However, I have done it with part of a circle of 19.  The 5/17-comma 
5ths are not exactly 1:1:1 (5th:M3:m3) equal-beating (would need to 
be 695.6304 cents), so I'll have to recalculate these figures when I 
have more time:

! secor_19wt.scl
!
George Secor's 19-tone well temperament with ten 5/17-comma fifths
 19
!
  69.40735
 131.54493
 191.25924
 260.66659
 317.95765
 382.51849
 451.92584
 504.37038
 573.77773
 638.33856
 695.62962
 765.03697
 824.57129
 886.88886
 956.29622
1011.16402
1078.14811
1145.13220
2/1

With C as 1/1, the 6:7:9 triads on F, C, G, and D are greatly 
improved over 19-ET, due to 7:9 having very low error.  I came up 
with the essential idea for this in 1978.

> > [oy:]
> > Uh, slow down please! And tell me what beat-rates, or BRATS are 
and what their significance is in your own words.
> [gs:]
> A beat-rate is simply the number of beats per second that occur in 
a tempered consonance.  A triad has synchronous beating when the 
ratios between the beat rates for its constituent intervals can be 
expressed with small integers.
> 
> [oy:]
> Okay, ratios between the beat-rates... as in the ratio of the beat 
rate of 5/4 to the beat rate of 3/2? In the case of 5/17-comma 
temperament, the major third beats 4.14 times per second and the 
fifth the same, so the ratio is 1/1. In the case of the 5/14-comma 
temperament, the BRAT of the consonant major triad is 10/5=2. I think 
I got it! The comma-fractions you gave yield multiples of beat-
ratios. Right?

Yep!

> [oy:]
> > ...
> > And what is the total error for this extra-ordinary temperament 
of yours?
> [gs:]
> For the 12 major triads: ~386.5 cents (in cell H21); since this 
isn't a well-temperament, that figure exceeds the theoretical minimum 
of ~375.4 cents for a curculating 12-tone temperament.
> 
> [oy:]
> That is because you have chosen to increase the error in the remote 
keys in favor of the more accustomed keys. Correct?

It's because I've chosen to increase the total absolute error in 
those triads over and above that of a pythagorean triad by using some 
fifths wider than just.

> [gs:]
> > Anyway, I hope that some others will try this tuning and give me 
> your reactions -- at least go through the triads in Scala using the 
> chord playing function in the keyboard clavier.  Convince me that 
this isn't all a dream!
> 
> And for those others who did: thank you for your feedback -- much 
> appreciated.
> 
> Best,
> 
> --George
> 
> [oy:]
> I have implemented your tuning on my Bechstein Imperial Grand just 
yesterday. Being the first to tune an acoustic instrument to your new 
temperament (my tuners almost demanded double the price for the extra 
effort!), let me also be the first to say that I simply loved the 
results. Good Work dear George!

I guess that's the ultimate compliment -- you've made my day!

Best,

--George

From: Gene Ward Smith (2005-09-06)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> That's unfortunate, because for plenty of important scales and 
> tunings, such as diminished, the identity-interval (the octave) is 
> *not* one of the generators.

It should be kept in mind anyway that the whole question of generators
is really distinct from the question of identity intevals. It makes
perfectly good sense to use 25/24 and 128/125 as generators for
meantone, for instance.

From: wallyesterpaulrus (2005-09-06)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> > [gs:]
> > Hello, again!  Sorry for my delay in replying -- I had to take some 
> > time off yesterday due to an illness in my family.  (And I have 
> time today to reply to only a few things.)
> > [oy:]
> > It's quite alright dear George. I have delayed to reply myself.
> 
> This has turned into even more of a delay -- I learned last Friday 
> afternoon that my mother-in-law (87 years old) died the previous 
> night,

My deepest condolences, George, and my best wishes to your family.

From: wallyesterpaulrus (2005-09-06)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 
> > That's unfortunate, because for plenty of important scales and 
> > tunings, such as diminished, the identity-interval (the octave) is 
> > *not* one of the generators.
> 
> It should be kept in mind anyway that the whole question of generators

Or periods!

> is really distinct from the question of identity intevals. It makes
> perfectly good sense to use 25/24 and 128/125 as generators for
> meantone, for instance.

Or, more immediately relevant to Western notation, 16;15 and 9;8 (or 
identically, 16;15 and 10;9) -- this tells you that the minor second 
and major second suffice as a basis from which one can construct any 
meantone interval. I use semicolons because these ratios are not to be 
taken in their JI sense but rather their meantone-tempered incarnations 
are meant, something you might have wanted to make clear in your post.

From: Ozan Yarman (2005-09-06)
Subject: Re: [tuning] Re: Suggestion for Ozan....1/7-comma meantone

Yes I have, but I'm not sure anymore about the accuracy of my synth after what George said.
  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 06 Eylül 2005 Salı 22:46 
  Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone



  > 
  > Now, George has formulated an excellent temperament which I have 
  >just yesterday implemented on my Bechstein Imperial Grand, and it 
  >sounds absolutely terrific.

  Ozan, have you tried either of Bob Wendell's Synchronous Well 
  Temperaments?

From: wallyesterpaulrus (2005-09-06)
Subject: Re: Suggestion for Ozan....1/7-comma meantone

Well I'm glad this point is finally making an impact on you. You can 
check your synth's accuracy at:

http://www.microtonal-synthesis.com/

Anyone -- Is Kyma listed here? If so, under what brand?


--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Yes I have, but I'm not sure anymore about the accuracy of my synth 
after what George said.
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 06 Eylül 2005 Salý 22:46 
>   Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone
> 
> 
> 
>   > 
>   > Now, George has formulated an excellent temperament which I 
have 
>   >just yesterday implemented on my Bechstein Imperial Grand, and 
it 
>   >sounds absolutely terrific.
> 
>   Ozan, have you tried either of Bob Wendell's Synchronous Well 
>   Temperaments?

From: Gene Ward Smith (2005-09-06)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Or, more immediately relevant to Western notation, 16;15 and 9;8 (or 
> identically, 16;15 and 10;9) -- this tells you that the minor second 
> and major second suffice as a basis from which one can construct any 
> meantone interval. 

I was thinking of Eytan Agmon when I picked that pair, since 25;24 is
one step of 12-et and 0 steps of 7 et, and 128;125 is 0 steps of 12 et
and one step of 7-et. Your pair leads to the two vals <2 3 4| and 
<5 8 12| (5-et.) 5 and 7 et together are then associated to 16;15 and
25;24, another reasonable pair of generators.

From: Gene Ward Smith (2005-09-07)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:

> I was thinking of Eytan Agmon when I picked that pair, since 25;24 is
> one step of 12-et and 0 steps of 7 et, and 128;125 is 0 steps of 12 et
> and one step of 7-et. Your pair leads to the two vals <2 3 4| and 
> <5 8 12| (5-et.) 5 and 7 et together are then associated to 16;15 and
> 25;24, another reasonable pair of generators.

I've posted more on this here:

http://groups.yahoo.com/group/tuning-math/message/12632

From: monz (2005-09-07)
Subject: Re: MOS at folding points other than the 2:1

Hi Paul,


--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:

> --- In [email protected], "monz" <monz@t...> wrote:
>
> > Thus, the Greek tunings are best
> > designed in Tonescape without an identity-interval.
> 
> I think this is just silly because the Greeks used similar 
> nomenclature for notes an octave apart, just as we do today.



Only for certain notes. For many others, the two different
octaves of what for us is one pitch-class, each had its
own totally unique symbol in both the "vocal" and the
"instrumental" notation.

As far as the "regular" nomenclature (_proslambanomenos_,
_mese_, etc.) -- no, there was no indication of octave
equivalence at all. The notes were named strictly
according to the periodicity of the 4/3 ratio.
Take a look at my page about the PIS:

http://tonalsoft.com/enc/p/pis.aspx



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Ozan Yarman (2005-09-07)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

Graham, can you give me some concrete examples from some famous recent
tunings? Like, Blackjack, Mavila, Metameantone...

Cordially,
Ozan


----- Original Message -----
From: "Graham Breed" <[email protected]>
To: <[email protected]>
Sent: 02 Eyl\ufffdl 2005 Cuma 19:53
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1


> Ozan Yarman wrote:
> > Identity interval??? What happened to generator and period?
>
> The period is an equal division of the identity interval.  Usually the
> identity interval is an octave.  The period is the interval by which the
> tuning repeats, and the identity interval is a higher level concept.
>
>
>                     Graham
>
>

From: Ozan Yarman (2005-09-07)
Subject: Re: [tuning] Re: Suggestion for Ozan....1/7-comma meantone

It is indeed making an impact on me since the beginning Paul, but I don't see Yamaha SW1000XG listed anywhere.
  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 07 Eylül 2005 Çarşamba 0:35 
  Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone


  Well I'm glad this point is finally making an impact on you. You can check your synth's accuracy at:

  http://www.microtonal-synthesis.com/

  Anyone -- Is Kyma listed here? If so, under what brand?

From: Graham Breed (2005-09-07)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

Ozan Yarman wrote:
> Graham, can you give me some concrete examples from some famous recent
> tunings? Like, Blackjack, Mavila, Metameantone...

Hello!  Yes, well, all the examples in your motley collection have the 
octave as the period.  An important one that doesn't is the diaschismic 
family, including that explained in Paul Erlich's 22 tone temperament 
paper.  I explain them in my website, I think at

http://www.microtonal.co.uk/diaschis.htm

The 10 note MOS is of the form

s s s s L s s s s L

in terms of small and large intervals.  In 22-equal, the small interval 
is 2 steps and the large one is 3 steps.  There's no way of defining 
this scale with the octave as a generator, because you lose half the 
notes.  Hence the period has to be set at a half octave, but for 
convenience you can still think of the scale as being 10 notes to an octave.

Another example is the octatonic scale in 12-equal:

1 2 1 2 1 2 1 2

This repeats 4 times in an octave, and can be generalized to give a 
linear temperament with a period of a quarter octave.


My linear temperament searches threw up a lot of examples where the 
period is quite small.  They are most useful in the higher limits, and 
so less famous because of the difficulties involved in making music with 
such beasties.  My favourite is the one I call "mystery" which does 
15-limit JI by dividing the octave into 29 equal parts, and using either 
0 or 1 of the other generator (around 16 cents) to get to each 15-limit 
interval.


Looked at from another direction, meantone always divides the major 
third into two equal parts.  If, for some strange reason, you decided 
that the major third was to be your identity interval, the whole tone 
would have to be your period.  Some regular temperaments happen to 
equally divide an interval that wouldn't be so divisible in JI.  If one 
of those intervals happens to be the octave, then the octave can't be 
the period.


                  Graham

From: George D. Secor (2005-09-07)
Subject: Re: Suggestion for Ozan....1/7-comma meantone

--- In [email protected], Aaron Krister Johnson <aaron@a...> 
wrote:
> On Monday 29 August 2005 7:34 pm, Ozan Yarman wrote:
> > ... Simply put, non-microtonal synths can perform much better with
> > George's tuning as default instead of the borin' ol' 12-tET in my 
opinion.
> >
> > I for one wish to tune my piano to this temperament straight away.
> 
> Ozan,
> 
> It sounds to me like you would also rather like 1/7-comma meantone. 
It has 
> some very sweet builtin beat ratios, and is at a kind of 'sweet 
spot' in the 
> space of meantones: the fifths are large enough to make it a 
circulating 
> temperament, the largest fifth is a tiny bit larger than 
Pythagorean, and 
> very close to being 19/15, and it's just subversive enough in sound 
to be not 
> the same old 12-equal, but you can play music of any Western era in 
it 
> without fundamentally 'breaking it'!
> 
> What I do is also distribute the wolf a bit between Ab-Eb and Eb-
Bb. The wolf 
> in the case of 1/7-comma is really samll enough that putting it 
between 2 
> fifths makes it sound like a very very mild distortion of a well-
temperament, 
> or rather, a very very mild temperament ordinaire.
> 
> Margo and I were talking about 1/7-comma for about an hour on the 
phone a few 
> days ago....check it out!
> 
> In the meantime, George's tuning does seem intriguing...
> ...
> Warmly,
> Aaron.

Aaron,

Curiousity got the better of me, so I decided to try my hand at a 1/7-
comma well-temperament (no wide 5ths) and came up with the following:

! Secor_WT1-7.scl
!
George Secor's 1/7-comma well-temperament
 12
!
93.97984
197.75601
297.88984
395.51202
501.12200
593.26802
698.87800
795.93484
869.63401
999.88484
1094.39002
2/1

Here are some numbers to compare with Tom Dent't 5/32-comma well-
temperament (ref. msgs. 60165, 60248):

Total absolute error (cents):
Major triad: Db Ab Eb Bb F. C. G. D. A. E. B. F#
Secor WT1-7: 42 36 29 25 25 25 25 25 28 34 41 43
Dent WT5-32: 36 35 31 27 23 23 27 31 35 35 37 37

Both temperaments have only 5 proportional-beating triads (F thru A 
in mine, C thru E in Tom's).  I haven't heard either of these yet, 
but I intend to try them out in Scala (even if the accuracy isn't up 
to exhibiting the proportional beating).

Considering that I was able to get 8 proportional-beating triads in 
my #1 temperament (extra)ordinaire and 7 in each of two other (5/23 
and 10/43-comma) temperaments ordinaire, I'm now more inclined to 
believe that Robert Wendell's equal-beating well-temperaments were 
indeed more difficult to construct (than temperaments ordinaires).

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> I'm under the impression that George is unfamiliar with a lot of 
the 
> work on equal-beating well-temperaments that Robert Wendell, you, 
and 
> others have done. So someone should start introducing him to it.

You're right about that, Paul.  Now that I've had a chance to 
experiment with a less-deviant-from-12-equal temperament, I'll have a 
greater appreciation for them.

--George

From: Gene Ward Smith (2005-09-07)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], Graham Breed <gbreed@g...> wrote:
> Ozan Yarman wrote:

> My linear temperament searches threw up a lot of examples where the 
> period is quite small.  They are most useful in the higher limits, and 
> so less famous because of the difficulties involved in making music
with 
> such beasties.  My favourite is the one I call "mystery" which does 
> 15-limit JI by dividing the octave into 29 equal parts, and using
either 
> 0 or 1 of the other generator (around 16 cents) to get to each 15-limit 
> interval.

Mystery is quite interesting if you are interested in higher limits,
and might be looked at from the point of view of someone interested in
maqams. I'll put in a plug for another temperament, ennealimmal, which
divides the octave into nine parts, and which by the time you get up
to mystery size, meaning 54 or 63 notes, is able to give a large
amount of harmony which has the interesting property of being sensibly
JI. Splitting the period in half, to 18 rather than 9 parts to an
octave allows something similar for the 11 limit rather than the 7 limit.

From: George D. Secor (2005-09-08)
Subject: Correction! Suggestion for Ozan....1/7-comma meantone

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> ...
> Curiousity got the better of me, so I decided to try my hand at a 
1/7-
> comma well-temperament (no wide 5ths) and came up with the 
following:
> ...

Oops!  I tried it last night in Scala and found that the "A" was way 
off (reversed a couple of digits) -- here's the corrected file:

! Secor_WT1-7.scl
!
George Secor's 1/7-comma well-temperament
 12
!
93.97984
197.75601
297.88984
395.51202
501.12200
593.26802
698.87800
795.93484
896.63401
999.88484
1094.39002
2/1

--George

From: wallyesterpaulrus (2005-09-08)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "monz" <monz@t...> wrote:
> Hi Paul,
> 
> 
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> 
> > --- In [email protected], "monz" <monz@t...> wrote:
> >
> > > Thus, the Greek tunings are best
> > > designed in Tonescape without an identity-interval.
> > 
> > I think this is just silly because the Greeks used similar 
> > nomenclature for notes an octave apart, just as we do today.
> 
> 
> 
> Only for certain notes. For many others, the two different
> octaves of what for us is one pitch-class, each had its
> own totally unique symbol in both the "vocal" and the
> "instrumental" notation.
> 
> As far as the "regular" nomenclature (_proslambanomenos_,
> _mese_, etc.) -- no, there was no indication of octave
> equivalence at all.

Oops! Thanks for correcting me, then.

> The notes were named strictly
> according to the periodicity of the 4/3 ratio.
> Take a look at my page about the PIS:
> 
> http://tonalsoft.com/enc/p/pis.aspx

I don't see it. How does this nomenclature fall "strictly according 
to the periodicity of the 4/3 ratio?" It seems that for the two upper 
tetrachords, the interval at which the names repeat can be either 4:3 
or 3:2, but neither of these intervals is repeated in such a 
capacity; and across the 'mese', there doesn't seem to be any 
indication of equivalence at all. Am I missing something?

From: Richard Eldon Barber (2005-09-08)
Subject: Kyma [Re: Suggestion for Ozan....1/7-comma meantone]

Kyma is under Symbolic Sound.   Although I wouldn't call Kyma the
synth, rather its external engine, the Capybara 320.

Anyone-- feel free to buy me one.

-Rick

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> Well I'm glad this point is finally making an impact on you. You can 
> check your synth's accuracy at:
> 
> http://www.microtonal-synthesis.com/
> 
> Anyone -- Is Kyma listed here? If so, under what brand?
> 
> 
> --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> > Yes I have, but I'm not sure anymore about the accuracy of my synth 
> after what George said.
> >   ----- Original Message ----- 
> >   From: wallyesterpaulrus 
> >   To: [email protected] 
> >   Sent: 06 Eylül 2005 Salý 22:46 
> >   Subject: [tuning] Re: Suggestion for Ozan....1/7-comma meantone
> > 
> > 
> > 
> >   > 
> >   > Now, George has formulated an excellent temperament which I 
> have 
> >   >just yesterday implemented on my Bechstein Imperial Grand, and 
> it 
> >   >sounds absolutely terrific.
> > 
> >   Ozan, have you tried either of Bob Wendell's Synchronous Well 
> >   Temperaments?

From: Ozan Yarman (2005-09-10)
Subject: Re: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

George,


  This has turned into even more of a delay -- I learned last Friday 
  afternoon that my mother-in-law (87 years old) died the previous 
  night, and I have had almost no time until today to catch up with the 
  discussion.


My condolences. May she rest in peace.




  > 
  > [oy:]
  > Now that is fascinating! But how does one go to a temperament where 
  the octave closes with equal-beating triads after, say, 30-40 tones?

  That's something I've never tried.  The first thing I would do is add 
  more rows to my spreadsheet, then take it from there.  You would have 
  to change some of the formulas for a 41-tone circulating system, 
  since you'd no longer be dealing mainly with narrow 5ths.


One could require as much as 70-80 rows there.


  However, I have done it with part of a circle of 19.  The 5/17-comma 
  5ths are not exactly 1:1:1 (5th:M3:m3) equal-beating (would need to 
  be 695.6304 cents), so I'll have to recalculate these figures when I 
  have more time:

  ! secor_19wt.scl
  !
  George Secor's 19-tone well temperament with ten 5/17-comma fifths
  19
  !
    69.40735
  131.54493
  191.25924
  260.66659
  317.95765
  382.51849
  451.92584
  504.37038
  573.77773
  638.33856
  695.62962
  765.03697
  824.57129
  886.88886
  956.29622
  1011.16402
  1078.14811
  1145.13220
  2/1


Neat! Many maqams fit into this scheme.



  With C as 1/1, the 6:7:9 triads on F, C, G, and D are greatly 
  improved over 19-ET, due to 7:9 having very low error.  I came up 
  with the essential idea for this in 1978.


Eventually, I have myself seen that equal temperaments provide only close approximations to the intended to tuning.


  > [oy:]
  > > ...
  > > And what is the total error for this extra-ordinary temperament 
  of yours?
  > [gs:]
  > For the 12 major triads: ~386.5 cents (in cell H21); since this 
  isn't a well-temperament, that figure exceeds the theoretical minimum 
  of ~375.4 cents for a curculating 12-tone temperament.


How exactly does this 375 cents accumulate? And how does it become a theoretical minimum?



  > 
  > [oy:]
  > I have implemented your tuning on my Bechstein Imperial Grand just 
  yesterday. Being the first to tune an acoustic instrument to your new 
  temperament (my tuners almost demanded double the price for the extra 
  effort!), let me also be the first to say that I simply loved the 
  results. Good Work dear George!

  I guess that's the ultimate compliment -- you've made my day!

  Best,

  --George


  The pleasure is all mine.

  Cordially,
  Ozan

From: Ozan Yarman (2005-09-10)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

Hello Graham!

----- Original Message -----
From: "Graham Breed" <[email protected]>
To: <[email protected]>
Sent: 07 Eyl\ufffdl 2005 \ufffdar\ufffdamba 23:22
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1
>
> Hello!  Yes, well, all the examples in your motley collection have the
> octave as the period.  An important one that doesn't is the diaschismic
> family, including that explained in Paul Erlich's 22 tone temperament
> paper.  I explain them in my website, I think at
>
> http://www.microtonal.co.uk/diaschis.htm
>
> The 10 note MOS is of the form
>
> s s s s L s s s s L
>
> in terms of small and large intervals.  In 22-equal, the small interval
> is 2 steps and the large one is 3 steps.  There's no way of defining
> this scale with the octave as a generator, because you lose half the
> notes.  Hence the period has to be set at a half octave, but for
> convenience you can still think of the scale as being 10 notes to an
octave.
>



So the identity interval is the octave as you said previously. Accordinly,
the generator interval is the period?


> Another example is the octatonic scale in 12-equal:
>
> 1 2 1 2 1 2 1 2
>
> This repeats 4 times in an octave, and can be generalized to give a
> linear temperament with a period of a quarter octave.
>


I remember a famous Lex Luther melody from Superman series based on this
very mode.


>
> My linear temperament searches threw up a lot of examples where the
> period is quite small.  They are most useful in the higher limits, and
> so less famous because of the difficulties involved in making music with
> such beasties.  My favourite is the one I call "mystery" which does
> 15-limit JI by dividing the octave into 29 equal parts, and using either
> 0 or 1 of the other generator (around 16 cents) to get to each 15-limit
> interval.
>


I was once interested in 29-EDO. But the more I investigated, the more I
discovered the need for higher cardinalities in Maqam Music.


>
> Looked at from another direction, meantone always divides the major
> third into two equal parts.


That had escaped me since now. Interesting!


> If, for some strange reason, you decided
> that the major third was to be your identity interval, the whole tone
> would have to be your period.


It's beginning to sink in ever so slowly.


> Some regular temperaments happen to
> equally divide an interval that wouldn't be so divisible in JI.  If one
> of those intervals happens to be the octave, then the octave can't be
> the period.


Unless the identity interval is 4/1?


>
>
>                   Graham
>
>
>


Cordially,
Ozan

From: Graham Breed (2005-09-10)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

> Hello Graham!

Hiya Ozan!

> So the identity interval is the octave as you said previously. Accordinly,
> the generator interval is the period?

The generator interval is the generator.  The period is a special kind 
of generator.  You need two generators to define a two-dimensional scale 
(or linear temperament, where the period-generator becomes invisible)


>>Another example is the octatonic scale in 12-equal:
>>
>>1 2 1 2 1 2 1 2
>>
>>This repeats 4 times in an octave, and can be generalized to give a
>>linear temperament with a period of a quarter octave.
> 
> I remember a famous Lex Luther melody from Superman series based on this
> very mode.

Could be, it gets around!

> I was once interested in 29-EDO. But the more I investigated, the more I
> discovered the need for higher cardinalities in Maqam Music.

Did you check 58?  It's the next step in the mystery chain.  Good for 
15-limit harmony, may or may not be what you want for Maqam music.


>>Some regular temperaments happen to
>>equally divide an interval that wouldn't be so divisible in JI.  If one
>>of those intervals happens to be the octave, then the octave can't be
>>the period.
> 
> Unless the identity interval is 4/1?

Yes, you can always define it that way, but most people would be 
confused by an identity interval that isn't the octave.  If a scale 
doesn't have an octave, this is unavoidable.  Otherwise, we usually 
divide the octave to get a new period.  It's a matter of convenience. 
That is, it's *supposed* to make it easier to understand.


              Graham

From: wallyesterpaulrus (2005-09-12)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Hello Graham!
> 
> ----- Original Message -----
> From: "Graham Breed" <gbreed@g...>
> To: <[email protected]>
> Sent: 07 Eylül 2005 Çarþamba 23:22
> Subject: Re: [tuning] Re: MOS at folding points other than the 2:1
> >
> > Hello!  Yes, well, all the examples in your motley collection 
have the
> > octave as the period.  An important one that doesn't is the 
diaschismic
> > family, including that explained in Paul Erlich's 22 tone 
temperament
> > paper.  I explain them in my website, I think at
> >
> > http://www.microtonal.co.uk/diaschis.htm
> >
> > The 10 note MOS is of the form
> >
> > s s s s L s s s s L
> >
> > in terms of small and large intervals.  In 22-equal, the small 
interval
> > is 2 steps and the large one is 3 steps.  There's no way of 
defining
> > this scale with the octave as a generator, because you lose half 
the
> > notes.  Hence the period has to be set at a half octave, but for
> > convenience you can still think of the scale as being 10 notes to 
an
> octave.
> >
> 
> 
> 
> So the identity interval is the octave as you said previously. 
Accordinly,
> the generator interval is the period?

If I may jump in . . .

The generator is not the period. In the case of the diatonic scale, 
the generator is the fifth (or fourth) and the period is the octave. 
In the case of the scale Graham refers to above:

s s s s L s s s s L

the period is a half-octave, or 4*s+L; the generator can be expressed 
as s, or as 4s (or as 5s+L or as 8s+L). Start with 1 note per period 
and then construct a chain of generators from each starting note. 
After iterating the generation four times, you will obtain the above 
as the arrangement of notes within each octave span.

> > Another example is the octatonic scale in 12-equal:
> >
> > 1 2 1 2 1 2 1 2
> >
> > This repeats 4 times in an octave, and can be generalized to give 
a
> > linear temperament with a period of a quarter octave.
> >
> 
> 
> I remember a famous Lex Luther melody from Superman series based on 
this
> very mode.

The generator here is 1 or 2 (or equivalently, any integer not 
divisible by 3) in steps of 12-equal.


> > My linear temperament searches threw up a lot of examples where 
the
> > period is quite small.  They are most useful in the higher 
limits, and
> > so less famous because of the difficulties involved in making 
music with
> > such beasties.  My favourite is the one I call "mystery" which 
does
> > 15-limit JI by dividing the octave into 29 equal parts, and using 
either
> > 0 or 1 of the other generator (around 16 cents) to get to each 15-
limit
> > interval.
> >
> 
> 
> I was once interested in 29-EDO. But the more I investigated, the 
more I
> discovered the need for higher cardinalities in Maqam Music.

Have you considered this system of Graham's though, with a generator 
of around 16 cents?

From: wallyesterpaulrus (2005-09-12)
Subject: Re: MOS at folding points other than the 2:1

Jumping in again (sorry . . .)

--- In [email protected], Graham Breed <gbreed@g...> wrote:

> >>Some regular temperaments happen to
> >>equally divide an interval that wouldn't be so divisible in JI.  If 
one
> >>of those intervals happens to be the octave, then the octave can't 
be
> >>the period.
> > 
> > Unless the identity interval is 4/1?
> 
> Yes, you can always define it that way,

It doesn't seem to me that that would help -- the octave still can't be 
the period, no matter what the identity interval, if one of the 
tempered JI intervals happens to be the octave.

> Otherwise, we usually 
> divide the octave to get a new period.

Right -- which wouldn't be 4/1, but rather would occur *twice as many 
times* in the 4/1 as it does in the octave.

From: Graham Breed (2005-09-13)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

wallyesterpaulrus wrote:

> It doesn't seem to me that that would help -- the octave still can't be 
> the period, no matter what the identity interval, if one of the 
> tempered JI intervals happens to be the octave.

Oh, yes, I clearly didn't understand the question there...


          Graham

From: George D. Secor (2005-09-13)
Subject: 19-tone Well-temperament (Was: Temperament (Extra)Ordinaire ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> [gs:]
>   However, I have done it with part of a circle of 19.  The 5/17-
comma 
>   5ths are not exactly 1:1:1 (5th:M3:m3) equal-beating (would need 
to 
>   be 695.6304 cents), so I'll have to recalculate these figures 
when I 
>   have more time:
> 
>   ! secor_19wt.scl
>   !
>   George Secor's 19-tone well temperament with ten 5/17-comma fifths
>   19
>   !
>     69.40735
>   131.54493
>   191.25924
>   260.66659
>   317.95765
>   382.51849
>   451.92584
>   504.37038
>   573.77773
>   638.33856
>   695.62962
>   765.03697
>   824.57129
>   886.88886
>   956.29622
>   1011.16402
>   1078.14811
>   1145.13220
>   2/1

> [oy:]
> Neat! Many maqams fit into this scheme.

>   With C as 1/1, the 6:7:9 triads on F, C, G, and D are greatly 
>   improved over 19-ET, due to 7:9 having very low error.  I came up 
>   with the essential idea for this in 1978.
> 
> Eventually, I have myself seen that equal temperaments provide only 
close approximations to the intended to tuning.

I find that 19-ET leaves much to be desired when it comes to the 7ths 
(all types).  The diatonic major 7th is rather low, so is not the 
most effective leading tone.  The minor 7th is rather high, so does 
not contribute to a very effective (diatonic) dominant 7th chord.  
The augmented 6th is rather low, so does not contribute to a very 
effective harmonic 7th chord.  This well-temperament does much to 
alleviate these problems, in a few keys at least.

Breaking news:  I've not only recalculated the 5/17-comma fifths -- 
I've also developed a 19-tone well-temperament with 15 proportional-
beating major triads.

There are, at the moment, 2 versions.  The first one has the 
proportional-beating triads in the sharp direction:

! secor_19wt1.scl
!
George Secor's 19-tone proportional-beating (5/17-comma) well 
temperament (v.1)
 19
!
  69.41306
 131.21719
 191.26088
 260.67394
 318.03803
 382.52175
 451.93481
 504.36956
 573.78263
 638.12678
 695.63044
 765.04350
 824.94573
 886.89131
 956.30438
1011.12652
1078.15219
1145.03265
2/1

Rather than put out an entire spreadsheet with all the figures, I'll 
summarize with the following:

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
Db... 2.6627 0.6693 1.7822 (not proportional)
Ab... 2.1159 1.4839 3.1399 (not proportional)
Eb... 1.6394 24.486 40.143 (not proportional)
Bb... 1.1830 1.3784 1.6307 (not proportional)
F.... 5.0000 10.000 2.0000
C.... 1.0000 1.0000 1.0000
G.... 1.0000 1.0000 1.0000
D.... 1.0000 1.0000 1.0000
A.... 1.0000 1.0000 1.0000
E.... 1.0000 1.0000 1.0000
B.... 1.0000 1.0000 1.0000
F#... 1.0000 1.0000 1.0000
C#... 1.0000 1.0000 1.0000
G#... 1.6667 ------ ------
D#... 2.3333 1.0000 2.3333
A#... 3.0000 0.5000 1.5000
E#/Fb 2.5000 0.8000 2.0000
B#/Cb 2.5000 0.8000 2.0000
Fx/Gb 2.5000 0.8000 2.0000
Total triad error:  330.864 cents

The second version was a little trickier to construct; it has 3 of 
the propotional-beating triads in the flat direction:

! secor_19wt2.scl
!
George Secor's 19-tone proportional-beating (5/17-comma) well 
temperament (v.2)
 19
!
  69.41306
 131.21719
 191.26088
 260.67394
 317.43304
 382.52175
 451.93481
 504.36956
 573.78263
 638.12678
 695.63044
 765.04350
 824.40909
 886.89131
 956.30438
1010.63200
1078.15219
1145.03265
2/1

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
Db... 2.5000 0.8000 2.0000
Ab... 2.0000 2.0000 4.0000
Eb... 1.5452 5.4877 8.4796 (not proportional)
Bb... 1.1538 1.3000 1.5000
F.... 5.0000 10.000 2.0000
C.... 1.0000 1.0000 1.0000
G.... 1.0000 1.0000 1.0000
D.... 1.0000 1.0000 1.0000
A.... 1.0000 1.0000 1.0000
E.... 1.0000 1.0000 1.0000
B.... 1.0000 1.0000 1.0000
F#... 1.0000 1.0000 1.0000
C#... 1.0000 1.0000 1.0000
G#... 1.6667 ------ ------
D#... 2.3333 1.0000 2.3333
A#... 3.0000 0.5000 1.5000
E#/Fb 2.6004 0.7139 1.8566 (not proportional)
B#/Cb 2.6132 0.7043 1.8406 (not proportional)
Fx/Gb 2.5925 0.7201 1.8668 (not proportional)
Total triad error:  331.853 cents

The two versions differ in the tuning of only 3 tones, with the 
differences on the order of ~1/2-cent each.  I haven't listened to 
them yet to see if I can tell any difference -- or whether either of 
these is really worth the trouble vs. the original temperament quoted 
at the top of this message (which has 5ths of only 2 different 
sizes).  :-)

The total triad error, BTW, is the sum of the total absolute error of 
all 19 major triads, which is smaller than that of the 12 major 
triads of 12-ET.

The rest of this discussion pertains to my 12-tone temperament (extra)
ordinaire:

> ...
>   > [oy:]
>   > > ...
>   > > And what is the total error for this extra-ordinary 
temperament of yours?
>   > [gs:]
>   > For the 12 major triads: ~386.5 cents (in cell H21); since this 
>   isn't a well-temperament, that figure exceeds the theoretical 
minimum 
>   of ~375.4 cents for a curculating 12-tone temperament.
> 
> How exactly does this 375 cents accumulate? And how does it become 
a theoretical minimum?

Take the sum of the total absolute errors of all 12 triads.  The 
theoretical minimum is the same as the sum-total for 12-ET, and you 
won't exceed this figure if you have no 5ths or minor 3rds wider than 
just and no major 3rds narrower than just.

Best,

--George

From: George D. Secor (2005-09-13)
Subject: Re: 19-tone Well-temperament (OOPS!)

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

Oops!!! The beat ratios for F major triad in the tables for both 
versions should be:

Major ----Beat Ratios----
Triad M3/5th 5th/m3 M3/m3
----- ------ ------ ------
F.... 1.0000 1.0000 1.0000

--George

From: Ozan Yarman (2005-09-16)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

> Hiya Ozan!
>

Hello again.


> The generator interval is the generator.  The period is a special kind
> of generator.  You need two generators to define a two-dimensional scale
> (or linear temperament, where the period-generator becomes invisible)
>

And it becomes visible in rank 3 temperaments?


> Did you check 58?  It's the next step in the mystery chain.  Good for
> 15-limit harmony, may or may not be what you want for Maqam music.
>


In 58, the cycle of fifths produce super-pythagorean intervals, which do not
very well suit my needs. My default is a meantone-pythagorean hybrid that
closes the cycle in between 31 or 55 jumps.


> Unless the identity interval is 4/1?
>
> Yes, you can always define it that way, but most people would be
> confused by an identity interval that isn't the octave.  If a scale
> doesn't have an octave, this is unavoidable.  Otherwise, we usually
> divide the octave to get a new period.  It's a matter of convenience.
> That is, it's *supposed* to make it easier to understand.
>


So, the `period` AKA the `interval of periodicity` is the interval where the
scale repeats itself using the exact same intervals. This generates the
`interval of equivalance`, as in the `octave`, or else, is squeezed inside
it any number of times.



>
>               Graham
>
>

Cordially,
Ozan

From: Ozan Yarman (2005-09-17)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

Paul, I did not find the oppurtunity to thank you for your informative post. Can you point to me a scala file for Graham's tuning where 16 cents is the generator?

Cordially,
Ozan

------------------------


> So the identity interval is the octave as you said previously.Accordinly,the generator interval is the period?


If I may jump in . . .

The generator is not the period. In the case of the diatonic scale, 
the generator is the fifth (or fourth) and the period is the octave. 
In the case of the scale Graham refers to above:

s s s s L s s s s L

the period is a half-octave, or 4*s+L; the generator can be expressed as s, or as 4s (or as 5s+L or as 8s+L). Start with 1 note per period and then construct a chain of generators from each starting note. After iterating the generation four times, you will obtain the above as the arrangement of notes within each octave span.


> I was once interested in 29-EDO. But the more I investigated, the 
more I discovered the need for higher cardinalities in Maqam Music.

Have you considered this system of Graham's though, with a generator 
of around 16 cents?

From: Ozan Yarman (2005-09-17)
Subject: Re: [tuning] 19-tone Well-temperament (Was: Temperament (Extra)Ordinaire ...)

Dear George,
  ----- Original Message ----- 
  From: George D. Secor 
  To: [email protected] 
  Sent: 13 Eylül 2005 Salı 21:58 
  Subject: [tuning] 19-tone Well-temperament (Was: Temperament (Extra)Ordinaire ...)
  SNIP


  I find that 19-ET leaves much to be desired when it comes to the 7ths (all types).  The diatonic major 7th is rather low, so is not the most effective leading tone.  The minor 7th is rather high, so does not contribute to a very effective (diatonic) dominant 7th chord. The augmented 6th is rather low, so does not contribute to a very effective harmonic 7th chord.  This well-temperament does much to alleviate these problems, in a few keys at least.


Your recent endeavour in equal-beating temperaments is truly commendable!



  Breaking news:  I've not only recalculated the 5/17-comma fifths -- 
  I've also developed a 19-tone well-temperament with 15 proportional-
  beating major triads.

  There are, at the moment, 2 versions.  The first one has the 
  proportional-beating triads in the sharp direction:

  ! secor_19wt1.scl
  !
  George Secor's 19-tone proportional-beating (5/17-comma) well 
  temperament (v.1)
  19
  !
    69.41306
  131.21719
  191.26088
  260.67394
  318.03803
  382.52175
  451.93481
  504.36956
  573.78263
  638.12678
  695.63044
  765.04350
  824.94573
  886.89131
  956.30438
  1011.12652
  1078.15219
  1145.03265
  2/1

  Rather than put out an entire spreadsheet with all the figures, I'll summarize with the following:

   Major ----Beat Ratios----
   Triad M3/5th 5th/m3 M3/m3
   ----- ------ ------ ------
   Db... 2.6627 0.6693 1.7822 (not proportional)
   Ab... 2.1159 1.4839 3.1399 (not proportional)
   Eb... 1.6394 24.486 40.143 (not proportional)
   Bb... 1.1830 1.3784 1.6307 (not proportional)
   F.... 1.0000 1.0000 1.0000
  C.... 1.0000 1.0000 1.0000
  G.... 1.0000 1.0000 1.0000
  D.... 1.0000 1.0000 1.0000
  A.... 1.0000 1.0000 1.0000
  E.... 1.0000 1.0000 1.0000
  B.... 1.0000 1.0000 1.0000
  F#... 1.0000 1.0000 1.0000
  C#... 1.0000 1.0000 1.0000
  G#... 1.6667 ------ ------
  D#... 2.3333 1.0000 2.3333
  A#... 3.0000 0.5000 1.5000
  E#/Fb 2.5000 0.8000 2.0000
  B#/Cb 2.5000 0.8000 2.0000
  Fx/Gb 2.5000 0.8000 2.0000
  Total triad error:  330.864 cents



This is pretty good.



  The second version was a little trickier to construct; it has 3 of 
  the propotional-beating triads in the flat direction:

  ! secor_19wt2.scl
  !
  George Secor's 19-tone proportional-beating (5/17-comma) well 
  temperament (v.2)
  19
  !
    69.41306
  131.21719
  191.26088
  260.67394
  317.43304
  382.52175
  451.93481
  504.36956
  573.78263
  638.12678
  695.63044
  765.04350
  824.40909
  886.89131
  956.30438
  1010.63200
  1078.15219
  1145.03265
  2/1

  Major ----Beat Ratios----
  Triad M3/5th 5th/m3 M3/m3
  ----- ------ ------ ------
  Db... 2.5000 0.8000 2.0000
  Ab... 2.0000 2.0000 4.0000
  Eb... 1.5452 5.4877 8.4796 (not proportional)
  Bb... 1.1538 1.3000 1.5000
  F.... 1.0000 1.0000 1.0000
  C.... 1.0000 1.0000 1.0000
  G.... 1.0000 1.0000 1.0000
  D.... 1.0000 1.0000 1.0000
  A.... 1.0000 1.0000 1.0000
  E.... 1.0000 1.0000 1.0000
  B.... 1.0000 1.0000 1.0000
  F#... 1.0000 1.0000 1.0000
  C#... 1.0000 1.0000 1.0000
  G#... 1.6667 ------ ------
  D#... 2.3333 1.0000 2.3333
  A#... 3.0000 0.5000 1.5000
  E#/Fb 2.6004 0.7139 1.8566 (not proportional)
  B#/Cb 2.6132 0.7043 1.8406 (not proportional)
  Fx/Gb 2.5925 0.7201 1.8668 (not proportional)
  Total triad error:  331.853 cents



Ditto.




  The two versions differ in the tuning of only 3 tones, with the 
  differences on the order of ~1/2-cent each.  I haven't listened to 
  them yet to see if I can tell any difference -- or whether either of 
  these is really worth the trouble vs. the original temperament quoted 
  at the top of this message (which has 5ths of only 2 different 
  sizes).  :-)

  The total triad error, BTW, is the sum of the total absolute error of 
  all 19 major triads, which is smaller than that of the 12 major 
  triads of 12-ET.


Exceptionally so! I assume that the total error in 12-tET is 12*(400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the minor thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum total error for the triads. Correct?




  The rest of this discussion pertains to my 12-tone temperament (extra)ordinaire:


  > How exactly does this 375 cents accumulate? And how does it become a theoretical minimum?

  Take the sum of the total absolute errors of all 12 triads.  The 
  theoretical minimum is the same as the sum-total for 12-ET, and you won't exceed this figure if you have no 5ths or minor 3rds wider than just and no major 3rds narrower than just.

  Best,

  --George




Cordially,
Ozan

From: George D. Secor (2005-09-19)
Subject: Re: 19-tone Well-temperament (Was: Temperament (Extra)Ordinaire ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear George,
>   ----- Original Message ----- 
>   From: George D. Secor 
>   To: [email protected] 
>   Sent: 13 Eylül 2005 Salý 21:58 
>   Subject: [tuning] 19-tone Well-temperament (Was: Temperament 
(Extra)Ordinaire ...)
>   SNIP
> [gs:]
>   I find that 19-ET leaves much to be desired when it comes to the 
7ths (all types).  The diatonic major 7th is rather low, so is not 
the most effective leading tone.  The minor 7th is rather high, so 
does not contribute to a very effective (diatonic) dominant 7th 
chord. The augmented 6th is rather low, so does not contribute to a 
very effective harmonic 7th chord.  This well-temperament does much 
to alleviate these problems, in a few keys at least.
> 
> [oy:]
> Your recent endeavour in equal-beating temperaments is truly 
commendable!
> 
> [gs:]
>   Breaking news:  I've not only recalculated the 5/17-comma fifths -
- 
>   I've also developed a 19-tone well-temperament with 15 
proportional-
>   beating major triads.
> 
>   There are, at the moment, 2 versions.  The first one has the 
>   proportional-beating triads in the sharp direction:
> 
>   ! secor_19wt1.scl
>   !
>   George Secor's 19-tone proportional-beating (5/17-comma) well 
>   temperament (v.1)
>   19
>   !
>   69.41306
>   131.21719
>   191.26088
>   260.67394
>   318.03803
>   382.52175
>   451.93481
>   504.36956
>   573.78263
>   638.12678
>   695.63044
>   765.04350
>   824.94573
>   886.89131
>   956.30438
>   1011.12652
>   1078.15219
>   1145.03265
>   2/1
> 
>   Rather than put out an entire spreadsheet with all the figures, 
I'll summarize with the following:
> 
>   Major ----Beat Ratios----
>   Triad M3/5th 5th/m3 M3/m3
>   ----- ------ ------ ------
>   Db... 2.6627 0.6693 1.7822 (not proportional)
>   Ab... 2.1159 1.4839 3.1399 (not proportional)
>   Eb... 1.6394 24.486 40.143 (not proportional)
>   Bb... 1.1830 1.3784 1.6307 (not proportional)
>   F.... 1.0000 1.0000 1.0000
>   C.... 1.0000 1.0000 1.0000
>   G.... 1.0000 1.0000 1.0000
>   D.... 1.0000 1.0000 1.0000
>   A.... 1.0000 1.0000 1.0000
>   E.... 1.0000 1.0000 1.0000
>   B.... 1.0000 1.0000 1.0000
>   F#... 1.0000 1.0000 1.0000
>   C#... 1.0000 1.0000 1.0000
>   G#... 1.6667 ------ ------
>   D#... 2.3333 1.0000 2.3333
>   A#... 3.0000 0.5000 1.5000
>   E#/Fb 2.5000 0.8000 2.0000
>   B#/Cb 2.5000 0.8000 2.0000
>   Fx/Gb 2.5000 0.8000 2.0000
>   Total triad error:  330.864 cents
> 
> 
> [oy:]
> This is pretty good.
> 
> [gs:]
>   The second version was a little trickier to construct; it has 3 
of 
>   the propotional-beating triads in the flat direction:
> 
>   ! secor_19wt2.scl
> 
>   (SNIP!)
> 
>   Total triad error:  331.853 cents
> 
> [oy:]
> Ditto.
> 
> [gs:]
>   The two versions differ in the tuning of only 3 tones, with the 
>   differences on the order of ~1/2-cent each.  I haven't listened 
to 
>   them yet to see if I can tell any difference -- or whether either 
of 
>   these is really worth the trouble vs. the original temperament 
quoted 
>   at the top of this message (which has 5ths of only 2 different 
>   sizes).  :-)

Now that I've listened to them, I find that I really can't hear any 
appreciable difference.  On the basis of the numbers I've decided 
that I'll advocate version 1 (for several reasons, but chiefly 
because it has the lowest total error in the 19 triads).

>   The total triad error, BTW, is the sum of the total absolute 
error of 
>   all 19 major triads, which is smaller than that of the 12 major 
>   triads of 12-ET.
> 
> [oy:]
> Exceptionally so! I assume that the total error in 12-tET is 12*
(400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds 
PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the minor 
thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum total 
error for the triads. Correct?

You forgot to add in the total error for the 5ths: 12*(701.9550 - 
700) = 23.4600, which gives a total absolute error of 375.3909 cents 
for 12 triads.

Best,

--George

From: Ozan Yarman (2005-09-19)
Subject: Re: [tuning] Re: 19-tone Well-temperament (Was: Temperament (Extra)Ordinaire ...)

> Exceptionally so! I assume that the total error in 12-tET is 12*
(400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds 
PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the minor thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum total 
error for the triads. Correct?

(GS)
You forgot to add in the total error for the 5ths: 12*(701.9550 - 
700) = 23.4600, which gives a total absolute error of 375.3909 cents for 12 triads.

-------

Ouch! You are right. But I did figure out the correct calculation, did I not?

--------

Best,

--George



Cordially,
Ozan

From: wallyesterpaulrus (2005-09-19)
Subject: Re: MOS at folding points other than the 2:1

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

> So, the `period` AKA the `interval of periodicity` is the interval 
where the
> scale repeats itself using the exact same intervals.

Yes.

> This generates the
> `interval of equivalance`, as in the `octave`, or else, is squeezed 
inside
> it any number of times.

The period normally generates the interval of equivalence, whether in 
1 step (when the two are the same) or in any whole number N steps 
(when one has to iterate the period N times in order to generate the 
interval of equivalence. And the interval of equivalence is normally 
taken as pretty much given a priori (some say pan-culturally) and 
understood to be the "octave" or (~)2:1 ratio, though experiments 
with 3:1 as the interval of equivalence certainly yield musically 
intriguing, if not convincing, results.

From: wallyesterpaulrus (2005-09-19)
Subject: Re: MOS at folding points other than the 2:1

I don't know of such a Scala file -- and Graham should provide one if 
he hasn't already -- but the basic idea is to take 29-equal, and 
superimpose another 29-equal set about 16 cents higher (and in theory 
another 29-equal set about 16 cents higher than that, and so on) -- 
this 58-note scale, I believe, is one of the most efficient 2D 
systems for acheiving lots of higher-limit harmonies.

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Paul, I did not find the oppurtunity to thank you for your 
informative post. Can you point to me a scala file for Graham's 
tuning where 16 cents is the generator?
> 
> Cordially,
> Ozan
> 
> ------------------------
> 
> 
> > So the identity interval is the octave as you said 
previously.Accordinly,the generator interval is the period?
> 
> 
> If I may jump in . . .
> 
> The generator is not the period. In the case of the diatonic scale, 
> the generator is the fifth (or fourth) and the period is the 
octave. 
> In the case of the scale Graham refers to above:
> 
> s s s s L s s s s L
> 
> the period is a half-octave, or 4*s+L; the generator can be 
expressed as s, or as 4s (or as 5s+L or as 8s+L). Start with 1 note 
per period and then construct a chain of generators from each 
starting note. After iterating the generation four times, you will 
obtain the above as the arrangement of notes within each octave span.
> 
> 
> > I was once interested in 29-EDO. But the more I investigated, the 
> more I discovered the need for higher cardinalities in Maqam Music.
> 
> Have you considered this system of Graham's though, with a 
generator 
> of around 16 cents?

From: Graham Breed (2005-09-19)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

wallyesterpaulrus wrote:
> I don't know of such a Scala file -- and Graham should provide one if 
> he hasn't already -- but the basic idea is to take 29-equal, and 
> superimpose another 29-equal set about 16 cents higher (and in theory 
> another 29-equal set about 16 cents higher than that, and so on) -- 
> this 58-note scale, I believe, is one of the most efficient 2D 
> systems for acheiving lots of higher-limit harmonies.

This is completely untested...

!mystery58.scl
Mystery temperament with 16 cent generator
  58
!
16.000
41.379
57.379
82.759
98.759
124.138
140.138
165.517
181.517
206.897
222.897
248.276
264.276
289.655
305.655
331.034
347.034
372.414
388.414
413.793
429.793
455.172
471.172
496.552
512.552
537.931
553.931
579.310
595.310
620.690
636.690
662.069
678.069
703.448
719.448
744.828
760.828
786.207
802.207
827.586
843.586
868.966
884.966
910.345
926.345
951.724
967.724
993.103
1009.103
1034.483
1050.483
1075.862
1091.862
1117.241
1133.241
1158.621
1174.621
1200.000


                 Graham

From: George D. Secor (2005-09-19)
Subject: Re: 19-tone Well-temperament (Was: Temperament (Extra)Ordinaire ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> > Exceptionally so! I assume that the total error in 12-tET is 12*
> (400 - 386.3137) = 12*13.6863 = 164.2354 cents for the major thirds 
> PLUS 12*(315.6413 - 300) = 12*15.6413 = 187.6954 cents for the 
minor thirds, which equals 187.6954 + 164.2354 = 351.9308 cents sum 
total 
> error for the triads. Correct?
> 
> (GS)
> You forgot to add in the total error for the 5ths: 12*(701.9550 - 
> 700) = 23.4600, which gives a total absolute error of 375.3909 
cents for 12 triads.
> 
> -------
> (OY)
> Ouch! You are right. But I did figure out the correct calculation, 
did I not?

Apart from the omission of the 5th, your calculation was correct.

Best,

--George

From: Ozan Yarman (2005-09-20)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

So the interval of equivalence differing from the period is only the result of the tuning jargon developed by the pioneers of microtonality and the efforts of the explorers on this list?

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 19 Eylül 2005 Pazartesi 22:23 
  Subject: [tuning] Re: MOS at folding points other than the 2:1


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:

  > So, the `period` AKA the `interval of periodicity` is the interval 
  where the
  > scale repeats itself using the exact same intervals.

  Yes.

  > This generates the
  > `interval of equivalance`, as in the `octave`, or else, is squeezed 
  inside
  > it any number of times.

  The period normally generates the interval of equivalence, whether in 
  1 step (when the two are the same) or in any whole number N steps 
  (when one has to iterate the period N times in order to generate the 
  interval of equivalence. And the interval of equivalence is normally 
  taken as pretty much given a priori (some say pan-culturally) and 
  understood to be the "octave" or (~)2:1 ratio, though experiments 
  with 3:1 as the interval of equivalence certainly yield musically 
  intriguing, if not convincing, results.

From: Ozan Yarman (2005-09-20)
Subject: Re: [tuning] Re: MOS at folding points other than the 2:1

Paul, I thought you had said to me that 29-tET was unsuitable for Western polyphony. What's so special about this particular 58?
  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 19 Eylül 2005 Pazartesi 23:02 
  Subject: [tuning] Re: MOS at folding points other than the 2:1


  I don't know of such a Scala file -- and Graham should provide one if he hasn't already -- but the basic idea is to take 29-equal, and superimpose another 29-equal set about 16 cents higher (and in theory another 29-equal set about 16 cents higher than that, and so on) -- this 58-note scale, I believe, is one of the most efficient 2D systems for acheiving lots of higher-limit harmonies.

From: wallyesterpaulrus (2005-09-21)
Subject: Re: MOS at folding points other than the 2:1

I wouldn't say that. There are plenty of examples in Western music, 
from Rimsky-Korsakov to Miles Davis, of scales like 3-1-3-1-3-1 and 1-
2-1-2-1-2-1-2, where the interval of equivalence is 2 or 3 or 4 times 
the period.

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> So the interval of equivalence differing from the period is only 
the result of the tuning jargon developed by the pioneers of 
microtonality and the efforts of the explorers on this list?
> 
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 19 Eylül 2005 Pazartesi 22:23 
>   Subject: [tuning] Re: MOS at folding points other than the 2:1
> 
> 
>   --- In [email protected], "Ozan Yarman" <ozanyarman@s...> 
wrote:
> 
>   > So, the `period` AKA the `interval of periodicity` is the 
interval 
>   where the
>   > scale repeats itself using the exact same intervals.
> 
>   Yes.
> 
>   > This generates the
>   > `interval of equivalance`, as in the `octave`, or else, is 
squeezed 
>   inside
>   > it any number of times.
> 
>   The period normally generates the interval of equivalence, 
whether in 
>   1 step (when the two are the same) or in any whole number N steps 
>   (when one has to iterate the period N times in order to generate 
the 
>   interval of equivalence. And the interval of equivalence is 
normally 
>   taken as pretty much given a priori (some say pan-culturally) and 
>   understood to be the "octave" or (~)2:1 ratio, though experiments 
>   with 3:1 as the interval of equivalence certainly yield musically 
>   intriguing, if not convincing, results.

From: wallyesterpaulrus (2005-09-21)
Subject: Re: MOS at folding points other than the 2:1

It's still unsuitable for Western polyphony, but extremely efficient 
for delivering lots of complete 13-limit and 15-limit near-JI 
harmonies. To my ear, these kind of harmonies (including variants where 
some notes are omitted) are the only places where near-JI tuning of 
ratios of 13 and 15 really matters aurally and is noticeably different 
from not-near-JI tuning of such intervals. So if you're interested in 
such all intervals and in polyphony, such a system seems like an 
excellent choice. So easy to notate too! Looks like Gene suggested a 
generator of 15.82578774 cents in one 13-odd-limit (so not considering 
ratios of 15) context, in case 16 cents wasn't exact enough for you :)

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> Paul, I thought you had said to me that 29-tET was unsuitable for 
Western polyphony. What's so special about this particular 58?
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 19 Eylül 2005 Pazartesi 23:02 
>   Subject: [tuning] Re: MOS at folding points other than the 2:1
> 
> 
>   I don't know of such a Scala file -- and Graham should provide one 
if he hasn't already -- but the basic idea is to take 29-equal, and 
superimpose another 29-equal set about 16 cents higher (and in theory 
another 29-equal set about 16 cents higher than that, and so on) -- 
this 58-note scale, I believe, is one of the most efficient 2D systems 
for acheiving lots of higher-limit harmonies.

From: Yahya Abdal-Aziz (2005-10-15)
Subject: RE: Temperament (Extra)ordinare! (Was: Further doubts ...)

Hi George,

On Mon, 29 Aug 2005, George D. Secor wrote: 
> 
> HEY EVERYONE!
> 
> Having been unable to ignore the discussion about equal-beating 
> temperaments lately, I have some startling news about a "temperament 
> (extra)ordinaire!"
... [snipt]

! secor12_1.scl
!
George Secor's 12-tone temperament ordinaire #1, proportional beating
 12
!
 86.53330
 194.55680
 294.12876
 389.11361
 499.91792
 585.54105
 697.27840
 789.37483
 891.83521
 997.96292
 1086.39201

 2/1

... [snipt]

> After listening and comparing, I've concluded that the 8 equal- and 
> proportional-beating major triads in this temperament sound as if 
> they have less total error than the numbers would indicate, with the 
> faster beats tending to be masked by the slower ones.  In conducting 
> some head-to-head comparisons of the C and G major triads with those 
> in 31-ET (12c total error), I found that they sound at least as good; 
> likewise the E and Eb major triads as compared with 12-ET.  (The 
> Scalatron permits me to jump back and forth instantly from one tuning 
> to another, which easily allows me to make comparisons using the same 
> timbres.)  I also found, for some reason, that the B and Ab major 
> triads sound only about as dissonant as pythagorean triads, even 
> though they have about 5 cents greater total error.
> 
> I was frankly a bit surprised at how quickly I was able to come up 
> with proportional beat rates for so many triads, thinking that it 
> should have been a lot harder to do this and beginning to experience 
> some feelings of doubt that there must be some room for improvement --
>  at this point I don't know.  (But the more I play around with it, 
> the less my doubt.)
> 
> Anyway, I hope that some others will try this tuning and give me your 
> reactions -- at least go through the triads in Scala using the chord 
> playing function in the keyboard clavier.  Convince me that this 
> isn't all a dream!


I expect you've had plenty of feedback by now?
Anyway, FWIW, here are my reactions to tuning up your
"extraordinary" temperament, then playing with it for
an hour or so using both piano and harpsichord samples
on my Roland keyboard.

Everything is fine - EXCEPT the sounds of the bare Db, 
Eb, E, F#, Ab and B major triads in root position with 
root in the octave above middle C.  Other inversions of 
these triads are (more or less) acceptable, while more 
open chords using these harmonies are fine.  Playing an 
accompanied melody, or improvised two- and three-part 
counterpoint, all sounds pleasant enough to me.  The only 
problem I have is with these triads in that range.

I checked out the actual sizes of the fifths and thirds
in all major chords.  The fifths are not a problem - by
construction, almost.  (I don't really like fifths as 
narrow as 697 cents, but that's the price we pay for
being able to play in all keys, I guess.)  The major thirds
in these chords, without exception, are a little wide to 
very wide, and I think this explains the worst of my 
difficulty with them.  Still, the E-G# is only 400.26122,
hardly different from the 12EDO third of 400 cents.
I can't really explain why the E major triad on E4 sounds
so wrong to me.  Is it the minor third of ~297 cents? It
can't be the fifth - the D, G and A major chords, which 
have the same size fifth, are all quite good; tho the D 
chord seems noticeably flat, the A sounds a little better
than in 12EDO and the G major chord sounds superb - 
much better than in 12EDO.

Perhaps all the beat frequencies for all my problem 
triads lie in a range that I react badly to.  Anyway, 
something about them sets my teeth on edge!

Regards,
Yahya


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From: wallyesterpaulrus (2005-10-17)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:

> I can't really explain why the E major triad on E4 sounds
> so wrong to me.  Is it the minor third of ~297 cents?

I think that may very well be your explanation.

> Perhaps all the beat frequencies for all my problem 
> triads lie in a range that I react badly to.

How did you implement this tuning, exactly? Did you check the tuning 
independently after you set it?

From: George D. Secor (2005-10-17)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> 
> Hi George,
> 
> On Mon, 29 Aug 2005, George D. Secor wrote: 
> > 
> > HEY EVERYONE!
> > 
> > Having been unable to ignore the discussion about equal-beating 
> > temperaments lately, I have some startling news about 
a "temperament 
> > (extra)ordinaire!"
> ... [snipt]
> > 
> > I was frankly a bit surprised at how quickly I was able to come 
up 
> > with proportional beat rates for so many triads, thinking that it 
> > should have been a lot harder to do this and beginning to 
experience 
> > some feelings of doubt that there must be some room for 
improvement --
> >  at this point I don't know.  (But the more I play around with 
it, 
> > the less my doubt.)

A couple of months have passed since I posted that, and I'm happy to 
say that I've not only improved it (proportional beating in the 
*minor* as well as the major triads), but after extensive 
experimentation involving dozens of tuning attempts, I've selected 
the five best tunings to be members of an entire suite of 
proportional-beating well-temperaments (WT's) and _temperaments 
ordinaires_ (TO's) -- details to be given at a later date (once I'm 
sure that I've settled on my final selection).

> > Anyway, I hope that some others will try this tuning and give me 
your 
> > reactions -- at least go through the triads in Scala using the 
chord 
> > playing function in the keyboard clavier.  Convince me that this 
> > isn't all a dream!
> 
> I expect you've had plenty of feedback by now?

Yes, but more is welcome.

> Anyway, FWIW, here are my reactions to tuning up your
> "extraordinary" temperament, then playing with it for
> an hour or so using both piano and harpsichord samples
> on my Roland keyboard.
> 
> Everything is fine - EXCEPT the sounds of the bare Db, 
> Eb, E, F#, Ab and B major triads in root position with 
> root in the octave above middle C.  Other inversions of 
> these triads are (more or less) acceptable, while more 
> open chords using these harmonies are fine.  Playing an 
> accompanied melody, or improvised two- and three-part 
> counterpoint, all sounds pleasant enough to me.  The only 
> problem I have is with these triads in that range.

They all have total absolute error greater than the triads of 12-ET, 
and the problem is mostly with the thirds.  You'll need to understand 
that the purpose of a _temperament ordinaire_ is to have high 
(meantone-like) consonance in as many triads as possible after taming 
the "wolf".  In my temperament (extra)ordinaire wolf fifths are 
completely eliminated.  The 3rds will of necessity be worse than in 
12-ET, but the objective is to have as few "worst" triads as 
possible.  I'm a bit surprised that you had problems with E and 
especially Eb -- I would judge them to be not significantly worse 
than 12-ET.  (However, see my comment about becoming "spoiled", 
below.)

> I checked out the actual sizes of the fifths and thirds
> in all major chords.  The fifths are not a problem - by
> construction, almost.  (I don't really like fifths as 
> narrow as 697 cents, but that's the price we pay for
> being able to play in all keys, I guess.)  

No, narrow fifths are the price we pay for being able to get the 
thirds closer to just in the best triads.

> The major thirds
> in these chords, without exception, are a little wide to 
> very wide, and I think this explains the worst of my 
> difficulty with them.  

Yes.

> Still, the E-G# is only 400.26122,
> hardly different from the 12EDO third of 400 cents.
> I can't really explain why the E major triad on E4 sounds
> so wrong to me.  Is it the minor third of ~297 cents? 

Probably.  (Those who think that the beating of minor 3rds aren't 
nearly as important as beating major 3rds should take note.)

Another possibility is that once you've gotten "spoiled" hearing how 
nice the triads from Bb thru A sound, you're jarred by the relative 
dissonance of triads only slightly worse than 12-ET.  In my latest 
experimentation with WT's and TO's I've listened and relistened (and 
compared and recompared) tunings with one another and have been 
occasionally surprised to find that I may experience different 
reactions at different times -- sometimes only a matter of 10 or 15 
minutes apart.  At one time a Pythagorean triad may sound only 
marginally acceptable and at another time a triad with even more 
total absolute error than that (such as the worst ones in the 
temperament extraordinaire) will sound more acceptable.  Likewise, 
sometimes a low-contrast modern well-temperament (with a relatively 
small difference in intonation between the best and worst triads) 
will sound as if all the triads are very close to 12-ET (and 
therefore probably not worth the effort), whereas at other times the 
difference between the best and worst will seem quite effective.

There seems to be some sort of "hearing fatigue" in operation here, 
and it has made me look at Aaron Johnson's "Killing the Buddha of 
well-temperament" message
http://groups.yahoo.com/group/tuning/message/60103
with a new perspective.

> It
> can't be the fifth - the D, G and A major chords, which 
> have the same size fifth, are all quite good; tho the D 
> chord seems noticeably flat, 

I assume you're referring primarily to the 3rd as "flat" -- that's 
what make the triad more consonant.

> the A sounds a little better
> than in 12EDO and the G major chord sounds superb - 
> much better than in 12EDO.

The G, D, and A major triads are significantly better than what 
you'll hear in most other circulating 12-tone temperaments.

> Perhaps all the beat frequencies for all my problem 
> triads lie in a range that I react badly to.  Anyway, 
> something about them sets my teeth on edge!

Yes, that's understandable.

You'll have to try out the other temperaments in my suite and let me 
know what you think.  All of the others will have less error in the 
worst triads, and one of them is a well-temperament that has 
proportional beating in *all 24* major & minor triads (a feat to 
behold!).

Best,

--George

From: Yahya Abdal-Aziz (2005-10-19)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Hi Paul,

On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote: 
> 
> --- In [email protected], "Yahya Abdal-Aziz" 
> <yahya@m...> wrote:
> 
> > I can't really explain why the E major triad on E4 sounds
> > so wrong to me.  Is it the minor third of ~297 cents?
> 
> I think that may very well be your explanation.
> 
> > Perhaps all the beat frequencies for all my problem 
> > triads lie in a range that I react badly to.
> 
> How did you implement this tuning, exactly?

Exactly:  I used the Keyboard Scale, + and - buttons of the 
Roland E-28 keyboard.  Holding the Keyboard Scale button 
down for two seconds changes the tuning mode to one in 
which you can adjust the tuning of each note name (octave
equivalent pitch class) independently by from -64 to +63
cents, in steps of 1 cent, away from 12-EDO.

It's also possible to toggle between the altered tuning and 
the standard 12-EDO tuning by tapping the Keyboard Scale
button.

>... Did you check the tuning 
> independently after you set it?

No.  I have no specialised tools for doing so.  But my 
impression is that the frequency change per tap of the + 
and - buttons is [log] linear.

Regards,
Yahya


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From: klaus schmirler (2005-10-19)
Subject: Re: [tuning] Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Yahya Abdal-Aziz wrote:

> Exactly:  I used the Keyboard Scale, + and - buttons of the 
> Roland E-28 keyboard.  Holding the Keyboard Scale button 
> down for two seconds changes the tuning mode to one in 
> which you can adjust the tuning of each note name (octave
> equivalent pitch class) independently by from -64 to +63
> cents, in steps of 1 cent, away from 12-EDO.
> 
> It's also possible to toggle between the altered tuning and 
> the standard 12-EDO tuning by tapping the Keyboard Scale
> button.
> 
> 
>>... Did you check the tuning 
>>independently after you set it?
> 
> 
> No.  I have no specialised tools for doing so.  But my 
> impression is that the frequency change per tap of the + 
> and - buttons is [log] linear.

I'd be suspicious when the settings are from -64 to 63 - I have a 
cheap old Kawai keyboard with these settings, and it doesn't work that 
way. My C+50 was way higher than the C#-50. Nor did I find any logic 
in the step sizes except that they seemed to increase towards the edges.

klaus

From: wallyesterpaulrus (2005-10-20)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> 
> Hi Paul,
> 
> On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote: 
> > 
> > --- In [email protected], "Yahya Abdal-Aziz" 
> > <yahya@m...> wrote:
> > 
> > > I can't really explain why the E major triad on E4 sounds
> > > so wrong to me.  Is it the minor third of ~297 cents?
> > 
> > I think that may very well be your explanation.
> > 
> > > Perhaps all the beat frequencies for all my problem 
> > > triads lie in a range that I react badly to.
> > 
> > How did you implement this tuning, exactly?
> 
> Exactly:  I used the Keyboard Scale, + and - buttons of the 
> Roland E-28 keyboard.  Holding the Keyboard Scale button 
> down for two seconds changes the tuning mode to one in 
> which you can adjust the tuning of each note name (octave
> equivalent pitch class) independently by from -64 to +63
> cents, in steps of 1 cent, away from 12-EDO.
> 
> It's also possible to toggle between the altered tuning and 
> the standard 12-EDO tuning by tapping the Keyboard Scale
> button.
> 
> >... Did you check the tuning 
> > independently after you set it?
> 
> No.  I have no specialised tools for doing so.  But my 
> impression is that the frequency change per tap of the + 
> and - buttons is [log] linear.

Be suspicious. Monz did a test where he found that, for at least one 
synth model, the increments were not at all equal (in his case, some 
were one cent and some were two cents), even though nominally, if you 
trusted the manufacturer, they should have been. The same is most 
definitely true of another synth, my Ensoniq. Synth manufacturers 
have had very little market pressure to implement accurate 
microtuning offsets. So you should *always* do an independent check 
when you care about 1 or 2 cents accuracy.

From: Yahya Abdal-Aziz (2005-10-21)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

On Wed, 19 Oct 2005, klaus schmirler wrote: 
> 
> Yahya Abdal-Aziz wrote:
> 
> > Exactly:  I used the Keyboard Scale, + and - buttons of the 
> > Roland E-28 keyboard.  Holding the Keyboard Scale button 
> > down for two seconds changes the tuning mode to one in 
> > which you can adjust the tuning of each note name (octave
> > equivalent pitch class) independently by from -64 to +63
> > cents, in steps of 1 cent, away from 12-EDO.
> > 
> > It's also possible to toggle between the altered tuning and 
> > the standard 12-EDO tuning by tapping the Keyboard Scale
> > button.
> > 
> > 
> >>... Did you check the tuning 
> >>independently after you set it?
> > 
> > 
> > No.  I have no specialised tools for doing so.  But my 
> > impression is that the frequency change per tap of the + 
> > and - buttons is [log] linear.
> 
> I'd be suspicious when the settings are from -64 to 63 - I have a 
> cheap old Kawai keyboard with these settings, and it doesn't work that 
> way. My C+50 was way higher than the C#-50. Nor did I find any logic 
> in the step sizes except that they seemed to increase towards the edges.

Klaus,

I've just checked.  My C+50 and C#-50 are identical.

Regards,
Yahya

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From: Yahya Abdal-Aziz (2005-10-21)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

On Thu, 20 Oct 2005, "wallyesterpaulrus" wrote: 
> 
> --- In [email protected], "Yahya Abdal-Aziz" 
> <yahya@m...> wrote:
> >
> > Hi Paul,
> > 
> > On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote: 
> > > 
> > > --- In [email protected], "Yahya Abdal-Aziz" 
> > > <yahya@m...> wrote:
> > > 
> > > > I can't really explain why the E major triad on E4 sounds
> > > > so wrong to me.  Is it the minor third of ~297 cents?
> > > 
> > > I think that may very well be your explanation.
> > > 
> > > > Perhaps all the beat frequencies for all my problem 
> > > > triads lie in a range that I react badly to.
> > > 
> > > How did you implement this tuning, exactly?
> > 
> > Exactly:  I used the Keyboard Scale, + and - buttons of the 
> > Roland E-28 keyboard.  Holding the Keyboard Scale button 
> > down for two seconds changes the tuning mode to one in 
> > which you can adjust the tuning of each note name (octave
> > equivalent pitch class) independently by from -64 to +63
> > cents, in steps of 1 cent, away from 12-EDO.
> > 
> > It's also possible to toggle between the altered tuning and 
> > the standard 12-EDO tuning by tapping the Keyboard Scale
> > button.
> > 
> > >... Did you check the tuning 
> > > independently after you set it?
> > 
> > No.  I have no specialised tools for doing so.  But my 
> > impression is that the frequency change per tap of the + 
> > and - buttons is [log] linear.
> 
> Be suspicious. Monz did a test where he found that, for at least one 
> synth model, the increments were not at all equal (in his case, some 
> were one cent and some were two cents), even though nominally, if you 
> trusted the manufacturer, they should have been. The same is most 
> definitely true of another synth, my Ensoniq. Synth manufacturers 
> have had very little market pressure to implement accurate 
> microtuning offsets. So you should *always* do an independent check 
> when you care about 1 or 2 cents accuracy.
 
Paul,

I've answered (elsewhere) the suggestion that the C +50c might not
equal C# -50c on some keyboards.  I've demonstrated clearly on my
Roland E-28 that the two are identical.

To be doubly sure, I've just now checked that C +63c equals C# -37c.

So I'm content to follow my ear on this one.

Regards,
Yahya


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From: Yahya Abdal-Aziz (2005-10-21)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Hi George,

On Mon, 17 Oct 2005, "George D. Secor" wrote:
>
> --- In [email protected], "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> >
> > Hi George,
> >
> > On Mon, 29 Aug 2005, George D. Secor wrote:
> > >
> > > HEY EVERYONE!
> > >
> > > Having been unable to ignore the discussion about equal-beating
> > > temperaments lately, I have some startling news about
> > > a "temperament (extra)ordinaire!"
 ... [ much snipt]

> A couple of months have passed since I posted that, and I'm happy to
> say that I've not only improved it (proportional beating in the
> *minor* as well as the major triads), but after extensive
> experimentation involving dozens of tuning attempts, I've selected
> the five best tunings to be members of an entire suite of
> proportional-beating well-temperaments (WT's) and _temperaments
> ordinaires_ (TO's) -- details to be given at a later date (once I'm
> sure that I've settled on my final selection).

I look forward to reading the details.


> > > Anyway, I hope that some others will try this tuning and give me
> > > your reactions -- at least go through the triads in Scala using
> > > the chord playing function in the keyboard clavier.  Convince me
> > > that this isn't all a dream!
> >
> > I expect you've had plenty of feedback by now?
>
> Yes, but more is welcome.
>
> > Anyway, FWIW, here are my reactions to tuning up your
> > "extraordinary" temperament, then playing with it for
> > an hour or so using both piano and harpsichord samples
> > on my Roland keyboard.
> >
> > Everything is fine - EXCEPT the sounds of the bare Db,
> > Eb, E, F#, Ab and B major triads in root position with
> > root in the octave above middle C.  Other inversions of
> > these triads are (more or less) acceptable, while more
> > open chords using these harmonies are fine.  Playing an
> > accompanied melody, or improvised two- and three-part
> > counterpoint, all sounds pleasant enough to me.  The only
> > problem I have is with these triads in that range.
>
> They all have total absolute error greater than the triads of 12-ET,
> and the problem is mostly with the thirds.  You'll need to understand
> that the purpose of a _temperament ordinaire_ is to have high
> (meantone-like) consonance in as many triads as possible after taming
> the "wolf".  In my temperament (extra)ordinaire wolf fifths are
> completely eliminated.  The 3rds will of necessity be worse than in
> 12-ET, but the objective is to have as few "worst" triads as
> possible.

Thanks for that explanation; until now, I had no idea what
the goals of a "temp\ufffdrament ordinaire" were.


>  ... I'm a bit surprised that you had problems with E and
> especially Eb -- I would judge them to be not significantly worse
> than 12-ET.  (However, see my comment about becoming "spoiled",
> below.)

I can rule this out;  I toggled between the two tunings several times,
playing identical passages in each.


> > I checked out the actual sizes of the fifths and thirds
> > in all major chords.  The fifths are not a problem - by
> > construction, almost.  (I don't really like fifths as
> > narrow as 697 cents, but that's the price we pay for
> > being able to play in all keys, I guess.)
>
> No, narrow fifths are the price we pay for being able to get the
> thirds closer to just in the best triads.

Hmmm.  So the nature of the compromise in these tunings
is that (unlike any EDO) some keys MUST be worse than
others.


> > The major thirds
> > in these chords, without exception, are a little wide to
> > very wide, and I think this explains the worst of my
> > difficulty with them.
>
> Yes.


> > Still, the E-G# is only 400.26122,
> > hardly different from the 12EDO third of 400 cents.
> > I can't really explain why the E major triad on E4 sounds
> > so wrong to me.  Is it the minor third of ~297 cents?
>
> Probably.  (Those who think that the beating of minor 3rds aren't
> nearly as important as beating major 3rds should take note.)


> Another possibility is that once you've gotten "spoiled" hearing how
> nice the triads from Bb thru A sound, you're jarred by the relative
> dissonance of triads only slightly worse than 12-ET.  In my latest
> experimentation with WT's and TO's I've listened and relistened (and
> compared and recompared) tunings with one another and have been
> occasionally surprised to find that I may experience different
> reactions at different times -- sometimes only a matter of 10 or 15
> minutes apart.

I noticed the same thing, but over much shorter time intervals, mostly
when I was contrasting the effect of the two tunings (your extra-TO
and 12-EDO) by playing a short phrase of perhaps two bars, first in
one and then the other, then repeating both - in the pattern 1 2 1 2.


> ...  At one time a Pythagorean triad may sound only
> marginally acceptable and at another time a triad with even more
> total absolute error than that (such as the worst ones in the
> temperament extraordinaire) will sound more acceptable.  Likewise,
> sometimes a low-contrast modern well-temperament (with a relatively
> small difference in intonation between the best and worst triads)
> will sound as if all the triads are very close to 12-ET (and
> therefore probably not worth the effort), whereas at other times the
> difference between the best and worst will seem quite effective.
>
> There seems to be some sort of "hearing fatigue" in operation here,

Perhaps my even shorter-term "hearing fatigue" arose from the
fact I have Chronic Fatigue Syndrome ... ? :-)  Yes, I thought much
the same thing, so I took a rest break and came back to it.  The first
time I repeated the fourfold pattern 1 2 1 2 as above, I still heard
the representations of the first tuning differently.  So the effect I
heard probably had more to do with the different contrasts, as
follows:

  Tuning          1      2      1      2
  Contrast     0-1   1-2  2-1  1-2

All of which makes me even more certain that I will never listen
to an adaptive tuning without a distinct feeling of nausea!  (Slight
OT: Is barbershop tuning necessarily adaptive?  It certainly sounds
so to me.  And yes, although I do love the sweet harmonies I hear,
that style of singing usually does leave me feeling queasy.)


> ... and it has made me look at Aaron Johnson's "Killing the Buddha of
> well-temperament" message
> http://groups.yahoo.com/group/tuning/message/60103
> with a new perspective.

I'll look it up.


> > It
> > can't be the fifth - the D, G and A major chords, which
> > have the same size fifth, are all quite good; tho the D
> > chord seems noticeably flat,
>
> I assume you're referring primarily to the 3rd as "flat" -- that's
> what make the triad more consonant.

Yes, but the whole chord is noticeably "squashed" when compared
to the 12-EDO.


> > > the A sounds a little better
> > than in 12EDO and the G major chord sounds superb -
> > much better than in 12EDO.
>
> The G, D, and A major triads are significantly better than what
> you'll hear in most other circulating 12-tone temperaments.
>
> > Perhaps all the beat frequencies for all my problem
> > triads lie in a range that I react badly to.  Anyway,
> > something about them sets my teeth on edge!
>
> Yes, that's understandable.
>
> You'll have to try out the other temperaments in my suite and let me
> know what you think.  All of the others will have less error in the
> worst triads, and one of them is a well-temperament that has
> proportional beating in *all 24* major & minor triads (a feat to
> behold!).

I await with breath abated ... :-)

>
> Best,
> --George

Likewise,
Yahya


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From: monz (2005-10-21)
Subject: harware tuning resolution (was: Temperament (Extra)ordinare!)

Hi Yahya and Paul,


--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
> >
> > > ... Did you check the tuning independently after you set it?
> > 
> > No.  I have no specialised tools for doing so.  But my 
> > impression is that the frequency change per tap of the + 
> > and - buttons is [log] linear.
> 
> Be suspicious. Monz did a test where he found that, for
> at least one synth model, the increments were not at all
> equal (in his case, some were one cent and some were two
> cents), even though nominally, if you trusted the
> manufacturer, they should have been. The same is most 
> definitely true of another synth, my Ensoniq. Synth
> manufacturers have had very little market pressure to
> implement accurate microtuning offsets. So you should
> *always* do an independent check when you care about
> 1 or 2 cents accuracy.


Those tests were done on my previous and current computer 
soundcards. Their tuning resolutions, as usual for MIDI 
hardware, are quantized to 6 bits, which gives 64 divisions
per semitone and thus 768 per octave. 

*But*, while my old soundcard did give 768-edo resolution,
my current one doesn't. The current one is quantized to a
768-tone subset of 1200-edo, so that each tuning "zone"
encompasses either 1 or 2 cents. Obviously, this makes it
fiendishly difficult to determine the precise tuning of
any MIDI i do on my computer.


Here are the old posts:

http://launch.groups.yahoo.com/group/tuning/message/45522

http://launch.groups.yahoo.com/group/tuning/message/45533



Thankfully, Tonescape (which is now the only software i
use to compose music) has now entered the phase of 
development where we are able to create mp3's without
using MIDI, and our tuning resolution uses floating point
cents values with so many decimal places that i can't
remember how many there are. If we get 8 decimal places,
then it's (1.2 * 10^11)-edo, which is precise enough
for me. ;-)

The question which still remains for me is: is the
tuning resolution i discovered for my soundcard relevant
only to MIDI, or to any sound it produces? I did the
test using MIDI software, so i have no idea what resolution
the soundcard gives for non-MIDI audio.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: George D. Secor (2005-10-21)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
> 
> Hi George,
> 
> On Mon, 17 Oct 2005, "George D. Secor" wrote:
> >...
> > You'll have to try out the other temperaments in my suite and let 
me
> > know what you think.  All of the others will have less error in 
the
> > worst triads, and one of them is a well-temperament that has
> > proportional beating in *all 24* major & minor triads (a feat to
> > behold!).
> 
> I await with breath abated ... :-)

Please don't hold it for more than a few seconds at a time, because 
I'm not able to reply at length right now.  ;-)

When I do (hopefully by the end of next week), I think I should make 
a separate file describing the 5 temperaments in the suite (including 
the purpose of each) and include a link to it so it will be easier to 
locate later.

Best,

--George

From: Aaron Krister Johnson (2005-10-21)
Subject: the simplest pan-proportionally beating 12-tone temperament

it seems that all this exploration into equal-beating well temperaments has 
given me a great question:

what is the simplest possible 12-note temperament where all 24 major and minor 
triads have rationally proportional beating? here 'simplest' means that the 
brats (beat ratios for the un-initiated) are the lowest numbers in the 
numerator and denominator that they can be.....

anyone? this is ripe for george or gene to explore.

-aaron.

From: Gene Ward Smith (2005-10-21)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], Aaron Krister Johnson <aaron@a...> wrote:
>
> 
> it seems that all this exploration into equal-beating well
temperaments has 
> given me a great question:
> 
> what is the simplest possible 12-note temperament where all 24 major
and minor 
> triads have rationally proportional beating?

If you are asking for a tuning where the brats are all rational, any
tuning with rational values, producing rational values of fifths and
thirds, will do.

From: Carl Lumma (2005-10-21)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > it seems that all this exploration into equal-beating well
> temperaments has 
> > given me a great question:
> > 
> > what is the simplest possible 12-note temperament where all
> > 24 major and minor 
> > triads have rationally proportional beating?
> 
> If you are asking for a tuning where the brats are all rational,
> any tuning with rational values, producing rational values of
> fifths and thirds, will do.

Bob Wendell's Natural Synchronous Well claimed to be this tuning.
Brats of 2 or 1.5 or all 24 triads (except a few of the values are
only very close).  Bob made great claims for this scale, and it
apparently got the attention of some piano tuners, and I've
tried it on my MIDI keyboard with Scala.  I wasn't immediately
bowled over.

Gene seemed to agree that this was the simplest tuning in this
sense at the time, but more recently he seemed to weaken those
claims...

-Carl

From: Gene Ward Smith (2005-10-21)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> Gene seemed to agree that this was the simplest tuning in this
> sense at the time, but more recently he seemed to weaken those
> claims...

Bob's scale does not precisely fit Aaron's requirements; I first need
to get clear if all the beat ratios actually need to be exact. The 12
note Pythagorean scale, for instance, has all brats of exactly 3/2,
except for the triad with the wolf fifth, which is rational but hardly
simple.

From: Gene Ward Smith (2005-10-21)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:

> Bob's scale does not precisely fit Aaron's requirements; I first need
> to get clear if all the beat ratios actually need to be exact. The 12
> note Pythagorean scale, for instance, has all brats of exactly 3/2,
> except for the triad with the wolf fifth, which is rational but hardly
> simple.

From out of the Scala archives, here is my rational number version of
Wendell Well. It has all-rational beat ratios. Ten of the major brats
are simple, being either 3/2 or 2. The other two are close to 2.

! wendell1r.scl
!
Rational version of wendell1.scl by Gene Ward Smith
 12
!
 24000/22729
 76460/68187
 27000/22729
 85600/68187
 4/3
 32100/22729
 102140/68187
 36000/22729
 114400/68187
 40500/22729
 42800/22729
 2/1

From: Yahya Abdal-Aziz (2005-10-23)
Subject: RE:  harware tuning resolution (was: Temperament (Extra)ordinare!)

Hi Monz,

On Fri, 21 Oct 2005, "monz" wrote:
> 
> Hi Yahya and Paul,
> 
> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Yahya Abdal-Aziz" 
> > <yahya@m...> wrote:
> > >
> > > > ... Did you check the tuning independently after you set it?
> > > 
> > > No.  I have no specialised tools for doing so.  But my 
> > > impression is that the frequency change per tap of the + 
> > > and - buttons is [log] linear.
> > 
> > Be suspicious. Monz did a test where he found that, for
> > at least one synth model, the increments were not at all
> > equal (in his case, some were one cent and some were two
> > cents), even though nominally, if you trusted the
> > manufacturer, they should have been. The same is most 
> > definitely true of another synth, my Ensoniq. Synth
> > manufacturers have had very little market pressure to
> > implement accurate microtuning offsets. So you should
> > *always* do an independent check when you care about
> > 1 or 2 cents accuracy.
> 
> 
> Those tests were done on my previous and current computer 
> soundcards. Their tuning resolutions, as usual for MIDI 
> hardware, are quantized to 6 bits, which gives 64 divisions
> per semitone and thus 768 per octave. 
> 
> *But*, while my old soundcard did give 768-edo resolution,
> my current one doesn't. The current one is quantized to a
> 768-tone subset of 1200-edo, so that each tuning "zone"
> encompasses either 1 or 2 cents. Obviously, this makes it
> fiendishly difficult to determine the precise tuning of
> any MIDI i do on my computer.
> 
> Here are the old posts:
> 
> http://launch.groups.yahoo.com/group/tuning/message/45522
> 
> http://launch.groups.yahoo.com/group/tuning/message/45533

I'm having no luck retrieving either message; 
I tried yesterday and have retried today.

Would you please briefly describe your test method?


> Thankfully, Tonescape (which is now the only software i
> use to compose music) has now entered the phase of 
> development where we are able to create mp3's without
> using MIDI, and our tuning resolution uses floating point
> cents values with so many decimal places that i can't
> remember how many there are. If we get 8 decimal places,
> then it's (1.2 * 10^11)-edo, which is precise enough
> for me. ;-)

A resolution that probably exceeds the capabilities of any
(sentient) creature on Earth!

 
> The question which still remains for me is: is the
> tuning resolution i discovered for my soundcard relevant
> only to MIDI, or to any sound it produces? I did the
> test using MIDI software, so i have no idea what resolution
> the soundcard gives for non-MIDI audio.

Sorry, I can't help you with an answer to this.

The only thoughts I have on the matter are, that since 
the soundcard is a digital device, it necessarily DOES 
have a fixed resolution that will depend on the chip and
logic design.  Thus, the manufacturer would probably be
best able to advise you on the resolution they intended
the card to have.

Regards,
Yahya

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From: Yahya Abdal-Aziz (2005-10-23)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

George,

On Fri, 21 Oct 2005 "George D. Secor" wrote: 
> 
> --- In [email protected], "Yahya Abdal-Aziz" 
> <yahya@m...> wrote:
> > 
> > Hi George,
> > 
> > On Mon, 17 Oct 2005, "George D. Secor" wrote:
> > >...
> > > You'll have to try out the other temperaments in my suite and let 
> > > me
> > > know what you think.  All of the others will have less error in 
> > > the
> > > worst triads, and one of them is a well-temperament that has
> > > proportional beating in *all 24* major & minor triads (a feat to
> > > behold!).
> > 
> > I await with breath abated ... :-)
> 
> Please don't hold it for more than a few seconds at a time, because 
> I'm not able to reply at length right now.  ;-)

Just when I was "Blue turning gray" ...


> When I do (hopefully by the end of next week), I think I should make 
> a separate file describing the 5 temperaments in the suite (including 
> the purpose of each) and include a link to it so it will be easier to 
> locate later.
> 
> Best,
> 
> --George

Looking forward to it.

Regards,
Yahya


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From: monz (2005-10-23)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

Hi Yahya,


--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:

> On Fri, 21 Oct 2005, "monz" wrote:


> > Here are the old posts:
> > 
> > http://launch.groups.yahoo.com/group/tuning/message/45522
> > 
> > http://launch.groups.yahoo.com/group/tuning/message/45533
> 
> I'm having no luck retrieving either message; 
> I tried yesterday and have retried today.


They are there. Just keep trying.

 
> Would you please briefly describe your test method?


Very briefly ... 

Using Cakewalk 9.03, i probably used the "ocarina" 
general-MIDI sound because it's the closest thing MIDI
gives to a sine wave. I set two identical tracks to
a long sustained note at MIDI-note 69 (A-440 Hz).

Then, while keeping the pitch of Track 1 constant,
and using the finest pitch-bend resolution in Cakewalk -- 
the 12mu (dodekamu) level, 4096 steps per semitone
-- i incremented the pitch-bend values of Track 2 until
i heard the notes go out of phase, indicating that
the pitch of Track 2 had changed minutely.


On my old soundcard, the pitch change occurred each
time i incremented the 12mu values by 64, which indicates
that the soundcard was giving 6mu (hexamu) resolution,
that is, 4096/64 = 64 steps per semitone, which equals 
768-edo.

But on my new soundcard, the pitch change occurred at 
unequally-spaced intervals which incremented by either
40 or 41 (12mu) steps, or 80 or 81 steps, which means
that each pitch-change "zone" is quantized to either
1 or 2 cents:

4096/40 or 4096/41 = ~1 cent
4096/80 or 4096/81 = ~2 cents

So in effect, what i get is a 768-tone subset of 1200-edo.
 

> The only thoughts I have on the matter are, that since 
> the soundcard is a digital device, it necessarily DOES 
> have a fixed resolution that will depend on the chip and
> logic design.  Thus, the manufacturer would probably be
> best able to advise you on the resolution they intended
> the card to have.


Without bothering to get any more scientific about it
than i already have, i just think it's easiest and safest
to assume that digital audio hardware is not going to
give resolution better than 2 cents. 

So essentially, while it's interesting to do tuning theory
to any level of resolution you like, in practice, doing 
music on a computer, it's probably useless to even 
contemplate anything finer than 600-edo.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Jon Szanto (2005-10-23)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

Monz,

--- In [email protected], "monz" <monz@t...> wrote:
> Without bothering to get any more scientific about it
> than i already have, i just think it's easiest and safest
> to assume that digital audio hardware is not going to
> give resolution better than 2 cents. 
> 
> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing 
> music on a computer, it's probably useless to even 
> contemplate anything finer than 600-edo.

I think you need to really re-think a statement like the above. While
that may be true for some audio cards, and mainly if you are staying
within the limitations of midi (and pitch bends), which these days I
don't know why people would, it doesn't apply at all if you want to
utilize other digital sound/music tools. For instance, I think you can
get very accurate tunings using Csound, and while I don't know how
many people have done extremely careful testing, my suspicion is that
softsynths with built-in tuning capacity will get better pitch
accuracy than relying on pitch bends.

I bet a Csound guru would know for sure.

Cheers,
Jon

From: Gene Ward Smith (2005-10-23)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "monz" <monz@t...> wrote:

> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing 
> music on a computer, it's probably useless to even 
> contemplate anything finer than 600-edo.

That assumes music on a computer must use a soundcard.

From: Gene Ward Smith (2005-10-23)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "Jon Szanto" <jszanto@c...> wrote:

> I bet a Csound guru would know for sure.

Csound tuning has nothing to do with either pitch bends or soundcards.
The waveform is computed, and the tuning is given in frequency terms.

From: Jon Szanto (2005-10-24)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "Jon Szanto" <jszanto@c...> wrote:
> 
> > I bet a Csound guru would know for sure.
> 
> Csound tuning has nothing to do with either pitch bends or soundcards.
> The waveform is computed, and the tuning is given in frequency terms.

Ah, yes, that was my point. By rendering a file you aren't locked into
any hardware paradigm at all. I'm just not exactly sure of Csound's
tuning resolution, but I'd bet the farm it is a lot better than pitch
bending.

Of course, you do give up the fun of interacting with the music in a
live manner, but for some people that isn't an issue. BTW, Gene, are
there currently any really good midi file -> Csound .sco utilities?

Cheers,
Jon

From: Gene Ward Smith (2005-10-24)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "Jon Szanto" <jszanto@c...> wrote:

> Of course, you do give up the fun of interacting with the music in a
> live manner, but for some people that isn't an issue. BTW, Gene, are
> there currently any really good midi file -> Csound .sco utilities?

There's a utility which has been around for years, but it isn't much
use. What is needed is a good suite of GM instruments for Csound.

From: monz (2005-10-24)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

Hi Jon,


--- In [email protected], "Jon Szanto" <jszanto@c...> wrote:
>
> Monz,
> 
> --- In [email protected], "monz" <monz@t...> wrote:
> > Without bothering to get any more scientific about it
> > than i already have, i just think it's easiest and safest
> > to assume that digital audio hardware is not going to
> > give resolution better than 2 cents. 
> > 
> > So essentially, while it's interesting to do tuning theory
> > to any level of resolution you like, in practice, doing 
> > music on a computer, it's probably useless to even 
> > contemplate anything finer than 600-edo.
> 
> I think you need to really re-think a statement like the
> above. While that may be true for some audio cards, and
> mainly if you are staying within the limitations of midi
> (and pitch bends), which these days I don't know why
> people would, it doesn't apply at all if you want to
> utilize other digital sound/music tools. For instance,
> I think you can get very accurate tunings using Csound,
> and while I don't know how many people have done extremely
> careful testing, my suspicion is that softsynths with
> built-in tuning capacity will get better pitch accuracy
> than relying on pitch bends.
> 
> I bet a Csound guru would know for sure.


Agreed, and i'm glad you wrote this ... i'm not sure why
i didn't qualify what i wrote myself, because i had already
written in a previous post that this resolution applies
to MIDI, but that i have no idea of the resolution of any
soundcards when doing non-MIDI digital audio. In fact,
i'm immersed in a study of Csound myself right now. I'll
see if i can find that answer.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: monz (2005-10-24)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "Jon Szanto" <jszanto@c...> wrote:
> 
> > Of course, you do give up the fun of interacting with
> > the music in a live manner, but for some people that
> > isn't an issue. BTW, Gene, are there currently any really
> > good midi file -> Csound .sco utilities?
> 
> There's a utility which has been around for years, but it
> isn't much use. What is needed is a good suite of GM
> instruments for Csound.



And amazingly, after all these years, it seems that still
it doesn't exist. That's exactly what i'm deep into right
now: studying how Csound .orc files would do GM.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Yahya Abdal-Aziz (2005-10-24)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

Hi Monz,

On Sun, 23 Oct 2005, "monz" wrote: 
> Hi Yahya,
> 
> --- In [email protected], "Yahya Abdal-Aziz" 
> <yahya@m...> wrote:
> 
> > On Fri, 21 Oct 2005, "monz" wrote:
> > > Here are the old posts:
> > > 
> > > http://launch.groups.yahoo.com/group/tuning/message/45522
> > > 
> > > http://launch.groups.yahoo.com/group/tuning/message/45533
> > 
> > I'm having no luck retrieving either message; 
> > I tried yesterday and have retried today.
> 
> They are there. Just keep trying.

Success!


> > Would you please briefly describe your test method?
> 
> Very briefly ... 
> 
> Using Cakewalk 9.03, i probably used the "ocarina" 
> general-MIDI sound because it's the closest thing MIDI
> gives to a sine wave. I set two identical tracks to
> a long sustained note at MIDI-note 69 (A-440 Hz).
> 
> Then, while keeping the pitch of Track 1 constant,
> and using the finest pitch-bend resolution in Cakewalk -- 
> the 12mu (dodekamu) level, 4096 steps per semitone
> -- i incremented the pitch-bend values of Track 2 until
> i heard the notes go out of phase, indicating that
> the pitch of Track 2 had changed minutely.
> 
> On my old soundcard, the pitch change occurred each
> time i incremented the 12mu values by 64, which indicates
> that the soundcard was giving 6mu (hexamu) resolution,
> that is, 4096/64 = 64 steps per semitone, which equals 
> 768-edo.
> 
> But on my new soundcard, the pitch change occurred at 
> unequally-spaced intervals which incremented by either
> 40 or 41 (12mu) steps, or 80 or 81 steps, which means
> that each pitch-change "zone" is quantized to either
> 1 or 2 cents:
> 
> 4096/40 or 4096/41 = ~1 cent
> 4096/80 or 4096/81 = ~2 cents
> 
> So in effect, what i get is a 768-tone subset of 1200-edo.

Thank you, that is quite clear.  So to test the MIDI 
frequency resolution of our sound sources (card or external 
MIDI source), using whatever MIDI software we have to 
hand, we could also try to find the smallest MIDI Pitch Bend
 that audibly changes the simplest timbre.  Obviously, only 
software with fine PB resolution would be useful in such an 
exercise.


> > The only thoughts I have on the matter are, that since 
> > the soundcard is a digital device, it necessarily DOES 
> > have a fixed resolution that will depend on the chip and
> > logic design.  Thus, the manufacturer would probably be
> > best able to advise you on the resolution they intended
> > the card to have.
> 
> Without bothering to get any more scientific about it
> than i already have, i just think it's easiest and safest
> to assume that digital audio hardware is not going to
> give resolution better than 2 cents. 

I don't know about that ...  I suspect the actual limits
to the frequency resolution of digital audio hardware 
will be much better than that.  In practice, this will
probably only matter in music with slow sustained sounds
or very slow changes in pitch.  For "most" music, your
resolution of 2 cents is pretty much the limit of human 
pitch discrimination, isn't it?


> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing 
> music on a computer, it's probably useless to even 
> contemplate anything finer than 600-edo.

Even if we do slowly evolving sine textures in a La Monte
Young kind of way?


Regards,
Yahya

-- 
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

From: Jon Szanto (2005-10-24)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "monz" <monz@t...> wrote:
> And amazingly, after all these years, it seems that still
> it doesn't exist. That's exactly what i'm deep into right
> now: studying how Csound .orc files would do GM.

I don't think it is that amazing, really, as most Csounders don't seem
to be working with midi (or GM, for that matter). Why don't you post
an open question, and see if Dave Seidel or Prent Rodgers or Bill
Sethares  or ??? can help you?

Cheers,
Jon

P.S. I'm assuming you have "The Csound Book"...

From: Carl Lumma (2005-10-24)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

> Monz,
> 
> --- In [email protected], "monz" <monz@t...> wrote:
> > Without bothering to get any more scientific about it
> > than i already have, i just think it's easiest and safest
> > to assume that digital audio hardware is not going to
> > give resolution better than 2 cents. 
> > 
> > So essentially, while it's interesting to do tuning theory
> > to any level of resolution you like, in practice, doing 
> > music on a computer, it's probably useless to even 
> > contemplate anything finer than 600-edo.
> 
> I think you need to really re-think a statement like the above.

Indeed.  Today's softsynths write audio data directly to
audio devices, and thus can be as accurate as their authors
cared to make them.

-Carl

From: George D. Secor (2005-10-24)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "Gene Ward Smith" <gwsmith@s...> 
wrote:
> 
> > Bob's scale does not precisely fit Aaron's requirements; I first 
need
> > to get clear if all the beat ratios actually need to be exact. 
The 12
> > note Pythagorean scale, for instance, has all brats of exactly 
3/2,
> > except for the triad with the wolf fifth, which is rational but 
hardly
> > simple.
> 
> From out of the Scala archives, here is my rational number version 
of
> Wendell Well. It has all-rational beat ratios. Ten of the major 
brats
> are simple, being either 3/2 or 2. The other two are close to 2.
> 
> ! wendell1r.scl
> !
> Rational version of wendell1.scl by Gene Ward Smith
>  12
> !
>  24000/22729
>  76460/68187
>  27000/22729
>  85600/68187
>  4/3
>  32100/22729
>  102140/68187
>  36000/22729
>  114400/68187
>  40500/22729
>  42800/22729
>  2/1

Gene, I'm impressed!

But, OTOH, I hope you realize that there are reasonably simple brats 
in only 5 minor triads (the ones with just 5ths), and the rest are 
pretty much hit-or-miss (mostly the latter) -- much like what I got 
in my _temperament (extra)ordinaire_ a couple months back.

In attempting to construct synchronous temperaments (with irrational 
intervals) with both proportional *major and minor* triads, I found 
that, unless the 5ths are exact, the minor-triad brats will at best 
only approximate simple ratios when the majors are exact (or vice 
versa), or you can compromise and make both inexact by about half and 
thereby get most of the brats to within 0.02 of simple numbers -- 
close enough, I would say, considering that the required adjustment 
is very small. In the following example, each of the tempered fifths 
is modified by less than 0.002 cents (i.e., -3.9114 vs. -3.9132 cents 
for an exact 2/11-comma narrow fifth).

Here's a sneak preview of my 2/11-comma well-temperament (with beat 
synchrony in virtually all 24 major & minor triads).  Bear in mind 
that this is an honest-to-goodness well-temperament that meets *all* 
of Jorgensen's requirements, with key contrasts that are very similar 
to the Valotti-Young (1/6-pythagorean-comma) temperament:

! secor_WT2-11.scl
!
George Secor's 2/11-comma well-temperament, proportional beating
 12
!
 92.13884
 196.08721
 296.01562
 392.17442
 499.92562
 590.20835
 698.04361
 794.06062
 894.13082
 997.97062
 1090.21803
 2/1

As to how close it comes to perfection, judge for yourself:

Major -----Beat Ratios-----
Triad  M3/5th  m3/5th m3/M3
----- ------- ------- -----
Ab... ------- -------  1.50
Eb... ------- -------  1.50
Bb... ------- -------  1.50
F....    7.01   13.02  1.86
C....    2.50    6.26  2.50
G....    2.50    6.26  2.50
D....    3.34    7.51  2.25
A....    5.01   10.01  5.00
E....    6.67   12.51  1.87
B…...   16.63   27.45  1.65
F#... 1467.47 2203.71  1.50
C#... 1084.06 1628.58  1.50

Minor -----Beat Ratios-----
Triad  M3/5th  m3/5th m3/M3
----- ------- ------- -----
Ab... ------- -------  1.00
Eb... ------- -------  1.00
Bb... ------- -------  1.00
F....   20.74   22.74  1.10
C....    7.99    9.99  1.25
G....    6.01    8.01  1.33
D....    4.02    6.02  1.50
A....    2.99    4.99  1.67
E....    2.99    4.99  1.67
B…...    7.93    9.93  1.25
F#...  950.96  952.96  1.00
C#...  933.34  935.34  1.00

The F# and C# triads have 5ths that are nearly just, so you can 
consider those 3- and 4-digit numbers as approaching infinity.  As 
for B major, it's your call as to whether 1.65 is close enough to 5/3 
to qualify as proportional; or for F minor, whether 11/10 is simple 
enough.

Anyway, it turns out that I came up with at least two other 
synchonous temperaments (one well, the other _ordinaire_) that I like 
better than this one.  I'll share them once I've settled on the best 
way to categorize the whole collection.

--George

From: Gene Ward Smith (2005-10-25)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

> Gene, I'm impressed!
> 
> But, OTOH, I hope you realize that there are reasonably simple brats 
> in only 5 minor triads (the ones with just 5ths), and the rest are 
> pretty much hit-or-miss (mostly the latter) -- much like what I got 
> in my _temperament (extra)ordinaire_ a couple months back.

The beat ratios are Wendell's targets; what I did was precisely solve
for them. The brat for 12-et is 1.7133, so a mixture of 1.5 and 2.0
brats makes sense. The brats for 1.5 are 3/2, 0, infinity and their
inverses, which is nice. The 2 brats are not so good, being 2, -5,
-1/10 and their inverses. However, you need to go up to a brat of 4 to
get something simpler, giving 4, -1, -1/4, and you then need to
balance off the flatter fifths with either more pure fifths, or even
fifths with brats less than 3/2. Wendell liked the sound of these brat
2 triads, and they represent the easiest way to tackle the problem.

> In attempting to construct synchronous temperaments (with irrational 
> intervals) with both proportional *major and minor* triads, I found 
> that, unless the 5ths are exact, the minor-triad brats will at best 
> only approximate simple ratios when the majors are exact (or vice 
> versa), or you can compromise and make both inexact by about half and 
> thereby get most of the brats to within 0.02 of simple numbers -- 
> close enough, I would say, considering that the required adjustment 
> is very small.

That's correct. I tend to think making the major triad brats exact and
letting the minor triad brats be a little bit off makes the most sense.

> As to how close it comes to perfection, judge for yourself:
> 
> Major -----Beat Ratios-----
> Triad  M3/5th  m3/5th m3/M3
> ----- ------- ------- -----
> Ab... ------- -------  1.50
> Eb... ------- -------  1.50
> Bb... ------- -------  1.50
> F....    7.01   13.02  1.86
> C....    2.50    6.26  2.50
> G....    2.50    6.26  2.50
> D....    3.34    7.51  2.25
> A....    5.01   10.01  5.00
> E....    6.67   12.51  1.87
> B…...   16.63   27.45  1.65
> F#... 1467.47 2203.71  1.50
> C#... 1084.06 1628.58  1.50

You didn't like Wendell's brats of 2, but I can't see this is an
improvement. A 5/2 brat has a brat orbit of 5/2, -5/2, -4/25, which
doesn't seem as good as 2. The 5 brat has an orbit 5, -5/7, -7/25, and
I would urge you to dump it in favor of one or more brats of 4
instead. I don't know where the 1.86 and 1.85 come from, but a brat
of 13/7 has an orbit 13/7, -7, -1/13. 9/4 is another not very terrific
looking brat, with an orbit of 9/4, -10/3, -2/15. 

I recommend using as many of the "magic" brats as possible; load up on
3/2 and 4, and fill in a gap with 2 or 3.

From: wallyesterpaulrus (2005-10-25)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> 
> On Thu, 20 Oct 2005, "wallyesterpaulrus" wrote: 
> > 
> > --- In [email protected], "Yahya Abdal-Aziz" 
> > <yahya@m...> wrote:
> > >
> > > Hi Paul,
> > > 
> > > On Mon, 17 Oct 2005, "wallyesterpaulrus" wrote: 
> > > > 
> > > > --- In [email protected], "Yahya Abdal-Aziz" 
> > > > <yahya@m...> wrote:
> > > > 
> > > > > I can't really explain why the E major triad on E4 sounds
> > > > > so wrong to me.  Is it the minor third of ~297 cents?
> > > > 
> > > > I think that may very well be your explanation.
> > > > 
> > > > > Perhaps all the beat frequencies for all my problem 
> > > > > triads lie in a range that I react badly to.
> > > > 
> > > > How did you implement this tuning, exactly?
> > > 
> > > Exactly:  I used the Keyboard Scale, + and - buttons of the 
> > > Roland E-28 keyboard.  Holding the Keyboard Scale button 
> > > down for two seconds changes the tuning mode to one in 
> > > which you can adjust the tuning of each note name (octave
> > > equivalent pitch class) independently by from -64 to +63
> > > cents, in steps of 1 cent, away from 12-EDO.
> > > 
> > > It's also possible to toggle between the altered tuning and 
> > > the standard 12-EDO tuning by tapping the Keyboard Scale
> > > button.
> > > 
> > > >... Did you check the tuning 
> > > > independently after you set it?
> > > 
> > > No.  I have no specialised tools for doing so.  But my 
> > > impression is that the frequency change per tap of the + 
> > > and - buttons is [log] linear.
> > 
> > Be suspicious. Monz did a test where he found that, for at least 
one 
> > synth model, the increments were not at all equal (in his case, 
some 
> > were one cent and some were two cents), even though nominally, if 
you 
> > trusted the manufacturer, they should have been. The same is most 
> > definitely true of another synth, my Ensoniq. Synth manufacturers 
> > have had very little market pressure to implement accurate 
> > microtuning offsets. So you should *always* do an independent 
check 
> > when you care about 1 or 2 cents accuracy.
>  
> Paul,
> 
> I've answered (elsewhere) the suggestion that the C +50c might not
> equal C# -50c on some keyboards.  I've demonstrated clearly on my
> Roland E-28 that the two are identical.
> 
> To be doubly sure, I've just now checked that C +63c equals C# -37c.
> 
> So I'm content to follow my ear on this one.
> 
> Regards,
> Yahya

Neither of your checks does anything to ensure that the 100 steps per 
semitone are equal -- and as Monz found with another synth, they may 
very well be unequal.

From: Graham Breed (2005-10-25)
Subject: Re: [tuning] Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

Jon Szanto wrote:
> --- In [email protected], "monz" <monz@t...> wrote:
> 
>>And amazingly, after all these years, it seems that still
>>it doesn't exist. That's exactly what i'm deep into right
>>now: studying how Csound .orc files would do GM.
> 
> I don't think it is that amazing, really, as most Csounders don't seem
> to be working with midi (or GM, for that matter). Why don't you post
> an open question, and see if Dave Seidel or Prent Rodgers or Bill
> Sethares  or ??? can help you?

Enough Csounders are working with MIDI to get the MIDI support working 
and do a lot of work on optimizing the real-time support.  Besides, a 
standard instrument library would be useful even if you don't use MIDI. 
  I think the problem here arises from the academic nature of the 
project.   Programming a real-time synthesis engine is an interesting 
project.  Writing a user-friendly front-end is interesting. 
Implementing a novel form of synthesis is interesting.  But getting some 
boring, general purpose instruments for newbies isn't interesting, and 
so it gets neglected.

It's a worthy project, certainly.  I don't know how I could help with it 
now as I'm still a relative newcomer.  I have some legato insruments 
with custom timbres that work well, but they're not very generic.  If 
anybody has Sound Fonts that they like, you can try duplicating the 
envelopes and whatever else gets left out of Csound.  The trouble is, 
all you will have is some wrapped up Sound Fonts and so the results 
won't be much different from any other Sound Font based synthesizer. 
Except for the tuning, which makes it a niche application.

I looked at the Fluid opcodes.  They wrap up a MIDI-based synthesizer. 
So you get full Sound Font support, but you still have to address it 
with MIDI note numbers, not frequencies :(  Retuning it will be as 
difficult as any MIDI synth.

There's also a Moog emulator in there.


If you post an open question, you should probably attract Stephen Yi's 
attention as well.  He may have a way of abstracting instruments without 
tuning for Blue.  You should also check Blue's feature set before 
claiming Csound doesn't do something ;)


> Cheers,
> Jon
> 
> P.S. I'm assuming you have "The Csound Book"...

Not that useful here, because it's also academically oriented, and so 
about interesting projects rather than quickly adapting your MIDI 
skills.  Also a lot of it doesn't consider real-time use.

There's also a CD of instruments you can order, and some of them are 
good, but they're all instruments that somebody else cooked up for some 
reason; they're either academically interesting or specifically designed 
for some piece of music.  It also takes a heck of a long time to go 
through them all to see if one of them is what you want.


                       Graham

From: Jon Szanto (2005-10-25)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], Graham Breed <gbreed@g...> wrote:
> Enough Csounders are working with MIDI to get the MIDI support working 
> and do a lot of work on optimizing the real-time support.

I'm always willing to stand corrected; I guess it is just that
virtually all of the Csound "end results" I have heard weren't
anything related to MIDI, but more custom crafted scores and
freer-form experimental pieces (say, for instance, like Prent for the
former and Dave Seidel for the latter). I know of the midi support,
but it just didn't seem that many were working in that direction.


> If you post an open question, you should probably attract Stephen Yi's 
> attention as well.  He may have a way of abstracting instruments
without 
> tuning for Blue.  You should also check Blue's feature set before 
> claiming Csound doesn't do something ;)

Oh, Blue is veeery cool, one of the few things that *might* get me
into Csound some day. :)

Cheers,
Jon

From: wallyesterpaulrus (2005-10-25)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:

> All of which makes me even more certain that I will never listen
> to an adaptive tuning without a distinct feeling of nausea!

Have you listened to any of John deLaubenfels's adaptive tuning 
examples? How about ones that use the Vicentino approach? I found that 
retunings of 11 cents give me a queasy feeling, but these approaches 
typically, for pre-Beethoven music, don't exceed 6 cent retuning 
motions, and my tummy is not disturbed by those. And you?

From: wallyesterpaulrus (2005-10-25)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "monz" <monz@t...> wrote:

> The question which still remains for me is: is the
> tuning resolution i discovered for my soundcard relevant
> only to MIDI, or to any sound it produces? 

Only to MIDI -- it obviously wouldn't apply to a .wav file!

From: wallyesterpaulrus (2005-10-25)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > it seems that all this exploration into equal-beating well
> > temperaments has 
> > > given me a great question:
> > > 
> > > what is the simplest possible 12-note temperament where all
> > > 24 major and minor 
> > > triads have rationally proportional beating?
> > 
> > If you are asking for a tuning where the brats are all rational,
> > any tuning with rational values, producing rational values of
> > fifths and thirds, will do.
> 
> Bob Wendell's Natural Synchronous Well claimed to be this tuning.
> Brats of 2 or 1.5 or all 24 triads (except a few of the values are
> only very close).  Bob made great claims for this scale, and it
> apparently got the attention of some piano tuners, and I've
> tried it on my MIDI keyboard with Scala.  I wasn't immediately
> bowled over.

Few MIDI keyboards provide the necessary resolution to acheive aural 
synchrony between the beat rates.

From: Dave Seidel (2005-10-25)
Subject: Csound and MIDI (was: harware tuning resolution)

My use of MIDI with Csound has so far been limited to using hardware 
MIDI controllers to manipulate Csound in real time (i.e., for 
performance).  Csound does have plenty of other MIDI capabilities, but I 
haven't explored them yet myself.  I know it is capable of emitting MIDI 
events as well as consuming them.  I've also seen, but not tried, .orc 
files that take realtime input from a MIDI source and output a new 
Csound .orc file.

I'm not sure what Monz means by "doing GM".  If he means replicating the 
full set of instruments represented in GM, that seems like an enormous 
amount of work, not sure if it's worth the trouble.  OTOH, if he means 
producing GM-compatible MIDI messages, that's probably a piece of cake, 
relatively speaking.

One combination I'm interested in exploring (I understand others have 
already done stuff like this) is a combination of Csound and Keykit, 
where KeyKit streams MIDI to Csound for rendering.  With Csound's 
essentially infinite tuning capabilities, could be very powerful.

- Dave


Jon Szanto wrote:
> --- In [email protected], "monz" <monz@t...> wrote:
> 
>>And amazingly, after all these years, it seems that still
>>it doesn't exist. That's exactly what i'm deep into right
>>now: studying how Csound .orc files would do GM.
> 
> 
> I don't think it is that amazing, really, as most Csounders don't seem
> to be working with midi (or GM, for that matter). Why don't you post
> an open question, and see if Dave Seidel or Prent Rodgers or Bill
> Sethares  or ??? can help you?
> 
> Cheers,
> Jon
> 
> P.S. I'm assuming you have "The Csound Book"...

From: wallyesterpaulrus (2005-10-25)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "monz" <monz@t...> wrote:

> Without bothering to get any more scientific about it
> than i already have, i just think it's easiest and safest
> to assume that digital audio hardware is not going to
> give resolution better than 2 cents. 
> 
> So essentially, while it's interesting to do tuning theory
> to any level of resolution you like, in practice, doing 
> music on a computer, it's probably useless to even 
> contemplate anything finer than 600-edo.

Monz, you're equating computer music with MIDI or keyboard synths. Bad 
equation. I can create .wav files for you with as much resolution as 
mathematics (and a sampling rate of 44.1 KHz or whatever) will allow.

From: Carl Lumma (2005-10-25)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > > If you are asking for a tuning where the brats are all rational,
> > > any tuning with rational values, producing rational values of
> > > fifths and thirds, will do.
> > 
> > Bob Wendell's Natural Synchronous Well claimed to be this tuning.
> > Brats of 2 or 1.5 or all 24 triads (except a few of the values
> > are only very close).  Bob made great claims for this scale, and
> > it apparently got the attention of some piano tuners, and I've
> > tried it on my MIDI keyboard with Scala.  I wasn't immediately
> > bowled over.
> 
> Few MIDI keyboards provide the necessary resolution to acheive aural 
> synchrony between the beat rates.

The pitch bend method I used gives, in theory, 4096 steps per
semitone.  Many synths discard least significant bits of bend,
I'm told, and I haven't measure the performance of mine.  But
since it's a digital additive synth, there's no reason for it
not to be accurate.  I haven't tried to measure its accuracy,
but I suspect it's very accurate.

-Carl

From: Carl Lumma (2005-10-25)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

> I'm not sure what Monz means by "doing GM".  If he means
> replicating the full set of instruments represented in GM,

That's what we mean.

> that seems like an enormous amount of work, not sure if
> it's worth the trouble.

???

It's absolutely fundamental to have some sort of standard
orchestra.

How hard could it be?  Even if they were just bad wavetables,
you've got to have some form of standard orchestra.  What a
joke.

-Carl

From: George D. Secor (2005-10-25)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@y...> 
wrote:
> 
> > Gene, I'm impressed!
> > 
> > But, OTOH, I hope you realize that there are reasonably simple 
brats 
> > in only 5 minor triads (the ones with just 5ths), and the rest 
are 
> > pretty much hit-or-miss (mostly the latter) -- much like what I 
got 
> > in my _temperament (extra)ordinaire_ a couple months back.
> 
> The beat ratios are Wendell's targets; what I did was precisely 
solve
> for them. The brat for 12-et is 1.7133, so a mixture of 1.5 and 2.0
> brats makes sense. The brats for 1.5 are 3/2, 0, infinity and their
> inverses, which is nice. The 2 brats are not so good, being 2, -5,
> -1/10 and their inverses. However, you need to go up to a brat of 4 
to
> get something simpler, giving 4, -1, -1/4, and you then need to
> balance off the flatter fifths with either more pure fifths, or even
> fifths with brats less than 3/2. Wendell liked the sound of these 
brat
> 2 triads, and they represent the easiest way to tackle the problem.

I didn't take a really close look at his temperaments until after I 
did most of the work on mine, and I see that my approach is quite 
different.

> > In attempting to construct synchronous temperaments (with 
irrational 
> > intervals) with both proportional *major and minor* triads, I 
found 
> > that, unless the 5ths are exact, the minor-triad brats will at 
best 
> > only approximate simple ratios when the majors are exact (or vice 
> > versa), or you can compromise and make both inexact by about half 
and 
> > thereby get most of the brats to within 0.02 of simple numbers -- 
> > close enough, I would say, considering that the required 
adjustment 
> > is very small.
> 
> That's correct. I tend to think making the major triad brats exact 
and
> letting the minor triad brats be a little bit off makes the most 
sense.

I agree, and that's exactly what I did with all of the other 
synchronous temperaments I worked on over the past couple of months.  
With this one there was actually a completely different reason for 
making both the major and minor brat ratios approximate: if the tones 
in a chain from Ab thru G# are pitched so that that brat ratios for 
major triads Ab thru E are exact then, G# is 0.01661 cents lower in 
pitch than Ab; widening all of the tempered fifths (spanning F thru 
G#) by 1/9 of that miniscule amount makes them the same pitch.  
(Hmmm, I just noticed that this improves some of the minor 3rd brat 
ratios but worsens others, but fortunately it's only a slight change -
- oh, well!)

> > As to how close it comes to perfection, judge for yourself:
> > 
> > Major -----Beat Ratios-----
> > Triad  M3/5th  m3/5th m3/M3
> > ----- ------- ------- -----
> > Ab... ------- -------  1.50
> > Eb... ------- -------  1.50
> > Bb... ------- -------  1.50
> > F....    7.01   13.02  1.86
> > C....    2.50    6.26  2.50
> > G....    2.50    6.26  2.50
> > D....    3.34    7.51  2.25
> > A....    5.01   10.01  2.00 [corrected from 5.00]
> > E....    6.67   12.51  1.87
> > B…...   16.63   27.45  1.65
> > F#... 1467.47 2203.71  1.50
> > C#... 1084.06 1628.58  1.50
> 
> You didn't like Wendell's brats of 2, but I can't see this is an
> improvement. 

Oh, I liked it okay, because I used it in my A major triad (note that 
I had the wrong figure in the 3rd column of the above table, which 
I've now noted as corrected).

> A 5/2 brat has a brat orbit of 5/2, -5/2, -4/25, which
> doesn't seem as good as 2. The 5 brat has an orbit 5, -5/7, -7/25, 
and
> I would urge you to dump it in favor of one or more brats of 4
> instead. 

My primary objective was to keep the total absolute error of the 
major triads for F, C, and G under 20 cents and Bb and D under 24 
cents (which is to say, 5 triads significantly better than 12-ET; 
Wendell has only 3).  I therefore selected brat ratios that would not 
only accomplish this, but also create a smooth change in intonation 
as one moves about the circle of fifths (whereas Wendell has all the 
major triads from D to B slightly worse than 12-ET).

In other words, my priorities were different.  Synchronized beating 
is a very subtle effect, and it requires considerable tuning 
precision.  The error of the individual intervals from just 
intonation is much more obvious, so I have selected brats that will 
produce a decent well-temperament.

> I don't know where the 1.86 and 1.85 come from, but a brat
> of 13/7 has an orbit 13/7, -7, -1/13. 

The untweaked value for 1.86 is 1.85714285714..., or 13/7.

I didn't have 1.85 anywhere; did you mean 1.87?  Its untweaked value 
is 1.875.

> 9/4 is another not very terrific
> looking brat, with an orbit of 9/4, -10/3, -2/15. 
> 
> I recommend using as many of the "magic" brats as possible; load up 
on
> 3/2 and 4, and fill in a gap with 2 or 3.

Okay, I'll take another look at this. I just think that a lot of the 
minor 3rd brats won't be as simple as we would like.

--George

From: Jon Szanto (2005-10-25)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

--- In [email protected], Dave Seidel <dave@s...> wrote:
> One combination I'm interested in exploring (I understand others have 
> already done stuff like this) is a combination of Csound and Keykit, 
> where KeyKit streams MIDI to Csound for rendering.  With Csound's 
> essentially infinite tuning capabilities, could be very powerful.

Cool. I'm using KeyKit and routing it to VST instruments that already
are microtuneable. I haven't done much new coding in KeyKit, but it
seems like a good environment to be doing some basic gymnastics.
Please report if you get your implementation working (I'm guessing it
could be done with the CsoundVST?).

Cheers,
Jon

From: Jon Szanto (2005-10-25)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> It's absolutely fundamental to have some sort of standard orchestra.

Fundamental only if you want to make orchestral mockups. When I
suggested that velocity sensitivity was fundamental in a keyboard, you
pointed out a couple areas where it wasn't necessary; I'd say that
there are a few areas of music where an orchestra isn't necessary. :)

> How hard could it be?

Maybe not hard, but time consuming? The question then becomes, who
will sit down and do it?

> Even if they were just bad wavetables,
> you've got to have some form of standard orchestra.  What a
> joke.

When I hear another 'orchestral' piece with bad sounds, that is when
the comedy starts!

Cheers,
Jon

From: Dave Seidel (2005-10-25)
Subject: Re: [tuning] Re: Csound and MIDI (was: harware tuning resolution)

I understand what you're saying, but you need to keep in mind the 
history of Csound, and what it is (and has been) very often used for, 
which is instrument design from scratch and experimentation with new 
forms of synthesis.

Does Max/MSP supply a standard GM instrument library?  Does PD?  What 
makes you think Csound is or should be a general purpose synth?

Here's the first sentence from "What is Csound?" 
(http://www.csounds.com/whatis/index.html): "Csound is a programming 
language designed and optimized for sound rendering and signal processing."

- Dave


Carl Lumma wrote:
>>I'm not sure what Monz means by "doing GM".  If he means
>>replicating the full set of instruments represented in GM,
> 
> 
> That's what we mean.
> 
> 
>>that seems like an enormous amount of work, not sure if
>>it's worth the trouble.
> 
> 
> ???
> 
> It's absolutely fundamental to have some sort of standard
> orchestra.
> 
> How hard could it be?  Even if they were just bad wavetables,
> you've got to have some form of standard orchestra.  What a
> joke.
> 
> -Carl

From: Carl Lumma (2005-10-25)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

> > It's absolutely fundamental to have some sort of standard
> > orchestra.
> 
> Fundamental only if you want to make orchestral mockups.

I menat "orchestra" in a general sense here.  Note that GM
contains things that aren't traditional elements of an
orchestra, like funk bass.

> > Even if they were just bad wavetables,
> > you've got to have some form of standard orchestra.  What a
> > joke.
> 
> When I hear another 'orchestral' piece with bad sounds, that is
> when the comedy starts!

At least it would (presumably) have the tunability of csound.
Bad orchestral sounds are still great for mock-ups.

-Carl

From: Carl Lumma (2005-10-25)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

> I understand what you're saying, but you need to keep in mind the 
> history of Csound, and what it is (and has been) very often used
> for, which is instrument design from scratch and experimentation
> with new forms of synthesis.
> 
> Does Max/MSP supply a standard GM instrument library?  Does PD?
> What makes you think Csound is or should be a general purpose
> synth?

I guess I'm just shocked that nobody's done it.

-Carl

From: Dave Seidel (2005-10-25)
Subject: Re: [tuning] Re: Csound and MIDI (was: harware tuning resolution)

I don't do any VST, so I'm not sure yet how it will work.  I think that 
the command-line version of Csound has a way of receiving events on a 
pipe or something.  Let you know if and when I get there.  :-)

- Dave


Jon Szanto wrote:
> Cool. I'm using KeyKit and routing it to VST instruments that already
> are microtuneable. I haven't done much new coding in KeyKit, but it
> seems like a good environment to be doing some basic gymnastics.
> Please report if you get your implementation working (I'm guessing it
> could be done with the CsoundVST?).
> 
> Cheers,
> Jon

From: Gene Ward Smith (2005-10-25)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], Graham Breed <gbreed@g...> wrote:

>   I think the problem here arises from the academic nature of the 
> project.   Programming a real-time synthesis engine is an interesting 
> project.  Writing a user-friendly front-end is interesting. 
> Implementing a novel form of synthesis is interesting.  But getting
some 
> boring, general purpose instruments for newbies isn't interesting, and 
> so it gets neglected.

In other words, most Csound people are interested in making weird
sounds, not music. Music mostly requires normal sounds, which
apparently are boring. However, there *is* interest in emulating
acoustic instruments, and I find it surprising this hasn't led to a
suite of GM instruments by now.

From: Gene Ward Smith (2005-10-25)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

--- In [email protected], Dave Seidel <dave@s...> wrote:

> I'm not sure what Monz means by "doing GM".  If he means replicating
the 
> full set of instruments represented in GM, that seems like an enormous 
> amount of work, not sure if it's worth the trouble. 

It depends on whether Csound is supposed to be used for music or not,
I suppose.

From: Jon Szanto (2005-10-25)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
> In other words, most Csound people are interested in making weird
> sounds, not music. Music mostly requires normal sounds, which
> apparently are boring.

Another quote to add to my collection. Amazing.

Cheers,
Jon

From: Dave Seidel (2005-10-26)
Subject: Re: [tuning] Re: Csound and MIDI (was: harware tuning resolution)

Are you serious, Gene?  Are you really that close-minded, or are these 
gratuitous remarks just a grumpy reflex?

You're welcome to think what you wish of my music, of course, although I 
don't assume that you've heard any of it (if you're curious, which I 
doubt, go to http://mysterybear.net).  In point of fact, I use Csound 
when I want to make certain kinds of music, and I use MIDI-driven 
emulated instruments when I want to make other kinds of music. 
Different tools are suited for different jobs, and it's ridiculous to 
try to make a square peg fit in a round hole.

You might be (genuinely) interested in the music of Prent Rodgers 
(http://prodgers13.home.comcast.net/), who uses Csound to manipulate 
sampled instruments, with marvelous results.  No crappy emulation, either.

- Dave


Gene Ward Smith wrote:
 > In other words, most Csound people are interested in making weird
 > sounds, not music. Music mostly requires normal sounds, which
 > apparently are boring. However, there *is* interest in emulating
 > acoustic instruments, and I find it surprising this hasn't led to a
 > suite of GM instruments by now.


Gene Ward Smith wrote:
> It depends on whether Csound is supposed to be used for music or not,
> I suppose.

From: Yahya Abdal-Aziz (2005-10-26)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Hi Paul,

On Tue, 25 Oct 2005, "wallyesterpaulrus" wrote: 
[much snipt] 
> > I've answered (elsewhere) the suggestion that the C +50c might not
> > equal C# -50c on some keyboards.  I've demonstrated clearly on my
> > Roland E-28 that the two are identical.
> > 
> > To be doubly sure, I've just now checked that C +63c equals C# -37c.
> > So I'm content to follow my ear on this one.
> 
> Neither of your checks does anything to ensure that the 100 steps per 
> semitone are equal -- and as Monz found with another synth, they may 
> very well be unequal.

By doing a few more of these checks (say about 24 more), I could
at best show that the steps on the "down" side equal those on the 
"up", at least in the region of overlap; that is, that each of the notes
C+37c to C+63c match the notes C#-63c to C#-37c.

So you're right, of course; I could easily have alternating steps of
sizes, say 2/3 and 4/3 of a cent.  But I repeat my assertion that
the steps seem equal to me.  I have no measurements to back this 
up, just my ear.

Regards,
Yahya

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From: Yahya Abdal-Aziz (2005-10-26)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote: 
> 
> --- In [email protected], "Yahya Abdal-Aziz" 
> <yahya@m...> wrote:
> 
> > All of which makes me even more certain that I will never listen
> > to an adaptive tuning without a distinct feeling of nausea!
> 
> Have you listened to any of John deLaubenfels's adaptive tuning 
> examples? How about ones that use the Vicentino approach? I found that 
> retunings of 11 cents give me a queasy feeling, but these approaches 
> typically, for pre-Beethoven music, don't exceed 6 cent retuning 
> motions, and my tummy is not disturbed by those. And you?

Those delights yet await me.  I should do so.  Have you links?

Regards,
Yahya


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From: Gene Ward Smith (2005-10-26)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

--- In [email protected], Dave Seidel <dave@s...> wrote:
>
> Are you serious, Gene?  Are you really that close-minded, or are these 
> gratuitous remarks just a grumpy reflex?

I'm pretty much of a grumpy reflex guy, but the only real music I know
of produced with Csound is by Prent Rodgers, and he uses it in a
highly idiosyncratic way. Then again, I haven't heard *your* music,
which I will immediately listen to now.

However, it's clear historically that Csound started out as an MIT
engineering project, and it seems to me that judging from things scuh
as the Csound book and the Csound mailing list, sound engineering and
not music is the real interest of most of the people involved.

> You might be (genuinely) interested in the music of Prent Rodgers 
> (http://prodgers13.home.comcast.net/), who uses Csound to manipulate 
> sampled instruments, with marvelous results.  No crappy emulation,
either.

I love Prent's music. We've exchanged emails and CDs, and one thing is
clear--Prent is interested in Csound as a means of making *music*. I
don't see a lot of that in the Csound world. Or at least, that is my
grumpy reflex at the moment. It seems to me that if there was more
interest in music anong the Soundrs there would obviously be more
Csound instruments with musical qualities publically available.

From: Aaron Krister Johnson (2005-10-26)
Subject: Re: [tuning] Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

On Tuesday 25 October 2005 5:12 pm, Gene Ward Smith wrote:

> In other words, most Csound people are interested in making weird
> sounds, not music. Music mostly requires normal sounds, which
> apparently are boring. However, there *is* interest in emulating
> acoustic instruments, and I find it surprising this hasn't led to a
> suite of GM instruments by now.


should a the output moog modular synth not be considered 'music'?

-aaron.

From: monz (2005-10-26)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], Graham Breed <gbreed@g...> wrote:
> 
> > I think the problem here arises from the academic nature
> > of the  project.   Programming a real-time synthesis engine
> > is an interesting project.  Writing a user-friendly
> > front-end is interesting. Implementing a novel form of
> > synthesis is interesting.  But getting some boring, 
> > general purpose instruments for newbies isn't interesting,
> > and so it gets neglected.
> 
> In other words, most Csound people are interested in
> making weird sounds, not music. Music mostly requires
> normal sounds, which apparently are boring. However,
> there *is* interest in emulating acoustic instruments,
> and I find it surprising this hasn't led to a suite 
> of GM instruments by now.



Unfortunately, my perceptions agree exactly with what
both of you wrote here. I was amazed, after doing a *lot*
of searching, that there is no catalog of GM instruments
for Csound ... but you can easily find *thousands* of
weird "experiemental" instruments.

It's taken me several weeks to create a decent Csound 
nylon-string guitar instrument, which i *finally*
accomplished today to my satisfaction.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: monz (2005-10-26)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Hi Yahya,

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:

> On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote: 
> > 
> > --- In [email protected], "Yahya Abdal-Aziz" 
> > <yahya@m...> wrote:
> > 
> > > All of which makes me even more certain that I will never listen
> > > to an adaptive tuning without a distinct feeling of nausea!
> > 
> > Have you listened to any of John deLaubenfels's
> > adaptive tuning examples? How about ones that use the
> > Vicentino approach? I found that retunings of 11 cents
> > give me a queasy feeling, but these approaches typically,
> > for pre-Beethoven music, don't exceed 6 cent retuning 
> > motions, and my tummy is not disturbed by those. And you?
> 
> Those delights yet await me.  I should do so.  Have you links?


John deLaubenfels's music page is here:

http://personalpages.bellsouth.net/j/d/jdelaub/jstudio.htm


Note also that he has retuned a lot of my own 12-edo
stuff, and my MIDI files of pieces such as Schoenberg's
_Verklaerte Nacht_, into adaptive-tuning.


For Vicentino's adaptive-JI, i provide this:

http://tonalsoft.com/enc/a/adaptive-ji.aspx

If you click on the first graphic, it takes you to a
MIDI of a Lasso piece which i specifically chose because
Easley Blackwood uses it to demonstrate the unsuitability
of strict-JI for Renaissance music.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Yahya Abdal-Aziz (2005-10-26)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

On Tue, 25 Oct 2005, "Carl Lumma" wrote: 
> 
> > I'm not sure what Monz means by "doing GM".  If he means
> > replicating the full set of instruments represented in GM,
> 
> That's what we mean.
> 
> > that seems like an enormous amount of work, not sure if
> > it's worth the trouble.
> 
> ???
> 
> It's absolutely fundamental to have some sort of standard
> orchestra.
> 
> How hard could it be?  Even if they were just bad wavetables,
> you've got to have some form of standard orchestra.  What a
> joke.


Hi Carl,

That's only true, of course, if you're trying to make orchestral 
music.  Personally, I detest the sound of large numbers of
strings played together, and find very few uses for them in
my music.  I much prefer the texture of, say, a string quartet,
or other small ensemble.  

But perhaps you would regard a collection of the usual chamber 
music instruments as "some form of standard orchestra".

Regards,
Yahya

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From: Yahya Abdal-Aziz (2005-10-26)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

Hi George and Gene,

On Tue, 25 Oct 2005 "Gene Ward Smith" wrote: 
> 
> --- In [email protected], "George D. Secor" 
> <gdsecor@y...> wrote:
> 
> > Gene, I'm impressed!
> > 
> > But, OTOH, I hope you realize that there are reasonably simple brats 
> > in only 5 minor triads (the ones with just 5ths), and the rest are 
> > pretty much hit-or-miss (mostly the latter) -- much like what I got 
> > in my _temperament (extra)ordinaire_ a couple months back.
> 
> The beat ratios are Wendell's targets; what I did was precisely solve
> for them. The brat for 12-et is 1.7133, so a mixture of 1.5 and 2.0
> brats makes sense. The brats for 1.5 are 3/2, 0, infinity and their
> inverses, which is nice. The 2 brats are not so good, being 2, -5,
> -1/10 and their inverses. However, you need to go up to a brat of 4 to
> get something simpler, giving 4, -1, -1/4, and you then need to
> balance off the flatter fifths with either more pure fifths, or even
> fifths with brats less than 3/2. Wendell liked the sound of these brat
> 2 triads, and they represent the easiest way to tackle the problem.
> 
> > In attempting to construct synchronous temperaments (with irrational 
> > intervals) with both proportional *major and minor* triads, I found 
> > that, unless the 5ths are exact, the minor-triad brats will at best 
> > only approximate simple ratios when the majors are exact (or vice 
> > versa), or you can compromise and make both inexact by about half and 
> > thereby get most of the brats to within 0.02 of simple numbers -- 
> > close enough, I would say, considering that the required adjustment 
> > is very small.
> 
> That's correct. I tend to think making the major triad brats exact and
> letting the minor triad brats be a little bit off makes the most sense.
> 
> > As to how close it comes to perfection, judge for yourself:
> > 

[snipt]

I've tried to follow this thread, but without much success.
I see you both doing some maths to make the ratios of beats
("brats") proportional - but still don't understand why.  So
please tell me -

  What is the *musical* purpose of having proportional brats?

Regards,
Yahya


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From: Yahya Abdal-Aziz (2005-10-26)
Subject: Re: Temperament (Extra)ordinare! (Was: Further doubts ...)

Hi Monz,

On Wed, 26 Oct 2005 "monz" wrote: 
> 
> Hi Yahya,
> 
> --- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:
> 
> > On Tue, 25 Oct 2005 "wallyesterpaulrus" wrote: 
> > > 
> > > --- In [email protected], "Yahya Abdal-Aziz" 
> > > <yahya@m...> wrote:
> > > 
> > > > All of which makes me even more certain that I will never listen
> > > > to an adaptive tuning without a distinct feeling of nausea!
> > > 
> > > Have you listened to any of John deLaubenfels's
> > > adaptive tuning examples? How about ones that use the
> > > Vicentino approach? I found that retunings of 11 cents
> > > give me a queasy feeling, but these approaches typically,
> > > for pre-Beethoven music, don't exceed 6 cent retuning 
> > > motions, and my tummy is not disturbed by those. And you?
> > 
> > Those delights yet await me.  I should do so.  Have you links?
> 
> 
> John deLaubenfels's music page is here:
> 
> http://personalpages.bellsouth.net/j/d/jdelaub/jstudio.htm
> 
> 
> Note also that he has retuned a lot of my own 12-edo
> stuff, and my MIDI files of pieces such as Schoenberg's
> _Verklaerte Nacht_, into adaptive-tuning.
> 
> 
> For Vicentino's adaptive-JI, i provide this:
> 
> http://tonalsoft.com/enc/a/adaptive-ji.aspx
> 
> If you click on the first graphic, it takes you to a
> MIDI of a Lasso piece which i specifically chose because
> Easley Blackwood uses it to demonstrate the unsuitability
> of strict-JI for Renaissance music.

Thank you very much for these links!

Regards,
Yahya

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From: Yahya Abdal-Aziz (2005-10-26)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

Monz,

On Wed, 26 Oct 2005, "monz" wrote: 
> 
> [snip] ... I was amazed, after doing a *lot*
> of searching, that there is no catalog of GM instruments
> for Csound ... but you can easily find *thousands* of
> weird "experiemental" instruments.
> 
> It's taken me several weeks to create a decent Csound 
> nylon-string guitar instrument, which i *finally*
> accomplished today to my satisfaction.

Now *there's* an achievement!  Well done ...
Is this perhaps the first in a usable "orchestra" of
CSound emulations of traditional instruments?

Regards,
Yahya


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From: Aaron Krister Johnson (2005-10-26)
Subject: csound instruments

On Wednesday 26 October 2005 8:43 am, Yahya Abdal-Aziz wrote:
> Monz,
>
> On Wed, 26 Oct 2005, "monz" wrote:
> > [snip] ... I was amazed, after doing a *lot*
> > of searching, that there is no catalog of GM instruments
> > for Csound ... but you can easily find *thousands* of
> > weird "experiemental" instruments.
> >
> > It's taken me several weeks to create a decent Csound
> > nylon-string guitar instrument, which i *finally*
> > accomplished today to my satisfaction.
>
> Now *there's* an achievement!  Well done ...
> Is this perhaps the first in a usable "orchestra" of
> CSound emulations of traditional instruments?

it appears that Prent Rodgers uses samples of trad instruments to good effect 
in csound.

-aaron.

From: Jon Szanto (2005-10-26)
Subject: Re: csound instruments

--- In [email protected], Aaron Krister Johnson <aaron@a...> wrote:
> it appears that Prent Rodgers uses samples of trad instruments to
good effect 
> in csound.

Yes. Prent has done a lot of work on them over the years, so it
definitely wasn't "plug and play". I believe many of his initial
samples came from the McGill University sample collection. Lots of
people have worked with samples for years (obviously); for a GM set
for use in Csound, it would take someone some time and initiative to
get a set ready. There is a fairly complete set of orchestral
instrument samples available one could start with from the University
of Iowa:

http://theremin.music.uiowa.edu/MIS.html

Cheers,
Jon

From: Dave Seidel (2005-10-26)
Subject: Re: [tuning] Re: csound instruments

BTW, Csound includes SoundFont support as well.

- Dave


Jon Szanto wrote:
> --- In [email protected], Aaron Krister Johnson <aaron@a...> wrote:
> 
>>it appears that Prent Rodgers uses samples of trad instruments to
> 
> good effect 
> 
>>in csound.
> 
> 
> Yes. Prent has done a lot of work on them over the years, so it
> definitely wasn't "plug and play". I believe many of his initial
> samples came from the McGill University sample collection. Lots of
> people have worked with samples for years (obviously); for a GM set
> for use in Csound, it would take someone some time and initiative to
> get a set ready. There is a fairly complete set of orchestral
> instrument samples available one could start with from the University
> of Iowa:
> 
> http://theremin.music.uiowa.edu/MIS.html
> 
> Cheers,
> Jon
> 
> 
> 
> 
> 
> 
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
>   [email protected] - join the tuning group.
>   [email protected] - leave the group.
>   [email protected] - turn off mail from the group.
>   [email protected] - set group to send daily digests.
>   [email protected] - set group to send individual emails.
>   [email protected] - receive general help information.
>  
> Yahoo! Groups Links
> 
> 
> 
>  
> 
> 
>

From: Gene Ward Smith (2005-10-26)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], Aaron Krister Johnson <aaron@a...> wrote:

> should a the output moog modular synth not be considered 'music'?

It's exactly like the output of a cello--it would depend on how it was
composed.

From: Gene Ward Smith (2005-10-26)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "monz" <monz@t...> wrote:

> Unfortunately, my perceptions agree exactly with what
> both of you wrote here. I was amazed, after doing a *lot*
> of searching, that there is no catalog of GM instruments
> for Csound ... but you can easily find *thousands* of
> weird "experiemental" instruments.

I did warn you.

> It's taken me several weeks to create a decent Csound 
> nylon-string guitar instrument, which i *finally*
> accomplished today to my satisfaction.

What's Tonalsoft going to do with these? Anything independently of
Tonescape?

From: Gene Ward Smith (2005-10-26)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Yahya Abdal-Aziz" <yahya@m...> wrote:

>   What is the *musical* purpose of having proportional brats?

It's a very subtle effect which requires correct tuning, but Wendell
believes it amelorates the mistuning of triads, and in some cases I
think that is true. Unfortunately, the most interesting of these brats
is -1, what I've taken to calling in a more general context (higher
prime limits) "synch tuning", and that isn't well adapated to
circulating temperaments of twelve notes based on meantone. Of course,
Kraig Grady can tell you about Wilson and his Mt Maru scales, and
there's twelve notes of Wilson meantone ("wilwolf") etc to consider.

I think proportional beating is pretty interesting in the ill-tuned,
low complexity temperaments, and Herman has made a few experiments
along those lines.

From: Carl Lumma (2005-10-26)
Subject: Re: Csound and MIDI (was: harware tuning resolution)

> > It's absolutely fundamental to have some sort of standard
> > orchestra.
> > 
> > How hard could it be?  Even if they were just bad wavetables,
> > you've got to have some form of standard orchestra.  What a
> > joke.
> 
> Hi Carl,
> 
> That's only true, of course, if you're trying to make orchestral 
> music.  Personally, I detest the sound of large numbers of
> strings played together, and find very few uses for them in
> my music.  I much prefer the texture of, say, a string quartet,
> or other small ensemble.  

I happen to agree totally, and have written only chamber and
keyboard music.  I used GM to specify instrumentation anyway.

> But perhaps you would regard a collection of the usual chamber 
> music instruments as "some form of standard orchestra".

As I said in a later message: yes.

-Carl

From: Carl Lumma (2005-10-26)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> >   What is the *musical* purpose of having proportional brats?
> 
> It's a very subtle effect which requires correct tuning, but
> Wendell believes it amelorates the mistuning of triads, and in
> some cases I think that is true.

I've done some brat comparisons between similarly-mistuned
chords with Cool Edit, and I can't agree.  It's a very subtle
effect, quite different from overall mistuning.

> Unfortunately, the most interesting of these brats is -1,

Why is it most interesting?  Did you determine this by listening?

> Kraig Grady can tell you about Wilson and his Mt Maru
> scales, and there's twelve notes of Wilson meantone ("wilwolf")
> etc to consider.

I believe those scales also are intended to work with
combination tones.

> I think proportional beating is pretty interesting in the
> ill-tuned, low complexity temperaments, and Herman has made
> a few experiments along those lines.

These were not controlled for key.  I'd be shocked if the
improvement noticed in the porcupine overture could be
attributed to brats.  Herman, can you tell us what key you
used in blackbeat15.scl, and if it was one of its better
ones?

-Carl

From: Gene Ward Smith (2005-10-26)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> > Unfortunately, the most interesting of these brats is -1,
> 
> Why is it most interesting?  Did you determine this by listening?

I do find the sound of it interesting, but this is really a
theory-based claim. Synch beating is the special case of everything
beating at the same rate.

From: monz (2005-10-26)
Subject: Re: harware tuning resolution (was: Temperament (Extra)ordinare!)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In [email protected], "monz" <monz@t...> wrote:

> > It's taken me several weeks to create a decent Csound 
> > nylon-string guitar instrument, which i *finally*
> > accomplished today to my satisfaction.
> 
> What's Tonalsoft going to do with these? Anything
> independently of Tonescape?


Tonescape has taken the big leap from MIDI to self-produced
WAV files. We're writing a synthesis engine which does 
direct digital synthesis. So i'm using Csound as a 
learning tool right now.

We do have plans (for pretty far in the future, i guess)
to have other products besides Tonescape, and so this
synthesis engine is an important part of several of them.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: wallyesterpaulrus (2005-10-26)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> 
> > > Unfortunately, the most interesting of these brats is -1,
> > 
> > Why is it most interesting?  Did you determine this by listening?
> 
> I do find the sound of it interesting, but this is really a
> theory-based claim. Synch beating is the special case of everything
> beating at the same rate.

Specifically, when and only when the "brat" is -1, the three intervals 
in the close-voiced root-position major triad (major third, minor 
third, perfect fifth) all produce the same rate of beating. All major 
triads in Wilson's metameantone have this property . . .

From: Gene Ward Smith (2005-10-26)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Specifically, when and only when the "brat" is -1, the three intervals 
> in the close-voiced root-position major triad (major third, minor 
> third, perfect fifth) all produce the same rate of beating. All major 
> triads in Wilson's metameantone have this property . . .

And it isn't just the closed-voice root position triads which do this,
either.

From: Carl Lumma (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > Specifically, when and only when the "brat" is -1, the three
> > intervals in the close-voiced root-position major triad (major
> > third, minor third, perfect fifth) all produce the same rate
> > of beating. All major triads in Wilson's metameantone have this
> > property . . .
>
> And it isn't just the closed-voice root position triads which do
> this, either.

I'm still not sure why this is supposed to be desirable, other
than it makes creating bearing plans much easier.

-Carl

From: wallyesterpaulrus (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 
> > Specifically, when and only when the "brat" is -1, the three 
intervals 
> > in the close-voiced root-position major triad (major third, minor 
> > third, perfect fifth) all produce the same rate of beating. All 
major 
> > triads in Wilson's metameantone have this property . . .
> 
> And it isn't just the closed-voice root position triads which do this,
> either.

Could you list all that do?

From: wallyesterpaulrus (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > Specifically, when and only when the "brat" is -1, the three
> > > intervals in the close-voiced root-position major triad (major
> > > third, minor third, perfect fifth) all produce the same rate
> > > of beating. All major triads in Wilson's metameantone have this
> > > property . . .
> >
> > And it isn't just the closed-voice root position triads which do
> > this, either.
> 
> I'm still not sure why this is supposed to be desirable, other
> than it makes creating bearing plans much easier.
> 
> -Carl

I'm not sure how you'd use this for a bearing plan -- Jorgenson likes 
tunings where different instances of an interval across the keyboard 
have the same rate of beating, but this is not at all the case for 
regular metameantone, where the beat rates are (only?) synchronized 
within a single major triad at a time.

But some have said that when all the beatings in each chord are 
synchronized, the effect is more coherent and less chaotic than when 
there are multiple, unrelated rates of beating all occuring at the 
same time.

From: Yahya Abdal-Aziz (2005-10-27)
Subject: RE: csound instruments

Aaron,

On Wed, 26 Oct 2005 you wrote: 
[edited]

[monz]
> > > It's taken me several weeks to create a decent Csound
> > > nylon-string guitar instrument, which i *finally*
> > > accomplished today to my satisfaction.

[Yahya]
> > Now *there's* an achievement!  Well done ...
> > Is this perhaps the first in a usable "orchestra" of
> > CSound emulations of traditional instruments?

[aaron]
> it appears that Prent Rodgers uses samples of trad 
> instruments to good effect in csound.

My bad.  Prent was kind enough to send me a CD of
his music, and I must agree that the sounds are good.
I'd forgotten he's a CSound user.

What I had in mind was that perhaps CSound users
might begin to *share* the instruments that they've 
created, so that other users might get a leg up.

At what price?  Since it obviously takes quite a deal
of effort to create them, they've got to be worth 
something to new users.  Still, the community of 
CSound users might think it worthwhile to give some
instruments away, as an inducement to get others
using it, thus swelling their ranks ...

Regards,
Yahya


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From: Yahya Abdal-Aziz (2005-10-27)
Subject: Re: csound instruments

On Wed, 26 Oct 2005 "Jon Szanto" wrote: 
> 
> --- In [email protected], Aaron Krister Johnson 
> <aaron@a...> wrote:
> > it appears that Prent Rodgers uses samples of trad 
> > instruments to good effect in csound.
> 
> Yes. Prent has done a lot of work on them over the years, so it
> definitely wasn't "plug and play". I believe many of his initial
> samples came from the McGill University sample collection. Lots of
> people have worked with samples for years (obviously); for a GM set
> for use in Csound, it would take someone some time and initiative to
> get a set ready. There is a fairly complete set of orchestral
> instrument samples available one could start with from the University
> of Iowa:
> 
> http://theremin.music.uiowa.edu/MIS.html

Jon,
This looks like it has good coverage of the commoner 
instruments.  One thing that concerns me is that the 
samples are in .aiff format.  Didn't someone recently
(you, perhaps) comment on list that AIF files don't
have the best resolution available, and that some other
format would be preferable?  From the web page you
linked to:

"All samples are in mono, 16 bit, 44.1 kHz, AIFF format."

According to the Nyquist limit theorem, 44.1 kHz would
give adequate sampling of all frequencies up to 22.05 
kHz, right?  Which pretty much covers the musical 
spectrum.  Am I missing something here?

Regards,
Yahya


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From: Herman Miller (2005-10-27)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Carl Lumma wrote:
> 
> These were not controlled for key.  I'd be shocked if the
> improvement noticed in the porcupine overture could be
> attributed to brats.  Herman, can you tell us what key you
> used in blackbeat15.scl, and if it was one of its better
> ones?

Since there wasn't any indication of what was supposed to be the 
"better" key of the scale, I just tuned C (MIDI note 60) to the usual 
261.63 Hz and used that as the first note of the scale. The music 
actually starts out in the key of "MIDI note 64" (more or less "E flat") 
minor, but it happens to be in C major from around 1:55 to around 2:10 
and then again briefly around 2:30. It keeps getting back around to "Eb" 
minor.

From: Carl Lumma (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > And it isn't just the closed-voice root position triads which
> > do this, either.
> 
> Could you list all that do?

With pure octaves, all inversions/voicings will, right?

-Carl

From: Carl Lumma (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > I'm still not sure why this is supposed to be desirable, other
> > than it makes creating bearing plans much easier.
> > 
> > -Carl
> 
> I'm not sure how you'd use this for a bearing plan -- Jorgenson
> likes tunings where different instances of an interval across
> the keyboard have the same rate of beating, but this is not at
> all the case for regular metameantone, where the beat rates are
> (only?) synchronized within a single major triad at a time.

Bearing plans often include things like, 'tune the E until
the major third C-E beats at the same rate as the minor third
E-G'.

-Carl

From: Carl Lumma (2005-10-27)
Subject: Re: csound instruments

> > http://theremin.music.uiowa.edu/MIS.html
> 
> Jon,
> This looks like it has good coverage of the commoner 
> instruments.  One thing that concerns me is that the 
> samples are in .aiff format.  Didn't someone recently
> (you, perhaps) comment on list that AIF files don't
> have the best resolution available, and that some other
> format would be preferable?  From the web page you
> linked to:
> 
> "All samples are in mono, 16 bit, 44.1 kHz, AIFF format."
> 
> According to the Nyquist limit theorem, 44.1 kHz would
> give adequate sampling of all frequencies up to 22.05 
> kHz, right?  Which pretty much covers the musical 
> spectrum.  Am I missing something here?

Nope, AIFF is just Apple's version of WAV (some Mac
person will probably claim it was the other way around,
but whatever).  It's uncompressed audio.  The resolution
depends on the resolution chosen when recording it.

For samples, though, there are a ton available for
free in SoundFont format...

http://www.hammersound.net/
http://www.personalcopy.com/

I believe there are free utilities that can extract
individual samples from SoundFont files.

But no need to do so -- Dave says Csound has
SoundFont support (I wonder if they remain as tunable
as a normal Csound instrument?).

Is there anything like the above sites for Csound
patches??

-Carl

From: Carl Lumma (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], Herman Miller <hmiller@I...> wrote:
>
> Carl Lumma wrote:
> > 
> > These were not controlled for key.  I'd be shocked if the
> > improvement noticed in the porcupine overture could be
> > attributed to brats.  Herman, can you tell us what key you
> > used in blackbeat15.scl, and if it was one of its better
> > ones?
> 
> Since there wasn't any indication of what was supposed to be the 
> "better" key of the scale, I just tuned C (MIDI note 60) to the
> usual 261.63 Hz and used that as the first note of the scale.
> The music actually starts out in the key of "MIDI note 64" (more
> or less "E flat") minor, but it happens to be in C major from
> around 1:55 to around 2:10 and then again briefly around 2:30.
> It keeps getting back around to "Eb" minor.

Is Eb the 4th line of the Scala file?  And C the first?

If so, the C major 3rd is 3 cents better than in 15-equal.
A small difference, but one audible to me in my well
temperament adventures (12-equal also has 400-cent 3rds).

The 5ths are all the same as 15-equal, but C and Eb are
2 of the 5 keys whose minor 3rds are 6 cents better, and
very near just.  This is especially big because the zone
above 6/5 gets dissonant very rapidly (it's why 15-equal
doesn't really have better minor 3rds than 12).

The 7:4s are all the same as in 15-equal.

-Carl

From: Gene Ward Smith (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> Could you list all that do?

"All" would require a lemma, I suppose, but start from

2:3:5
4:5:6
3:4:10
3:8:20
3:16:40
...
5:8:12
5:16:24
5:32:48
...

At least two infinite families.

From: Gene Ward Smith (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > And it isn't just the closed-voice root position triads which
> > > do this, either.
> > 
> > Could you list all that do?
> 
> With pure octaves, all inversions/voicings will, right?

Up to powers of two. Instead of getting 1,1,1 you might get 1,1,2, etc.

From: Graham Breed (2005-10-27)
Subject: Re: [tuning] Re: csound instruments

Carl Lumma wrote:

> Nope, AIFF is just Apple's version of WAV (some Mac
> person will probably claim it was the other way around,
> but whatever).  It's uncompressed audio.  The resolution
> depends on the resolution chosen when recording it.

I'm not sure what exactly they are.  Wav files are a generic wrapper for 
any encoding that's registered with Windows, but many applications that 
work with Wav files aren't this flexible.  AIFF files, as usually 
understood, allow you to set loop points while Wav files don't.  So AIFF 
is more appropriate for sample libraries.

> For samples, though, there are a ton available for
> free in SoundFont format...
> 
> http://www.hammersound.net/
> http://www.personalcopy.com/
> 
> I believe there are free utilities that can extract
> individual samples from SoundFont files.
> 
> But no need to do so -- Dave says Csound has
> SoundFont support (I wonder if they remain as tunable
> as a normal Csound instrument?).

Yes.  SoundFont support is at the opcode level, so you write your own 
instrument to use them.  From my copy of the documentation (online 
somewhere, try Google):

"""
  When iflag = 0, inotenum sets the frequency of the output according to 
the MIDI note number used, and xfreq is used as a multiplier. When iflag 
= 1, the frequency of the output, is set directly by xfreq. This allows 
the user to use any kind of micro-tuning based scales. However, this 
method is designed to work correctly only with presets tuned to the 
default equal temperament. Attempts to use this method with a preset 
already having non-standard tunings, or with drum-kit-based presets, 
could give unexpected results.
"""

But it also says this, which is scarier:

"""
These opcodes only support the sample structure of SF2 files. The 
modulator structure of the SoundFont2 format is not supported in Csound. 
Any modulation or processing to the sample data is left to the Csound 
user, bypassing all restrictions forced by the SF2 standard.
"""


You can also play Sound Fonts using FluidSynth.  I still don't know 
exactly what it is, but it's set up as a multi-channel MIDI synth.  So 
the only way to tune it is to send pitch bend messages to the relevant 
channel.  Which pretty much misses the point.  I suppose you could run 
multiple instances of the Fluid engine to get around the 16-channel limit.


> Is there anything like the above sites for Csound
> patches??

I don't think so.  You can order a CD-ROM which has lots of instruments 
and musical examples on it.  There's also a User Defined Opcode (UDO) 
database online.


                       Graham

From: Carl Lumma (2005-10-27)
Subject: Re: csound instruments

> > Nope, AIFF is just Apple's version of WAV (some Mac
> > person will probably claim it was the other way around,
> > but whatever).  It's uncompressed audio.  The resolution
> > depends on the resolution chosen when recording it.
> 
> I'm not sure what exactly they are.  Wav files are a generic
> wrapper for  any encoding that's registered with Windows, but
> many applications that work with Wav files aren't this flexible.
> AIFF files, as usually understood, allow you to set loop points
> while Wav files don't.  So AIFF is more appropriate for sample
> libraries.

"Acidized WAVs" let you set loop points, and are back-compat.
with WAV.

> > But no need to do so -- Dave says Csound has
> > SoundFont support (I wonder if they remain as tunable
> > as a normal Csound instrument?).
> 
> Yes.  SoundFont support is at the opcode level, so you write
> your own instrument to use them.  From my copy of the
> documentation (online somewhere, try Google):
> 
> """
>   When iflag = 0, inotenum sets the frequency of the output
> according to the MIDI note number used, and xfreq is used as
> a multiplier. When iflag = 1, the frequency of the output,
> is set directly by xfreq. This allows the user to use any
> kind of micro-tuning based scales. However, this method is
> designed to work correctly only with presets tuned to the
> default equal temperament. Attempts to use this method with
> a preset already having non-standard tunings, or with
> drum-kit-based presets, could give unexpected results.
> """
> 
> But it also says this, which is scarier:
> 
> """
> These opcodes only support the sample structure of SF2 files.
> The modulator structure of the SoundFont2 format is not
> supported in Csound. Any modulation or processing to the
> sample data is left to the Csound user, bypassing all
> restrictions forced by the SF2 standard.
> """

Good work.

-Carl

From: Aaron Krister Johnson (2005-10-27)
Subject: Re: [tuning] Re: csound instruments

On Thursday 27 October 2005 7:02 am, Graham Breed wrote:

> You can also play Sound Fonts using FluidSynth.

and let's not forget timidity, which plays them, without distorting (unlike 
fluidsynth, which distorts terribly)

-aaron.

From: wallyesterpaulrus (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > I'm still not sure why this is supposed to be desirable, other
> > > than it makes creating bearing plans much easier.
> > > 
> > > -Carl
> > 
> > I'm not sure how you'd use this for a bearing plan -- Jorgenson
> > likes tunings where different instances of an interval across
> > the keyboard have the same rate of beating, but this is not at
> > all the case for regular metameantone, where the beat rates are
> > (only?) synchronized within a single major triad at a time.
> 
> Bearing plans often include things like, 'tune the E until
> the major third C-E beats at the same rate as the minor third
> E-G'.

But in this case, how would you

() get the minor third E-G in the first place

() get beyond the C major triad and tune the rest of the notes

?

From: wallyesterpaulrus (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> The 5ths are all the same as 15-equal, but C and Eb are
> 2 of the 5 keys whose minor 3rds are 6 cents better,

How is that possible? The "minor 3rd" or approximate 6:5 is only 4 
cents off JI in 15-equal!

> and
> very near just.  This is especially big because the zone
> above 6/5 gets dissonant very rapidly (it's why 15-equal
> doesn't really have better minor 3rds than 12).

It doesn't? I guess I better get my ears checked . . .

From: wallyesterpaulrus (2005-10-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 
> > Could you list all that do?
> 
> "All" would require a lemma, I suppose, but start from
> 
> 2:3:5
> 4:5:6
> 3:4:10
> 3:8:20
> 3:16:40
> ...
> 5:8:12
> 5:16:24
> 5:32:48
> ...
> 
> At least two infinite families.

Cool.

From: Graham Breed (2005-10-27)
Subject: Re: [tuning] Re: csound instruments

Aaron Krister Johnson wrote:

> and let's not forget timidity, which plays them, without distorting (unlike 
> fluidsynth, which distorts terribly)

But you can't host Timidity in Csound, can you?


                 Graham

From: Herman Miller (2005-10-28)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Carl Lumma wrote:
> --- In [email protected], Herman Miller <hmiller@I...> wrote:
>>Since there wasn't any indication of what was supposed to be the 
>>"better" key of the scale, I just tuned C (MIDI note 60) to the
>>usual 261.63 Hz and used that as the first note of the scale.
>>The music actually starts out in the key of "MIDI note 64" (more
>>or less "E flat") minor, but it happens to be in C major from
>>around 1:55 to around 2:10 and then again briefly around 2:30.
>>It keeps getting back around to "Eb" minor.
> 
> 
> Is Eb the 4th line of the Scala file?  And C the first?

The 5th line (the first is MIDI note 60, the note I'm calling "Eb" Is 
MIDI note 64).

From: Gene Ward Smith (2005-10-28)
Subject: Re: csound instruments

--- In [email protected], Aaron Krister Johnson <aaron@a...> wrote:
>
> On Thursday 27 October 2005 7:02 am, Graham Breed wrote:
> 
> > You can also play Sound Fonts using FluidSynth.
> 
> and let's not forget timidity, which plays them, without distorting
(unlike 
> fluidsynth, which distorts terribly)

SynthFont is another program which has been improving.

From: Dave Keenan (2005-10-28)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> I'm still not sure why this is supposed to be desirable, other
> than it makes creating bearing plans much easier.
> 
> -Carl

Small whole-number ratios between the beat rates of the different 
intervals making up a chord? I think it has a special sound. I call 
it "second-order JI".  La Monte Young's Dream House depends on it. 
That's the way to think about the Dream House -- not as two-hundred-
and-something limit (first-order) JI.

-- Dave

From: Ozan Yarman (2005-10-28)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

George, I liked your previous extraordinary temperament better. Maybe it's because of better 5-limit approximation of intervals?

  ----- Original Message ----- 
  From: George D. Secor 
  To: [email protected] 
  Sent: 24 Ekim 2005 Pazartesi 22:44 
  Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament


  SNIP

  > ! wendell1r.scl
  > !
  > Rational version of wendell1.scl by Gene Ward Smith
  >  12
  > !
  >  24000/22729
  >  76460/68187
  >  27000/22729
  >  85600/68187
  >  4/3
  >  32100/22729
  >  102140/68187
  >  36000/22729
  >  114400/68187
  >  40500/22729
  >  42800/22729
  >  2/1

  Gene, I'm impressed!

  But, OTOH, I hope you realize that there are reasonably simple brats 
  in only 5 minor triads (the ones with just 5ths), and the rest are 
  pretty much hit-or-miss (mostly the latter) -- much like what I got 
  in my _temperament (extra)ordinaire_ a couple months back.

  In attempting to construct synchronous temperaments (with irrational 
  intervals) with both proportional *major and minor* triads, I found 
  that, unless the 5ths are exact, the minor-triad brats will at best 
  only approximate simple ratios when the majors are exact (or vice 
  versa), or you can compromise and make both inexact by about half and 
  thereby get most of the brats to within 0.02 of simple numbers -- 
  close enough, I would say, considering that the required adjustment 
  is very small. In the following example, each of the tempered fifths 
  is modified by less than 0.002 cents (i.e., -3.9114 vs. -3.9132 cents 
  for an exact 2/11-comma narrow fifth).

  Here's a sneak preview of my 2/11-comma well-temperament (with beat 
  synchrony in virtually all 24 major & minor triads).  Bear in mind 
  that this is an honest-to-goodness well-temperament that meets *all* 
  of Jorgensen's requirements, with key contrasts that are very similar 
  to the Valotti-Young (1/6-pythagorean-comma) temperament:

  ! secor_WT2-11.scl
  !
  George Secor's 2/11-comma well-temperament, proportional beating
  12
  !
  92.13884
  196.08721
  296.01562
  392.17442
  499.92562
  590.20835
  698.04361
  794.06062
  894.13082
  997.97062
  1090.21803
  2/1

  As to how close it comes to perfection, judge for yourself:

  Major -----Beat Ratios-----
  Triad  M3/5th  m3/5th m3/M3
  ----- ------- ------- -----
  Ab... ------- -------  1.50
  Eb... ------- -------  1.50
  Bb... ------- -------  1.50
  F....    7.01   13.02  1.86
  C....    2.50    6.26  2.50
  G....    2.50    6.26  2.50
  D....    3.34    7.51  2.25
  A....    5.01   10.01  5.00
  E....    6.67   12.51  1.87
  B…...   16.63   27.45  1.65
  F#... 1467.47 2203.71  1.50
  C#... 1084.06 1628.58  1.50

  Minor -----Beat Ratios-----
  Triad  M3/5th  m3/5th m3/M3
  ----- ------- ------- -----
  Ab... ------- -------  1.00
  Eb... ------- -------  1.00
  Bb... ------- -------  1.00
  F....   20.74   22.74  1.10
  C....    7.99    9.99  1.25
  G....    6.01    8.01  1.33
  D....    4.02    6.02  1.50
  A....    2.99    4.99  1.67
  E....    2.99    4.99  1.67
  B…...    7.93    9.93  1.25
  F#...  950.96  952.96  1.00
  C#...  933.34  935.34  1.00

  The F# and C# triads have 5ths that are nearly just, so you can 
  consider those 3- and 4-digit numbers as approaching infinity.  As 
  for B major, it's your call as to whether 1.65 is close enough to 5/3 
  to qualify as proportional; or for F minor, whether 11/10 is simple 
  enough.

  Anyway, it turns out that I came up with at least two other 
  synchonous temperaments (one well, the other _ordinaire_) that I like 
  better than this one.  I'll share them once I've settled on the best 
  way to categorize the whole collection.

  --George

From: Carl Lumma (2005-10-28)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> >
> > > > I'm still not sure why this is supposed to be desirable, other
> > > > than it makes creating bearing plans much easier.
> > > > 
> > > > -Carl
> > > 
> > > I'm not sure how you'd use this for a bearing plan -- Jorgenson
> > > likes tunings where different instances of an interval across
> > > the keyboard have the same rate of beating, but this is not at
> > > all the case for regular metameantone, where the beat rates are
> > > (only?) synchronized within a single major triad at a time.
> > 
> > Bearing plans often include things like, 'tune the E until
> > the major third C-E beats at the same rate as the minor third
> > E-G'.
> 
> But in this case, how would you
> 
> () get the minor third E-G in the first place
> 
> () get beyond the C major triad and tune the rest of the notes
> 
> ?

Usually using other devices.  What is the point of this line
of questioning?

-Carl

From: Carl Lumma (2005-10-28)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > The 5ths are all the same as 15-equal, but C and Eb are
> > 2 of the 5 keys whose minor 3rds are 6 cents better,
> 
> How is that possible? The "minor 3rd" or approximate 6:5 is
> only 4 cents off JI in 15-equal!

That's a good point.  It's 2 cents better.

> > and
> > very near just.  This is especially big because the zone
> > above 6/5 gets dissonant very rapidly (it's why 15-equal
> > doesn't really have better minor 3rds than 12).
> 
> It doesn't? I guess I better get my ears checked . . .

Maybe you should, Paul, maybe you should.

-Carl

From: Carl Lumma (2005-10-28)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > --- In [email protected], Herman Miller <hmiller@I...> wrote:
> >>Since there wasn't any indication of what was supposed to be the 
> >>"better" key of the scale, I just tuned C (MIDI note 60) to the
> >>usual 261.63 Hz and used that as the first note of the scale.
> >>The music actually starts out in the key of "MIDI note 64" (more
> >>or less "E flat") minor, but it happens to be in C major from
> >>around 1:55 to around 2:10 and then again briefly around 2:30.
> >>It keeps getting back around to "Eb" minor.
> > 
> > 
> > Is Eb the 4th line of the Scala file?  And C the first?
> 
> The 5th line (the first is MIDI note 60, the note I'm calling
> "Eb" Is MIDI note 64).

397cents is Eb?

-Carl

From: Carl Lumma (2005-10-28)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > I'm still not sure why this is supposed to be desirable, other
> > than it makes creating bearing plans much easier.
> > 
> > -Carl
> 
> Small whole-number ratios between the beat rates of the different 
> intervals making up a chord? I think it has a special sound.

Can you give me some chord (cents values) to compare?

> I call it "second-order JI".  La Monte Young's Dream House
> depends on it.  That's the way to think about the Dream
> House -- not as two-hundred-and-something limit (first-order) JI.

While the beat rates in the Dream House may be rational, are
they simple?

When I was at the dream house I didn't notice any special
synchrony.  I wasn't listening for it either, but I heard lots
of different things as I moved throughout the space.

-Carl

From: Herman Miller (2005-10-29)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Carl Lumma wrote:
>>>Is Eb the 4th line of the Scala file?  And C the first?
>>
>>The 5th line (the first is MIDI note 60, the note I'm calling
>>"Eb" Is MIDI note 64).
> 
> 
> 397cents is Eb?

Hmm; you confused me when you asked if C is the first line (Scala files 
don't include the base note of the scale, since it's the reference for 
the other notes). With C tuned as 0.0 (which isn't in the Scala file), 
then "Eb" is the 4th line, 322.836732 cents.

From: Carl Lumma (2005-10-29)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> Carl Lumma wrote:
> >>>Is Eb the 4th line of the Scala file?  And C the first?
> >>
> >>The 5th line (the first is MIDI note 60, the note I'm calling
> >>"Eb" Is MIDI note 64).
> > 
> > 
> > 397cents is Eb?
> 
> Hmm; you confused me when you asked if C is the first line (Scala
> files don't include the base note of the scale, since it's the
> reference for the other notes). With C tuned as 0.0 (which isn't
> in the Scala file), then "Eb" is the 4th line, 322.836732 cents.

So I think my analysis was right.  If you render a performance with
this, we can see, though:

!
 Blackwood[15] with brats of -1, bad C and Eb keys.
 15
!
 82.83673
 165.67346
 240.00000
 322.83673
 405.67346
 480.00000
 562.83673
 645.67346
 720.00000
 802.83673
 885.67346
 960.00000
 1042.83673
 1125.67346
 1200.00000
!

-Carl

From: Gene Ward Smith (2005-10-29)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> So I think my analysis was right.  If you render a performance with
> this, we can see, though:

This makes the C major chord not synch beating. What does that test?

From: Carl Lumma (2005-10-29)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > So I think my analysis was right.  If you render a performance
> > with this, we can see, though:
> 
> This makes the C major chord not synch beating. What does that
> test?

The file comments say "with brats of -1".  Maybe they
should say "major triads in 2 out of 3 keys have -1 brats".

-Carl

From: Carl Lumma (2005-10-29)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > > So I think my analysis was right.  If you render a performance
> > > with this, we can see, though:
> > 
> > This makes the C major chord not synch beating. What does that
> > test?
> 
> The file comments say "with brats of -1".  Maybe they
> should say "major triads in 2 out of 3 keys have -1 brats".

This raises the question of how many synch triads are available.
It'd be cool to see a graph with third size from say 200 - 500
cents on the y axis and fifth size from 650 - 750 cents on the
x axis, with synch-tuned triads plotted as dots.

-Carl

From: Herman Miller (2005-10-30)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Carl Lumma wrote:
> So I think my analysis was right.  If you render a performance with
> this, we can see, though:

http://home.comcast.net/~teamouse/porcupine-absynth-blackwood-worse.mp3

Compare with:
http://home.comcast.net/~teamouse/porcupine-absynth-blackwood.mp3
http://home.comcast.net/~teamouse/porcupine-absynth.mp3 (15-ET)

> !
>  Blackwood[15] with brats of -1, bad C and Eb keys.
>  15
> !
>  82.83673
>  165.67346
>  240.00000
>  322.83673
>  405.67346
>  480.00000
>  562.83673
>  645.67346
>  720.00000
>  802.83673
>  885.67346
>  960.00000
>  1042.83673
>  1125.67346
>  1200.00000
> !

The effect is somewhat interesting in the familiar "comma pump" 
progression at the end. The one chord in the sequence that's actually 
closer to being in tune sounds out of place and quickly "resolves" to a 
more dissonant chord.

From: Dave Keenan (2005-10-31)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> --- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
> > Small whole-number ratios between the beat rates of the 
different 
> > intervals making up a chord? I think it has a special sound.
> 
> Can you give me some chord (cents values) to compare?

Hi Carl,

Ratios describe it more precisely than cents. And it has to be tuned 
very precisely to get the effect. You'd probably need something 
based on digital-counter frequency dividers or phase-locked-loop 
frequency multipliers (like the Dreamhouse or a Scalatron). Your 
standard MIDI synth isn't likely to be able to do it.

Consider an approximate 4:5:6 chord. Multiply each of those numbers 
by 128 to get 512:640:768 then temper it to 512:639:766 (i.e. 512 : 
640-1 : 768-2). If those numbers were hertz we'd have a 4 Hz beat 
between the lowest relevant harmonics in each of the three 
intervals. And if it's tuned precisely enough those beats will be 
synchronised so there will be only one simple beat.

This example results in narrowing of the fifth and major and minor 
thirds by approximately 4.5, 2.7 and 1.8 cents respectively.

Compare this to 512 : 640-1/phi : 768-phi which should have no 
discernable beat synchrony but similar cent errors of approximately 
3.7, 1.7 and 2.0 cents. phi = (sqrt(5)+1)/2

I don't have the equipment to generate these accurately enough, so 
I'm running on pure theory here in suggesting that these will sound 
more different from each other than you'd expect just based on the 
cent errors. Maybe someone out there can generate these for us all 
to listen to.

> > I call it "second-order JI".  La Monte Young's Dream House
> > depends on it.  That's the way to think about the Dream
> > House -- not as two-hundred-and-something limit (first-order) JI.
> 
> While the beat rates in the Dream House may be rational, are
> they simple?

I haven't analysed the numbers in depth, but my impression is that 
they would often be so.

> When I was at the dream house I didn't notice any special
> synchrony.  I wasn't listening for it either, but I heard lots
> of different things as I moved throughout the space.

I'm only referring to the temporal aspects not the spatial. I've 
only ever heard that software simulation of it that someone kindly 
posted a few months back.

If you've got lots of simultaneous frequencies at about the same 
amplitude, then if you can hear any kind of rhythmic beating, as 
opposed to a complete mess, there's got to be some beat synchrony 
going on, doesn't there?

-- Dave

From: George D. Secor (2005-10-31)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@y...> 
wrote:
> ...
> > As to how close it [secor_WT2-11.scl] comes to perfection, judge 
for yourself:
> > 
> > Major -----Beat Ratios-----
> > Triad  M3/5th  m3/5th m3/M3
> > ----- ------- ------- -----
> > Ab... ------- -------  1.50
> > Eb... ------- -------  1.50
> > Bb... ------- -------  1.50
> > F....    7.01   13.02  1.86
> > C....    2.50    6.26  2.50
> > G....    2.50    6.26  2.50
> > D....    3.34    7.51  2.25
> > A....    5.01   10.01  5.00
> > E....    6.67   12.51  1.87
> > B…...   16.63   27.45  1.65
> > F#... 1467.47 2203.71  1.50
> > C#... 1084.06 1628.58  1.50
> 
> You didn't like Wendell's brats of 2, but I can't see this is an
> improvement. A 5/2 brat has a brat orbit of 5/2, -5/2, -4/25, which
> doesn't seem as good as 2. The 5 brat has an orbit 5, -5/7, -7/25, 
and
> I would urge you to dump it in favor of one or more brats of 4
> instead. I don't know where the 1.86 and 1.85 come from, but a brat
> of 13/7 has an orbit 13/7, -7, -1/13. 9/4 is another not very 
terrific
> looking brat, with an orbit of 9/4, -10/3, -2/15. 
> 
> I recommend using as many of the "magic" brats as possible; load up 
on
> 3/2 and 4, and fill in a gap with 2 or 3.

So what do you think of this one?

! secor_TEO4.scl
!
George Secor's 12-tone temperament (extra)ordinaire, proportional 
beating (attempt #4)
 12
!
 87.83846
 194.22760
 294.41956
 389.19724
 499.66357
 585.88345
 697.13843
 789.79346
 890.82392
 997.70857
 1086.36510
 2/1

Here are the brats and total absolute error for each triad:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 20.5856 28.3784 1.3786 32.81
Bb... ------- ------- 1.5000 20.41 
F....  5.0000 10.0000 2.0000 12.93
C....  1.0000  4.0000 4.0000 15.40
G....  1.0000  4.0000 4.0000 15.56
D....  1.6667  5.0000 3.0000 21.40
A....  5.0000 10.0000 2.0000 28.57
E....  5.0000 10.0000 2.0000 38.14
B.... 14.9750 24.9625 1.6669 48.35
F#... ------- ------- 1.5000 51.02
C#... ------- ------- 1.5000 51.02
G#... 15.0000 20.0000 1.3333 47.79

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 37.9820 35.9820 0.9473 51.02
Bb... ------- ------- 1.0000 51.02
F.... 29.3069 31.3069 1.0682 51.02
C....  6.7704  8.7704 1.2954 42.44
G....  4.1765  6.1765 1.4789 30.14
D....  1.8036  3.8036 2.1089 20.41
A....  1.6071  3.6071 2.2444 12.93
E....  1.2143  3.2143 2.6471 15.40
B....  4.3750  6.3750 1.4571 15.56
F#... ------- ------- 1.0000 21.40
C#... ------- ------- 1.0000 28.57
G#... 16.1892 14.1892 0.8765 43.48

I found it *extremely* difficult to get reasonably simple brats for 
*all* 12 major triads (the Eb and B brats are very close to 11/8 and 
5/3, but not exact), and if you do, then it's not too likely that the 
brats for the minor triads with tempered fifths will be simple (as 
you'll see above).  I'll have to work on this some more, if that's 
what's required for a "simplest" temperament.  (It looks as if 3-
brats on the C and G major triads will give you 2-brats on A and E 
minor, so there's hope.)  

In order to get the 4-brats (in the majors) you need triads with 
relatively low total absolute error, something on the order of 
meantone temperament.  I used my 5/23-comma temperament (extra)
ordinaire as a guide for arriving at the above, since it already had 
some excellent brats in about half the major triads.  

http://groups.yahoo.com/group/tuning/message/59999

If you compare the two in a listening test, you'll be hard pressed to 
hear any difference.

--George

From: George D. Secor (2005-10-31)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
>
> George, I liked your previous extraordinary temperament better. Maybe 
it's because of better 5-limit approximation of intervals?

Yes, the best triads of the (extra)ordinaire are much better.  Please 
note the latest improvement on the (extra)ordinaire, which has 
proportional beating on all 12 major triads:

http://groups.yahoo.com/group/tuning/message/62004

--George

From: Gene Ward Smith (2005-10-31)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

> So what do you think of this one?

At first glance, it looks better.

> I found it *extremely* difficult to get reasonably simple brats for 
> *all* 12 major triads (the Eb and B brats are very close to 11/8 and 
> 5/3, but not exact), and if you do, then it's not too likely that the 
> brats for the minor triads with tempered fifths will be simple (as 
> you'll see above).  

Here's a try of mine, which I think has been posted before. The beat
ratios are simple, but one of the fifths is sharp. Nonetheless, it
could be considered circulating.

! brac.scl
circulating temperament with simple beat ratios
! beat ratios 4 3/2 4 3/2 2 2 177/176 4 3/2 2 3/2 2
12
!
56640/53701
60008/53701
63720/53701
67264/53701
71685/53701
75056/53701
80276/53701
84960/53701
89920/53701
95580/53701
100544/53701
2




I'll have to work on this some more, if that's 
> what's required for a "simplest" temperament.  (It looks as if 3-
> brats on the C and G major triads will give you 2-brats on A and E 
> minor, so there's hope.)  
> 
> In order to get the 4-brats (in the majors) you need triads with 
> relatively low total absolute error, something on the order of 
> meantone temperament.  I used my 5/23-comma temperament (extra)
> ordinaire as a guide for arriving at the above, since it already had 
> some excellent brats in about half the major triads.  
> 
> http://groups.yahoo.com/group/tuning/message/59999
> 
> If you compare the two in a listening test, you'll be hard pressed to 
> hear any difference.
> 
> --George
>

From: wallyesterpaulrus (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > I'm still not sure why this is supposed to be desirable, other
> > than it makes creating bearing plans much easier.
> > 
> > -Carl
> 
> Small whole-number ratios between the beat rates of the different 
> intervals making up a chord? I think it has a special sound. I call 
> it "second-order JI".  La Monte Young's Dream House depends on it.

Is that true? Why isn't that first-order JI, and why is the lack of 
any upper partials that might beat not a problem?
 
> That's the way to think about the Dream House -- not as two-hundred-
> and-something limit (first-order) JI.

What about the combinational tones, which I thought were supposer to 
be more significant? And what about the fact that a constant Hz shift 
of the whole sound ruined the effect according to Daniel Wolf?

From: wallyesterpaulrus (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > >
> > > > > I'm still not sure why this is supposed to be desirable, 
other
> > > > > than it makes creating bearing plans much easier.
> > > > > 
> > > > > -Carl
> > > > 
> > > > I'm not sure how you'd use this for a bearing plan -- 
Jorgenson
> > > > likes tunings where different instances of an interval across
> > > > the keyboard have the same rate of beating, but this is not at
> > > > all the case for regular metameantone, where the beat rates 
are
> > > > (only?) synchronized within a single major triad at a time.
> > > 
> > > Bearing plans often include things like, 'tune the E until
> > > the major third C-E beats at the same rate as the minor third
> > > E-G'.
> > 
> > But in this case, how would you
> > 
> > () get the minor third E-G in the first place
> > 
> > () get beyond the C major triad and tune the rest of the notes
> > 
> > ?
> 
> Usually using other devices.  What is the point of this line
> of questioning?

To find out if indeed "it makes creating bearing plans much easier", 
as you claim. If so, I'd be interested to learn how.

From: Gene Ward Smith (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

> So what do you think of this one?

Here is a rational-number version of it. 

! secorteo4.scl
rational version of secor_TEO4
12
!
17848/16965
10544/9425
4022/3393
106208/84825
566029/424125
71392/50895
126884/84825
8924/5655
141904/84825
2264116/1272375
158872/84825
2

From: Dave Keenan (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
> >
> > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > > I'm still not sure why this is supposed to be desirable, other
> > > than it makes creating bearing plans much easier.
> > > 
> > > -Carl
> > 
> > Small whole-number ratios between the beat rates of the 
different 
> > intervals making up a chord? I think it has a special sound. I 
call 
> > it "second-order JI".  La Monte Young's Dream House depends on 
it.
> 
> Is that true? 

Maybe not. But it seemed to be of major significance in the software 
simulation. I realise there's all kinds of other stuff happening in 
the _real_ Dream House.

> Why isn't that first-order JI,

Because (if you'll allow me a little hyperbole) beating is what it's 
all about.

> and why is the lack of 
> any upper partials that might beat not a problem?

When you have multiple sine-waves phase-locked to exact ratios you 
can think of the whole thing as a single note of a strange timbre or 
you can group the frequencies in various ways and think of each 
group as a note with partials (and possibly a missing fundamentals).

So they are _all_ partials, and some of them beat.

> > That's the way to think about the Dream House -- not as two-
hundred-
> > and-something limit (first-order) JI.
> 
> What about the combinational tones, which I thought were supposer 
to 
> be more significant?

OK. It's _one_ way to think about the Dream House, or at least the 
software simulation which, as I said, is all I've ever heard.

> And what about the fact that a constant Hz shift 
> of the whole sound ruined the effect according to Daniel Wolf?

Why should that affect my opinion. I never said that synched beats 
is the _only_ thing happening.

-- Dave Keenan

From: Carl Lumma (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

Hi Dave,

> > > Small whole-number ratios between the beat rates of the 
> > > different intervals making up a chord? I think it has a
> > > special sound.
> > 
> > Can you give me some chord (cents values) to compare?
> 
> Hi Carl,
> 
> Ratios describe it more precisely than cents. And it has to be
> tuned very precisely to get the effect. You'd probably need
> something based on digital-counter frequency dividers or
> phase-locked-loop frequency multipliers (like the Dreamhouse
> or a Scalatron). Your standard MIDI synth isn't likely to be
> able to do it.

I can render directly to WAV from Cool Edit.
 
> Consider an approximate 4:5:6 chord. Multiply each of those
> numbers by 128 to get 512:640:768 then temper it
> to 512:639:766 (i.e. 512 : 640-1 : 768-2). If those numbers
> were hertz we'd have a 4 Hz beat between the lowest relevant
> harmonics in each of the three intervals. And if it's tuned
> precisely enough those beats will be synchronised so there
> will be only one simple beat.
> 
> This example results in narrowing of the fifth and major and
> minor thirds by approximately 4.5, 2.7 and 1.8 cents
> respectively.
> 
> Compare this to 512 : 640-1/phi : 768-phi which should have no 
> discernable beat synchrony but similar cent errors of
> approximately 3.7, 1.7 and 2.0 cents. phi = (sqrt(5)+1)/2

This is exactly the kind of comparison I've been asking for
the data to do...

http://groups.yahoo.com/group/tuning-math/message/6339

I get...

766.38196601125010515179541316563
639.38196601125010515179541316563
512.0

...Hz. for the messy chord.  But hrm, I don't see any brats
of phi with "show /relative beats" in Scala.  Isn't that what
you intended?

> I don't have the equipment to generate these accurately enough,
> so I'm running on pure theory here in suggesting that these
> will sound more different from each other than you'd expect
> just based on the cent errors. Maybe someone out there can
> generate these for us all to listen to.

Here are the files:

http://lumma.org/tuning/A.wav
http://lumma.org/tuning/B.wav
(1MB total)

Anybody care to guess which is which?  Anybody feel one
sounds better than the other?

> > > I call it "second-order JI".  La Monte Young's Dream House
> > > depends on it.  That's the way to think about the Dream
> > > House -- not as two-hundred-and-something limit (first-order)
> > > JI.
> > 
> > While the beat rates in the Dream House may be rational, are
> > they simple?
> 
> I haven't analysed the numbers in depth, but my impression is
> that they would often be so.

I haven't analysed either, but I thought La Monte basically
came up with ratios of large primes in a rather off-the-cuff
manner.

> > When I was at the dream house I didn't notice any special
> > synchrony.  I wasn't listening for it either, but I heard lots
> > of different things as I moved throughout the space.
> 
> I'm only referring to the temporal aspects not the spatial.

I just mean I heard a lot of stuff, some of it simple,
some of it complex.

> I've only ever heard that software simulation of it that
> someone kindly posted a few months back.

Dave Seidel?  I thought I remembered this too, but on his
site I find several pieces inspired by La Monte installations,
but none by the Dream House.  Dave?

> If you've got lots of simultaneous frequencies at about the
> same amplitude, then if you can hear any kind of rhythmic
> beating, as opposed to a complete mess, there's got to be
> some beat synchrony going on, doesn't there?

I don't recall hearing recognizable rhythms in the beats.

-Carl

From: Magnus Jonsson (2005-11-01)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

On Tue, 1 Nov 2005, Carl Lumma wrote:

> http://lumma.org/tuning/A.wav
> http://lumma.org/tuning/B.wav
> (1MB total)
>
> Anybody care to guess which is which?  Anybody feel one
> sounds better than the other?

I prefer B.

From: Jon Szanto (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> Dave Seidel?  I thought I remembered this too, but on his
> site I find several pieces inspired by La Monte installations,
> but none by the Dream House.  Dave?

IIRC, it was posted by someone but actually created by an Italian
musician, and then the software on his site was removed due to
complaints from LMY himself. David Beardsly will probably remember
this better than I.

Cheers,
Jon

From: Carl Lumma (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > http://lumma.org/tuning/A.wav
> > http://lumma.org/tuning/B.wav
> > (1MB total)
> >
> > Anybody care to guess which is which?  Anybody feel one
> > sounds better than the other?
> 
> I prefer B.

Thanks for weighing in!

-Carl

From: George D. Secor (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@y...> 
wrote:
> 
> > So what do you think of this one?
> 
> At first glance, it looks better.

And it sounds as good as it looks.

> > I found it *extremely* difficult to get reasonably simple brats 
for 
> > *all* 12 major triads (the Eb and B brats are very close to 11/8 
and 
> > 5/3, but not exact), and if you do, then it's not too likely that 
the 
> > brats for the minor triads with tempered fifths will be simple 
(as 
> > you'll see above).  
> 
> Here's a try of mine, which I think has been posted before. The beat
> ratios are simple, 

An understatement  :-) -- they're about as simple as they can get 
(ignoring the minor triads, of course), at least for something 
reasonably useful.

> but one of the fifths is sharp. Nonetheless, it
> could be considered circulating.

Yes, but I would transpose it up a fifth, so the present F would 
become C.  That would put your two worst major triads on F# and C# 
(instead of B and F#) and put the wide fifth between F# and C#.

> ! brac.scl
> circulating temperament with simple beat ratios
> ! beat ratios 4 3/2 4 3/2 2 2 177/176 4 3/2 2 3/2 2
> 12
> !
> 56640/53701
> 60008/53701
> 63720/53701
> 67264/53701
> 71685/53701
> 75056/53701
> 80276/53701
> 84960/53701
> 89920/53701
> 95580/53701
> 100544/53701
> 2

But I still don't think either one of us has answered Aaron's 
original question:
http://groups.yahoo.com/group/tuning/message/61731
which I repeat here:

<<  what is the simplest possible 12-note temperament where all 24 
major and minor triads have rationally proportional beating? 
here 'simplest' means that the brats (beat ratios for the un-
initiated) are the lowest numbers in the numerator and denominator 
that they can be..... >>

I also assume the following additional requirements:

That it be a *circulating* temperament, and
That it should, at the very least, be reasonably useful from a 
musical standpoint -- or should I go beyond that by insisting that it 
should sound good (just so we don't overlook the most important point 
of this exercise)?

--George

From: wallyesterpaulrus (2005-11-01)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Dave Keenan" <d.keenan@b...> 
wrote:
> > >
> > > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > > > I'm still not sure why this is supposed to be desirable, other
> > > > than it makes creating bearing plans much easier.
> > > > 
> > > > -Carl
> > > 
> > > Small whole-number ratios between the beat rates of the 
> different 
> > > intervals making up a chord? I think it has a special sound. I 
> call 
> > > it "second-order JI".  La Monte Young's Dream House depends on 
> it.
> > 
> > Is that true? 
> 
> Maybe not. But it seemed to be of major significance in the 
software 
> simulation. I realise there's all kinds of other stuff happening in 
> the _real_ Dream House.
> 
> > Why isn't that first-order JI,
> 
> Because (if you'll allow me a little hyperbole) beating is what 
it's 
> all about.
>
> > and why is the lack of 
> > any upper partials that might beat not a problem?
> 
> When you have multiple sine-waves phase-locked to exact ratios you 
> can think of the whole thing as a single note of a strange timbre 

Sounds like an aspect of first-order JI, or maybe "zeroth order JI" 
where there's only one note with its harmonics. Isn't that very 
different from what "second-order JI" is supposed to be about -- 
slightly mistuned intervals with equal rates of beating?

> or 
> you can group the frequencies in various ways and think of each 
> group as a note with partials (and possibly a missing fundamentals).

Can you show me these groups please?

From: Ozan Yarman (2005-11-01)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

They both sound delightful, because of proportional beating. A is more energetic, B is calmer.

  ----- Original Message ----- 
  From: Carl Lumma 
  To: [email protected] 
  Sent: 01 Kasım 2005 Salı 21:05 
  Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament


  > > http://lumma.org/tuning/A.wav
  > > http://lumma.org/tuning/B.wav
  > > (1MB total)
  > >
  > > Anybody care to guess which is which?  Anybody feel one
  > > sounds better than the other?
  > 
  > I prefer B.

  Thanks for weighing in!

  -Carl

From: Carl Lumma (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

>   > > http://lumma.org/tuning/A.wav
>   > > http://lumma.org/tuning/B.wav
>   > > (1MB total)
>   > >
>   > > Anybody care to guess which is which?  Anybody feel one
>   > > sounds better than the other?
>   > 
>   > I prefer B.
>>
>>   Thanks for weighing in!
>
> They both sound delightful, because of proportional beating.
> A is more energetic, B is calmer.

Thanks Ozan.  I'll wait for Dave and others to reply before
discussing it further.

-Carl

From: Gene Ward Smith (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> Thanks Ozan.  I'll wait for Dave and others to reply before
> discussing it further.

You know, if these weren't wav files and it was clear what you wanted
more people might play.

From: Carl Lumma (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > Thanks Ozan.  I'll wait for Dave and others to reply before
> > discussing it further.
> 
> You know, if these weren't wav files and it was clear what you
> wanted more people might play.

Why are WAVs bad?  They're fairly small.

I don't want anything other than comments.  How do they sound?
Can you identify which one is which?

-Carl

From: Yahya Abdal-Aziz (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

On Wed, 02 Nov 2005, "Carl Lumma" wrote: 
> 
> > > > http://lumma.org/tuning/A.wav
> > > > http://lumma.org/tuning/B.wav
> > > > (1MB total)
> > > >
> > > > Anybody care to guess which is which?  Anybody feel one
> > > > sounds better than the other?
> > > 
> > > I prefer B.
> > >
> > >   Thanks for weighing in!
> >
> > They both sound delightful, because of proportional beating.
> > A is more energetic, B is calmer.
> 
> Thanks Ozan.  I'll wait for Dave and others to reply before
> discussing it further.
 

Hi Carl,

I prefer B.  A has a much steadier beat, and I find it annoying.
B has a sound that seems to continue evolving throughout, much
like that of a natural instrument, or a group of them.

Regards,
Yahya

-- 
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.362 / Virus Database: 267.12.7/154 - Release Date: 1/11/05

From: Dave Keenan (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> I get...
> 
> 766.38196601125010515179541316563
> 639.38196601125010515179541316563
> 512.0
> 
> ...Hz. for the messy chord.

Correct.

> But hrm, I don't see any brats
> of phi with "show /relative beats" in Scala.  Isn't that what
> you intended?

That would have been nice, but I couldn't figure out how to get it 
so I was satisfied with them being other noble numbers.

The beat ratios are somewhere between 3:4:5 and 4:5:7 -- more like 
7:9:12. I think these are complex enough to not to count as "nearly 
sychronised".

> Here are the files:
> 
> http://lumma.org/tuning/A.wav
> http://lumma.org/tuning/B.wav
> (1MB total)
> 
> Anybody care to guess which is which?  Anybody feel one
> sounds better than the other?

Thanks for doing this, Carl. I now think my earlier theory on beat 
rates is crap, as I would have realised if I'd (before now) set up a 
spreadsheet to look at the waveforms and their envelopes, even 
without listening to it.

There is a lot more to it. The waveform used for the single notes 
matters a lot (I used sawtooths, what did you use Carl?). And the 
initial phases of the three notes matters. If you choose the right 
initial phases and relative amplitudes for the three notes you may 
actually be able to get the beats to cancel each other out, or 
nearly so.

I think A has the 1:1:1 rates and B has those with no simple ratio. 
But I think B sound more restful. It simply sounds like it has 
slower beats (maybe 1.5 times slower) and a lower beat amplitude.

-- Dave Keenan

From: Herman Miller (2005-11-02)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Carl Lumma wrote:

> http://lumma.org/tuning/A.wav
> http://lumma.org/tuning/B.wav
> (1MB total)
> 
> Anybody care to guess which is which?  Anybody feel one
> sounds better than the other?

I prefer B with these isolated chords; the beating in A is distracting. 
It could be an interesting effect in a different context, though. But I 
think that B sounds better to me for much the same reason that I prefer 
tempered octaves for electronic sounds: the varied beating adds an extra 
richness to tones that would otherwise sound static.

From: Magnus Jonsson (2005-11-02)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

This is exactly the same reason I had for preferring B.

On Wed, 2 Nov 2005, Yahya Abdal-Aziz wrote:

> Hi Carl,
>
> I prefer B.  A has a much steadier beat, and I find it annoying.
>
> B has a sound that seems to continue evolving throughout, much
> like that of a natural instrument, or a group of them.
>
> Regards,
> Yahya
>
>

From: Carl Lumma (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> Carl Lumma wrote:
> > So I think my analysis was right.  If you render a performance
> > with this, we can see, though:
> 
> http://home.comcast.net/~teamouse/porcupine-absynth-blackwood-
> worse.mp3
> 
> Compare with:
> http://home.comcast.net/~teamouse/porcupine-absynth-blackwood.mp3
> http://home.comcast.net/~teamouse/porcupine-absynth.mp3 (15-ET)

It sounds worse alright, but see Gene's post that these off
keys are beat-synched.  So we have to turn somewhere else to
see if that's really helping.

-Carl

From: Carl Lumma (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > > So I think my analysis was right.  If you render a performance
> > > with this, we can see, though:
> > 
> > http://home.comcast.net/~teamouse/porcupine-absynth-blackwood-
> > worse.mp3
> > 
> > Compare with:
> > http://home.comcast.net/~teamouse/porcupine-absynth-blackwood.mp3
> > http://home.comcast.net/~teamouse/porcupine-absynth.mp3 (15-ET)
> 
> It sounds worse alright, but see Gene's post that these off
> keys are beat-synched.  So we have to turn somewhere else to
> see if that's really helping.

You'd think I'd be in danger of wearing out the piece with all
this testing.  But it's such a damn fine piece of music, it
hasn't happened yet!

-Carl

From: Carl Lumma (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > But hrm, I don't see any brats of phi with "show /relative
> > beats" in Scala.  Isn't that what you intended?
> 
> That would have been nice, but I couldn't figure out how to
> get it so I was satisfied with them being other noble numbers.

According to Gene, the M3/m3 brat is

brat = (6t - 5f)/(4t - 5)

And Phi is

(sqrt(5) + 1)/2

Using your M3

t = 639/512 = 383.60655 cents

That gives me a 5th of

f = sqrt(5)/1280 - 479/320 = 696.32288 cents

ignoring a sign (which I think is ok).  Is this right?

> The beat ratios are somewhere between 3:4:5 and 4:5:7 -- more
> like 7:9:12. I think these are complex enough to not to count
> as "nearly sychronised".

Agreed.

> > Here are the files:
> > 
> > http://lumma.org/tuning/A.wav
> > http://lumma.org/tuning/B.wav
> > (1MB total)
> > 
> > Anybody care to guess which is which?  Anybody feel one
> > sounds better than the other?
> 
> Thanks for doing this, Carl. I now think my earlier theory on
> beat rates is crap, as I would have realised if I'd (before now)
> set up a spreadsheet to look at the waveforms and their
> envelopes, even without listening to it.
> 
> There is a lot more to it. The waveform used for the single
> notes matters a lot (I used sawtooths, what did you use Carl?).

The least-jarring thing with decent harmonic content that
I could coax out of Cool Edit.  Which turned out to be the
"inverse squared sine" waveform, at, I believe, equal power
and phase for all three notes, with short fades in and out
added.

> I think A has the 1:1:1 rates and B has those with no simple
> ratio.

That's right.

> But I think B sound more restful. It simply sounds like
> it has slower beats (maybe 1.5 times slower) and a lower beat
> amplitude.

I'm not sure about amplitude, but slower beat rate is the
major difference to my ear.  Respondants so far included
Magnus, Herman, Yahya, and yourself.  All preferred B for
similar reasons.  But my wife prefers A.  Go figure.  :)

-Carl

From: Mark (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> >   > > http://lumma.org/tuning/A.wav
> >   > > http://lumma.org/tuning/B.wav
> >   > > (1MB total)
> >   > >
I prefer B too. The beating of A is more active and unsettling. If used 
with more movement perhaps it could be got away with, but the sensation 
of B is much more calm. 

I always like these interesting little tests - it just shows how 
sensitive hearing really is.

From: Gene Ward Smith (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> t = 639/512 = 383.60655 cents
> 
> That gives me a 5th of
> 
> f = sqrt(5)/1280 - 479/320 = 696.32288 cents
> 
> ignoring a sign (which I think is ok).  Is this right?

No, the fifth has to be a positive number! Change the sign, and you
get a brat of -phi.

From: Carl Lumma (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > t = 639/512 = 383.60655 cents
> > 
> > That gives me a 5th of
> > 
> > f = sqrt(5)/1280 - 479/320 = 696.32288 cents
> > 
> > ignoring a sign (which I think is ok).  Is this right?
> 
> No, the fifth has to be a positive number! Change the sign,
> and you get a brat of -phi.

Maybe this question about negative brats will help:
Does a brat of -1.5 still means one third beats thrice
for every two beatings of the other?

And, what's the fifth that gives a +phi brat with
the 639/512 Maj 3rd?

-Carl

From: Gene Ward Smith (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> Maybe this question about negative brats will help:
> Does a brat of -1.5 still means one third beats thrice
> for every two beatings of the other?

Right.

> And, what's the fifth that gives a +phi brat with
> the 639/512 Maj 3rd?

(1918+sqrt(5))/1280, about a fifth of a cent sharp.

From: Carl Lumma (2005-11-02)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > Maybe this question about negative brats will help:
> > Does a brat of -1.5 still means one third beats thrice
> > for every two beatings of the other?
> 
> Right.

Then I think my -phi triad will have the desired effect,
and my statement about it being ok to ignore the sign
was right.

> > And, what's the fifth that gives a +phi brat with
> > the 639/512 Maj 3rd?
> 
> (1918+sqrt(5))/1280, about a fifth of a cent sharp.

Thanks.  I'll try both.  But my triad is closer tuning-wise
to Dave's synch triad, and so it might make a better
comparison.

-Carl

From: Carl Lumma (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

Right, so Dave's original triad of
512:639:766 Hz.
is still at
http://lumma.org/tuning/A.wav
Gene, can you confirm if this is what you
would call "synch tuned"?

Now we can compare with this triad
512:639:765.505572809 Hz.
at
http://lumma.org/tuning/B2.wav
whose major and minor third beat rates are
related by the golden mean, phi.  If the
simple-ratio beating theory were right, we'd
expect this triad to sound very bad indeed,
since in some sense phi is the least rational
number.

Last but not least, check out
512:639:768.094427191 Hz.
at
http://lumma.org/tuning/B3.wav
which is the other phi-beating triad based
on Dave's major 3rd.  It's fifth is
farther from the Dave's than B2.wav's.

Comments?

For Gene: all four chords as mp3s, in a zip
file (295KB)...
http://lumma.org/tuning/brats3.zip

Interestingly, the quality loss of A.mp3, even
with LAME's --preset extreme encoding, is
immediately noticeable to me (with headphones).

I played with a parameter in Cool Edit called
"Start Phase" (in degrees).  It didn't have any
audible effect.

-Carl

From: Carl Lumma (2005-11-03)
Subject: re: brats test!

> Comments?
//
> I played with a parameter in Cool Edit called
> "Start Phase" (in degrees).  It didn't have any
> audible effect.

For me, the most important difference between these
chords is beat rate (slower is better).

I know how to calculate beat rates for dyads (assuming
harmonic timbres).  But how to calculate them
for triads....

Is the fastest beat the thing we hear?  Or is it
the most prominent beat that sticks out the most?

I need to do more listening, but it seems like Dave's
"messy" triad (B.wav) is my favorite.

My guess is that brats are not the right formulation.
The beat rate of the 5th also needs to be considered.
And the desideratum seems to be to stagger them as
much as possible (in other words, to *de*rationalize
them).

I worried earlier that the intervals in synch triads
would beat all together, and thus make the beats
louder.  Paul E. said that in real life, phase
differences would make this unlikely.

Cool Edit doesn't seem to be giving me very good
control over phase.  Dave, care to make your spreadsheet
public?

-Carl

From: Magnus Jonsson (2005-11-03)
Subject: Re: [tuning] brats test! (was Re: the simplest pan-proportionally beating...)

In order from best to worst sounding:

* B3 (nice, emphasis moves between harmonics in a lovely way. Slightly 
weird attack, i guess some cancellation happened at just the wrong 
moment.)
* B from the original question (similar to B2, but slower beating)
* B2 (wawawawawa, static.)
* A (too rhythmic, as if trying to pull me here and there. Sounds 
forced...)

On Thu, 3 Nov 2005, Carl Lumma wrote:

> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav
> Gene, can you confirm if this is what you
> would call "synch tuned"?
>
> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi.  If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.
>
> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd.  It's fifth is
> farther from the Dave's than B2.wav's.
>
> Comments?
>
> For Gene: all four chords as mp3s, in a zip
> file (295KB)...
> http://lumma.org/tuning/brats3.zip
>
> Interestingly, the quality loss of A.mp3, even
> with LAME's --preset extreme encoding, is
> immediately noticeable to me (with headphones).
>
> I played with a parameter in Cool Edit called
> "Start Phase" (in degrees).  It didn't have any
> audible effect.
>
> -Carl
>
>
>
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
>  [email protected] - join the tuning group.
>  [email protected] - leave the group.
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>  [email protected] - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>
>
>
>
>

From: Herman Miller (2005-11-03)
Subject: Re: [tuning] brats test! (was Re: the simplest pan-proportionally beating...)

Carl Lumma wrote:

> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi.  If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.

I think this one sounds worse than the others, but mainly because the 
beats seem faster.

> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd.  It's fifth is
> farther from the Dave's than B2.wav's.

This one's nice and smooth. The faster beats are in a higher register 
where they don't stand out so badly.

From: Gene Ward Smith (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> Gene, can you confirm if this is what you
> would call "synch tuned"?

Sounds that way, and it ought to be.

 If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.

It seems to me that is a ridiculous conclusion. Certainly, I can't
recall anyone claiming such a thing. Clearly however the kind of
beating sounds different in the two cases.

Is this what all this has been in aid of? Why did you pick these
particlar chords?

> For Gene: all four chords as mp3s, in a zip
> file (295KB)...
> http://lumma.org/tuning/brats3.zip

I only see three chords.

From: Dave Keenan (2005-11-03)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > Comments?
> //
> > I played with a parameter in Cool Edit called
> > "Start Phase" (in degrees).  It didn't have any
> > audible effect.

There should be separate start phases for the three notes. It should 
only make a difference when the beats are in simple rational ratios. 
In that case some phase combinations will give shallower beats as 
beats partly cancel each other.

> For me, the most important difference between these
> chords is beat rate (slower is better).

Right, but beat depth matters too.

> I know how to calculate beat rates for dyads (assuming
> harmonic timbres).  But how to calculate them
> for triads....

It seems the best way is just to simulate them. 

> Is the fastest beat the thing we hear?  Or is it
> the most prominent beat that sticks out the most?

It doesn't seem to be any of those. It doesn't seem to be simple to 
predict at all.

> Cool Edit doesn't seem to be giving me very good
> control over phase.  Dave, care to make your spreadsheet
> public?

Hopefully, by the time you read this I will have uploaded it to:
http://launch.groups.yahoo.com/group/tuning_files/files/Keenan/beats.
xls.zip

-- Dave Keenan

From: Dave Keenan (2005-11-03)
Subject: Re: brats test!

I wrote:
> Hopefully, by the time you read this I will have uploaded it to:
> http://launch.groups.yahoo.com/group/tuning_files/files/Keenan/beats.
> xls.zip

Nope. It seems tuning_files is too full. I deleted some of my old 
stuff but still couldn't upload this (1.9 MB). Maybe someone who has 
_lots_ of files stored there could delete some old stuff.

But in any case, thanks to my friend Andrew Smith, I've put it up here:
http://asmith.id.au/~dkeenan/beats.xls.zip

-- Dave Keenan

From: monz (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav
> Gene, can you confirm if this is what you
> would call "synch tuned"?
> 
> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi.  If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.
> 
> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd.  It's fifth is
> farther from the Dave's than B2.wav's.
> 
> Comments?


I don't understand how any of you guys can objectively
say that any of these are "better" or "worse". ?

To my ears, they all exhibit different beat patterns,
which are simply different rhythms, none especially
better or worse than the others.


-monz
http://tonalsoft.com
Tonescape microtonal music software

From: monz (2005-11-03)
Subject: new Yahoo group to hold our files: tuning_files2

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:

> It seems tuning_files is too full. I deleted some of my old 
> stuff but still couldn't upload this (1.9 MB). Maybe someone
> who has _lots_ of files stored there could delete some old stuff.


Don't delete old stuff!

I solved the problem the simple way -- created a new Yahoo
group!

http://launch.groups.yahoo.com/group/tuning_files2

I've already set up a bunch of folders in the "Files" section
of that group, for the people who generally upload the most
files.

*Please*, folks ... if you don't already have a folder and
want to upload files, create a folder for your own stuff!
Help to keep things organized ... it's bad enough that now
we need to have 3 separate Yahoo groups for our files!



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Gene Ward Smith (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "monz" <monz@t...> wrote:

> I don't understand how any of you guys can objectively
> say that any of these are "better" or "worse". ?
> 
> To my ears, they all exhibit different beat patterns,
> which are simply different rhythms, none especially
> better or worse than the others.

Better or worse isn't the relevant question. I'd phrase it as more
organized vs less organized. The synch beating example beats
coherently, which someone may or may not like the sound of. But I
think it can be helpful in making the ear think it *ought* to sound
like this. It has some of the characteristics of JI, without being JI.

From: Gene Ward Smith (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav

Of course, for circulating 12-note scales we've been discussing brats
of 2 and 4, not -1. We can get a brat of 2 from, for example, 

2560:3205:3838 = 512:641:767.6

and of 4 from, for example, 

512:641:766

From: Carl Lumma (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> > If the simple-ratio beating theory were right, we'd
> > expect this triad to sound very bad indeed,
> > since in some sense phi is the least rational
> > number.
> 
> It seems to me that is a ridiculous conclusion.

Why?

> Certainly, I can't recall anyone claiming such a thing.

The claim that simple-ratio beat rates sound better has
been made by Bob Wendell, among several others.  Including
you in the blackbeat15.scl thread.

> Is this what all this has been in aid of? Why did you pick
> these particlar chords?

Dave picked the initial two.  I used his major 3rd to find
the -phi tuning, and you found the phi tuning.

> > For Gene: all four chords as mp3s, in a zip
> > file (295KB)...
> > http://lumma.org/tuning/brats3.zip
> 
> I only see three chords.

A, B, B2 and B3 (.mp3) should all be there.

-Carl

From: wallyesterpaulrus (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:

> If you choose the right 
> initial phases and relative amplitudes for the three notes you may 
> actually be able to get the beats to cancel each other out, or 
> nearly so.

How is this possible? Each of the beatings occurs at a different pitch 
(that is, for a different pair of nearly coincident partials) so I have 
no idea how this could happen here. I'd love to be educated, 
particularly with a demonstration!

From: wallyesterpaulrus (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

I'm curious -- How does a chord specifically designed *not* to have 
proportional beating sound delightful *because* of proportional 
beating?

--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
>
> They both sound delightful, because of proportional beating. A is 
>more energetic, B is calmer.
> 
>   ----- Original Message ----- 
>   From: Carl Lumma 
>   To: [email protected] 
>   Sent: 01 Kasým 2005 Salý 21:05 
>   Subject: [tuning] Re: the simplest pan-proportionally beating 12-
tone temperament
> 
> 
>   > > http://lumma.org/tuning/A.wav
>   > > http://lumma.org/tuning/B.wav
>   > > (1MB total)
>   > >
>   > > Anybody care to guess which is which?  Anybody feel one
>   > > sounds better than the other?
>   > 
>   > I prefer B.
> 
>   Thanks for weighing in!
> 
>   -Carl
>

From: wallyesterpaulrus (2005-11-03)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> I worried earlier that the intervals in synch triads
> would beat all together, and thus make the beats
> louder.  Paul E. said that in real life, phase
> differences would make this unlikely.

No you must be thinking of what I said on some other topic. Here, all 
the beats occur at different pitches, so relative phase should have no 
effect on the individual or overall loudness of the beats. I think 
Dave's all wet on this, but want to see his response.

From: Ozan Yarman (2005-11-03)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Maybe I have misunderstood this from the beginning considering the subject, in that case I apologize Paul. Maybe Carl will be kind enough to remind me what the brats were.

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 03 Kasım 2005 Perşembe 21:27 
  Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament


  I'm curious -- How does a chord specifically designed *not* to have 
  proportional beating sound delightful *because* of proportional 
  beating?

  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  >
  > They both sound delightful, because of proportional beating. A is 
  >more energetic, B is calmer.
  >

From: wallyesterpaulrus (2005-11-03)
Subject: Re: brats test!

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> >
> > > Comments?
> > //
> > > I played with a parameter in Cool Edit called
> > > "Start Phase" (in degrees).  It didn't have any
> > > audible effect.
> 
> There should be separate start phases for the three notes. It 
should 
> only make a difference when the beats are in simple rational 
ratios. 
> In that case some phase combinations will give shallower beats as 
> beats partly cancel each other.

I still think this is impossible. Please prove me wrong.

>http://launch.groups.yahoo.com/group/tuning_files/files/Keenan/beats.
> xls.zip

The requested file or directory is not found on the server.

From: wallyesterpaulrus (2005-11-03)
Subject: Re: brats test!

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> I wrote:
> > Hopefully, by the time you read this I will have uploaded it to:
> > 
http://launch.groups.yahoo.com/group/tuning_files/files/Keenan/beats.
> > xls.zip
> 
> Nope. It seems tuning_files is too full. I deleted some of my old 
> stuff but still couldn't upload this (1.9 MB). Maybe someone who 
has 
> _lots_ of files stored there could delete some old stuff.
> 
> But in any case, thanks to my friend Andrew Smith, I've put it up 
here:
> http://asmith.id.au/~dkeenan/beats.xls.zip
> 
> -- Dave Keenan

Can you explain the calculations in this spreadsheet and what you're 
trying to represent with them? The graph makes no sense to me right 
now as a depiction of any audible quantity; I'd love to be filled in.

From: wallyesterpaulrus (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

There is no need to apologize. It's just that one chord was designed 
to be proportionally beating; the other was designed not to be 
proportionally beating. So I think it's clear from people's reactions 
here that proportional beating is not at all necessary for, or even a 
significant contributor to, the delightful sound of a slightly 
detuned consonant triad.


--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
>
> Maybe I have misunderstood this from the beginning considering the 
subject, in that case I apologize Paul. Maybe Carl will be kind 
enough to remind me what the brats were.
> 
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 03 Kasým 2005 Perþembe 21:27 
>   Subject: [tuning] Re: the simplest pan-proportionally beating 12-
tone temperament
> 
> 
>   I'm curious -- How does a chord specifically designed *not* to 
have 
>   proportional beating sound delightful *because* of proportional 
>   beating?
> 
>   --- In [email protected], "Ozan Yarman" <ozanyarman@s...> 
wrote:
>   >
>   > They both sound delightful, because of proportional beating. A 
is 
>   >more energetic, B is calmer.
>   >
>

From: Tom Dent (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> Right, so Dave's original triad of
> 512:639:766 Hz.
> is still at
> http://lumma.org/tuning/A.wav
> Gene, can you confirm if this is what you
> would call "synch tuned"?
> 
> Now we can compare with this triad
> 512:639:765.505572809 Hz.
> at
> http://lumma.org/tuning/B2.wav
> whose major and minor third beat rates are
> related by the golden mean, phi.  If the
> simple-ratio beating theory were right, we'd
> expect this triad to sound very bad indeed,
> since in some sense phi is the least rational
> number.
> 
> Last but not least, check out
> 512:639:768.094427191 Hz.
> at
> http://lumma.org/tuning/B3.wav
> which is the other phi-beating triad based
> on Dave's major 3rd.  It's fifth is
> farther from the Dave's than B2.wav's.
> 
> Comments?
> 

B3 is amazingly pleasant. Of course it is the nearest to pure, but
just hovering shy of exact purity. B next, then B2 (more like 1/5
comma meantone, still pleasant) then A last.

~~~T~~~

From: Ozan Yarman (2005-11-03)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

In my opinion, you are too impatient to reach such conclusions. You have not even ascertained objective impartiality in this experiment, nor evaluated all the necessary inputs. Now I see I have been careless and rash in listening to the beats due to the poor quality of my PC speakers. A is non-proportional beating, while B is proportional, and thus better compared to A in regards to harmonic, if not melodic, consonance.

I feel that prop-brats are significant indeed.

Cordially,
Ozan

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 03 Kasım 2005 Perşembe 21:55 
  Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament


  There is no need to apologize. It's just that one chord was designed 
  to be proportionally beating; the other was designed not to be 
  proportionally beating. So I think it's clear from people's reactions 
  here that proportional beating is not at all necessary for, or even a 
  significant contributor to, the delightful sound of a slightly 
  detuned consonant triad.


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  >
  > Maybe I have misunderstood this from the beginning considering the 
  subject, in that case I apologize Paul. Maybe Carl will be kind 
  enough to remind me what the brats were.

From: Carl Lumma (2005-11-03)
Subject: Re: brats test!

> Nope. It seems tuning_files is too full. I deleted some of my old 
> stuff but still couldn't upload this (1.9 MB). Maybe someone who
> has  _lots_ of files stored there could delete some old stuff.
> 
> But in any case, thanks to my friend Andrew Smith, I've put it up
> here: http://asmith.id.au/~dkeenan/beats.xls.zip
> 
> -- Dave Keenan

Thanks!

These groups only get something like 20MB of storage, which is
really lame.  But given that web hosting is so cheap ($2/month
for a 200MB at one provider I just checked), I can't see why
it's a problem for so many folks.

-Carl

From: Carl Lumma (2005-11-03)
Subject: Re: brats test!

> > I worried earlier that the intervals in synch triads
> > would beat all together, and thus make the beats
> > louder.  Paul E. said that in real life, phase
> > differences would make this unlikely.
> 
> No you must be thinking of what I said on some other topic.
> Here, all the beats occur at different pitches, so relative
> phase should have no effect on the individual or overall
> loudness of the beats.

Oops, that was George Secor.  Sorry!  (Any comments you have
on the msg below would still be appreciated.)

> > Hiya George (and Ozan),
> > 
> > Do you find you like the triads where all the intervals beat
> > at the same rate?  I would think it would be better to stagger
> > the beats, so that they do not result in large amplitude
> > changes.  Or does this not happen?
> > 
> > -Carl
//
> even in the 5/17-comma temperament with its 1:1:1 beat
> ratio (which I previously incorporated in the best keys
> of my 19-tone well-temperament -- a tuning I currently
> have stored in my Scalatron).  I expect that one would
> have to deliberately adjust the phase relationships
> between the tones in order for the beats to reinforce 
> one another so as to produce an amplitude large enough
> to be judged objectionable.

-Carl

From: Carl Lumma (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> Maybe I have misunderstood this from the beginning considering
> the subject, in that case I apologize Paul. Maybe Carl will be
> kind enough to remind me what the brats were.

A brat is a Beat RATio, usually defined for approximate
4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
beat rate.

-Carl

From: Carl Lumma (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Tom Dent" <stringph@g...> wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> >
> > Right, so Dave's original triad of
> > 512:639:766 Hz.
> > is still at
> > http://lumma.org/tuning/A.wav
> > Gene, can you confirm if this is what you
> > would call "synch tuned"?
> > 
> > Now we can compare with this triad
> > 512:639:765.505572809 Hz.
> > at
> > http://lumma.org/tuning/B2.wav
> > whose major and minor third beat rates are
> > related by the golden mean, phi.  If the
> > simple-ratio beating theory were right, we'd
> > expect this triad to sound very bad indeed,
> > since in some sense phi is the least rational
> > number.
> > 
> > Last but not least, check out
> > 512:639:768.094427191 Hz.
> > at
> > http://lumma.org/tuning/B3.wav
> > which is the other phi-beating triad based
> > on Dave's major 3rd.  It's fifth is
> > farther from the Dave's than B2.wav's.
> > 
> > Comments?
> > 
> 
> B3 is amazingly pleasant. Of course it is the nearest to pure, but
> just hovering shy of exact purity. B next, then B2 (more like 1/5
> comma meantone, still pleasant) then A last.
> 
> ~~~T~~~

Thanks for listening, Tom.  That's very close to what I had,
except I had B3 and B switched.

-Carl

From: Carl Lumma (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> In my opinion, you are too impatient to reach such conclusions.
> You have not even ascertained objective impartiality in this
> experiment, nor evaluated all the necessary inputs. Now I see
> I have been careless and rash in listening to the beats due to
> the poor quality of my PC speakers. A is non-proportional
> beating, while B is proportional, and thus better compared to
> A in regards to harmonic, if not melodic, consonance.
> 
> I feel that prop-brats are significant indeed.

Actually Ozan, it's the reverse.  B is the non-proportional
chord, yet everyone (including you) seems to prefer it.  That's
what's surprising here, and we're still trying to figure
out what's going on.

-Carl

From: Carl Lumma (2005-11-03)
Subject: re: brats test!

> > Right, so Dave's original triad of
> > 512:639:766 Hz.
> > is still at
> > http://lumma.org/tuning/A.wav
> 
> Of course, for circulating 12-note scales we've been discussing brats
> of 2 and 4, not -1. We can get a brat of 2 from, for example, 
> 
> 2560:3205:3838 = 512:641:767.6
> 
> and of 4 from, for example, 
> 
> 512:641:766

Here they are:

http://lumma.org/tuning/2.wav
http://lumma.org/tuning/4.wav

The 'spikiness' of the waveform display in Cool Edit of
these chords seems to be a very good guide (as you'd expect)
of how prominent the beats are.  What are some good
'spikiness' measures of graphs?

-Carl

From: Ozan Yarman (2005-11-03)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

Yes, I meant the brats of the chord examples you gave the links to.
  ----- Original Message ----- 
  From: Carl Lumma 
  To: [email protected] 
  Sent: 03 Kasım 2005 Perşembe 23:20 
  Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament


  > Maybe I have misunderstood this from the beginning considering
  > the subject, in that case I apologize Paul. Maybe Carl will be
  > kind enough to remind me what the brats were.

  A brat is a Beat RATio, usually defined for approximate
  4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
  beat rate.

  -Carl

From: Magnus Jonsson (2005-11-03)
Subject: re: [tuning] re: brats test!

On Thu, 3 Nov 2005, Carl Lumma wrote:

> The 'spikiness' of the waveform display in Cool Edit of
> these chords seems to be a very good guide (as you'd expect)
> of how prominent the beats are.  What are some good
> 'spikiness' measures of graphs?
>
> -Carl

I have an idea which might work:

1) Filter out all frequencies that we can't hear.
2) Square each sample, to get the momental energy output flow. If the 
signal has prominent beating, this energy output flow will probably be 
very uneven.
3) Lowpass filter this energy signal to get rid of energy flow variations 
are too quick for us to consider as beating.
4b) (optional) Highpass filter to get rid of variations that are too slow 
to be meaningful.
5) Calculate the standard deviation as a measurement of the unevenness of 
energy flow.

Here is the signal flow:

input -> bandpass -> square each sample -> lowpass(or bandpass) -> 
estimate standard variation

Note that I haven't tried this out in practice so I can't give any 
guarantees about it :)

- Magnus Jonsson

From: wallyesterpaulrus (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

Hate to have to say it, but you're saying (along with most other 
listeners here) that you yourself prefer the non-proportionally-
beating triad B (which you erroneously say is proportionally beating) 
to the proportionally-beating triad A (which you erroneously say is 
non-proportionally beating). So I think your conclusion is unfounded.


--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
>
> In my opinion, you are too impatient to reach such conclusions. You 
>have not even ascertained objective impartiality in this experiment, 
>nor evaluated all the necessary inputs. Now I see I have been 
>careless and rash in listening to the beats due to the poor quality 
>of my PC speakers. A is non-proportional beating, while B is 
>proportional, and thus better compared to A in regards to harmonic, 
>if not melodic, consonance.
> 
> I feel that prop-brats are significant indeed.
> 
> Cordially,
> Ozan
> 
>   ----- Original Message ----- 
>   From: wallyesterpaulrus 
>   To: [email protected] 
>   Sent: 03 Kasým 2005 Perþembe 21:55 
>   Subject: [tuning] Re: the simplest pan-proportionally beating 12-
tone temperament
> 
> 
>   There is no need to apologize. It's just that one chord was 
designed 
>   to be proportionally beating; the other was designed not to be 
>   proportionally beating. So I think it's clear from people's 
reactions 
>   here that proportional beating is not at all necessary for, or 
even a 
>   significant contributor to, the delightful sound of a slightly 
>   detuned consonant triad.
> 
> 
>   --- In [email protected], "Ozan Yarman" <ozanyarman@s...> 
wrote:
>   >
>   > Maybe I have misunderstood this from the beginning considering 
the 
>   subject, in that case I apologize Paul. Maybe Carl will be kind 
>   enough to remind me what the brats were.
>

From: Ozan Yarman (2005-11-03)
Subject: Re: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament

In that case I retract my statement. Obviously I am confused as to what I am talking about. Please disregard my words.

  ----- Original Message ----- 
  From: wallyesterpaulrus 
  To: [email protected] 
  Sent: 04 Kasım 2005 Cuma 0:47 
  Subject: [tuning] Re: the simplest pan-proportionally beating 12-tone temperament


  Hate to have to say it, but you're saying (along with most other 
  listeners here) that you yourself prefer the non-proportionally-
  beating triad B (which you erroneously say is proportionally beating) 
  to the proportionally-beating triad A (which you erroneously say is 
  non-proportionally beating). So I think your conclusion is unfounded.


  --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
  >
  > In my opinion, you are too impatient to reach such conclusions. You 
  >have not even ascertained objective impartiality in this experiment, 
  >nor evaluated all the necessary inputs. Now I see I have been 
  >careless and rash in listening to the beats due to the poor quality 
  >of my PC speakers. A is non-proportional beating, while B is 
  >proportional, and thus better compared to A in regards to harmonic, 
  >if not melodic, consonance.
  > 
  > I feel that prop-brats are significant indeed.
  > 
  > Cordially,
  > Ozan
  >

From: wallyesterpaulrus (2005-11-03)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > I worried earlier that the intervals in synch triads
> > > would beat all together, and thus make the beats
> > > louder.  Paul E. said that in real life, phase
> > > differences would make this unlikely.
> > 
> > No you must be thinking of what I said on some other topic.
> > Here, all the beats occur at different pitches, so relative
> > phase should have no effect on the individual or overall
> > loudness of the beats.
> 
> Oops, that was George Secor.  Sorry!  (Any comments you have
> on the msg below would still be appreciated.)
> 
> > > Hiya George (and Ozan),
> > > 
> > > Do you find you like the triads where all the intervals beat
> > > at the same rate?  I would think it would be better to stagger
> > > the beats, so that they do not result in large amplitude
> > > changes.  Or does this not happen?
> > > 
> > > -Carl
> //
> > even in the 5/17-comma temperament with its 1:1:1 beat
> > ratio (which I previously incorporated in the best keys
> > of my 19-tone well-temperament -- a tuning I currently
> > have stored in my Scalatron).  I expect that one would
> > have to deliberately adjust the phase relationships
> > between the tones in order for the beats to reinforce 
> > one another so as to produce an amplitude large enough
> > to be judged objectionable.
> 
> -Carl

This statement (of George's) makes sense -- of course you can get the 
beatings to be in phase with one another so that they all hit maximum 
amplitude in synch. But they'll each sound equally loud regardless of 
whether they're in phase or not. There's no way to get the beats to 
cancel one another out, as Dave Keenan seemingly suggested. And there 
will be plenty of other partials in the sound anyway so the overall 
amplitude of everything will still remain reasonably constant.

From: wallyesterpaulrus (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > In my opinion, you are too impatient to reach such conclusions.
> > You have not even ascertained objective impartiality in this
> > experiment, nor evaluated all the necessary inputs. Now I see
> > I have been careless and rash in listening to the beats due to
> > the poor quality of my PC speakers. A is non-proportional
> > beating, while B is proportional, and thus better compared to
> > A in regards to harmonic, if not melodic, consonance.
> > 
> > I feel that prop-brats are significant indeed.
> 
> Actually Ozan, it's the reverse.  B is the non-proportional
> chord, yet everyone (including you) seems to prefer it.  That's
> what's surprising here, and we're still trying to figure
> out what's going on.
> 
> -Carl

Maybe the beats in A should have been staggered . . . (they still 
wouldn't cancel each other out, but perhaps the effect would be more 
natural-sounding.)

From: Carl Lumma (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > > In my opinion, you are too impatient to reach such conclusions.
> > > You have not even ascertained objective impartiality in this
> > > experiment, nor evaluated all the necessary inputs. Now I see
> > > I have been careless and rash in listening to the beats due to
> > > the poor quality of my PC speakers. A is non-proportional
> > > beating, while B is proportional, and thus better compared to
> > > A in regards to harmonic, if not melodic, consonance.
> > > 
> > > I feel that prop-brats are significant indeed.
> > 
> > Actually Ozan, it's the reverse.  B is the non-proportional
> > chord, yet everyone (including you) seems to prefer it.  That's
> > what's surprising here, and we're still trying to figure
> > out what's going on.
> > 
> > -Carl
> 
> Maybe the beats in A should have been staggered . . . (they
> still wouldn't cancel each other out, but perhaps the effect
> would be more natural-sounding.)

You mean the phases staggered?  That's what I can't seem to
do in Cool Edit.  Dave's spreadsheet looks like it might
produce sound, but I didn't hear any when playing with it.
The waveform graph doesn't seem to change too much when
I play with the phases there, though.

-Carl

From: Dave Keenan (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In [email protected], "monz" <monz@t...> wrote:
> 
> > I don't understand how any of you guys can objectively
> > say that any of these are "better" or "worse". ?
> > 
> > To my ears, they all exhibit different beat patterns,
> > which are simply different rhythms, none especially
> > better or worse than the others.
> 
> Better or worse isn't the relevant question. I'd phrase it as more
> organized vs less organized. The synch beating example beats
> coherently, which someone may or may not like the sound of. But I
> think it can be helpful in making the ear think it *ought* to sound
> like this. It has some of the characteristics of JI, without being 
JI.
>

Monz,

I don't think anyone's claiming "objectivity" for this. Carl asked 
us to say which we preferred, and that's what we've been doing. I 
happen to agree with the majority so far in finding that if your aim 
is to make your temperament sound more Just, then synchronised beat 
ratios may _not_ necessarily be the way to go. Or if it is, then it 
may need careful attention to the relative phases of the notes, to 
minimise the beat depth (this idea hasn't been checked by experiment 
yet).

If you don't have the capability for actual phase locked synchrony 
(or maybe even if you do), then it may be better to have quite the 
opposite of beat synchrony, namely beats whose ratios are noble 
numbers, e.g. of the form (a + b*phi)/(c + d*phi) where a,b,c,d are 
small whole numbers. Or maybe it just doesn't matter either way.

I said earlier that it doesn't seem easy to predict, but I've just 
realised (duh!) that, as Carl suggested, it is the most prominent 
beat that sticks out the most (is that a tautology?).

A single variable that correlates well with the preferences of the 
majority here is simply the _error_in_the_fifth_. It's the beat rate 
of the fifth that dominates, since it involves lower harmonics (2 
and 3), which have greater amplitude.

The logical conclusion of this for meantones is, sad to say, equal 
temperament.

-- Dave Keenan

From: wallyesterpaulrus (2005-11-03)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > > In my opinion, you are too impatient to reach such 
conclusions.
> > > > You have not even ascertained objective impartiality in this
> > > > experiment, nor evaluated all the necessary inputs. Now I see
> > > > I have been careless and rash in listening to the beats due to
> > > > the poor quality of my PC speakers. A is non-proportional
> > > > beating, while B is proportional, and thus better compared to
> > > > A in regards to harmonic, if not melodic, consonance.
> > > > 
> > > > I feel that prop-brats are significant indeed.
> > > 
> > > Actually Ozan, it's the reverse.  B is the non-proportional
> > > chord, yet everyone (including you) seems to prefer it.  That's
> > > what's surprising here, and we're still trying to figure
> > > out what's going on.
> > > 
> > > -Carl
> > 
> > Maybe the beats in A should have been staggered . . . (they
> > still wouldn't cancel each other out, but perhaps the effect
> > would be more natural-sounding.)
> 
> You mean the phases staggered?

Yes.

> That's what I can't seem to
> do in Cool Edit.

If you can stagger the onset times, it's likely the phases will end 
up staggered too.

> Dave's spreadsheet looks like it might
> produce sound, but I didn't hear any when playing with it.
> The waveform graph doesn't seem to change too much when
> I play with the phases there, though.

It does for me, though I'm skeptical that this waveform graph is 
actually showing the waveform of anything meaningful.

From: wallyesterpaulrus (2005-11-03)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:

> A single variable that correlates well with the preferences of the 
> majority here is simply the _error_in_the_fifth_. It's the beat rate 
> of the fifth that dominates, since it involves lower harmonics (2 
> and 3), which have greater amplitude.
> 
> The logical conclusion of this for meantones is, sad to say, equal 
> temperament.

How do you arrive at that conclusion?

From: Gene Ward Smith (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > If the simple-ratio beating theory were right, we'd
> > > expect this triad to sound very bad indeed,
> > > since in some sense phi is the least rational
> > > number.
> > 
> > It seems to me that is a ridiculous conclusion.
> 
> Why?
> 
> > Certainly, I can't recall anyone claiming such a thing.
> 
> The claim that simple-ratio beat rates sound better has
> been made by Bob Wendell, among several others.  

Wendell did not say that non-simple ratios sound very bad indeed.
You've extrapolated wildly from his claims.

Including
> you in the blackbeat15.scl thread.

No, all I said in that thread is that I thought the blackbeat version
was considerably better than the 15edo version. That suggested to me
that synch beating might have proven useful, but of course it wasn't
the only thing which differed.

From: Gene Ward Smith (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> Actually Ozan, it's the reverse.  B is the non-proportional
> chord, yet everyone (including you) seems to prefer it. 

I neither prefer it nor fail to prefer it. I note that the synch
beating sounds quite different than the non-synch beating, producing a
different musical effect.

From: Gene Ward Smith (2005-11-04)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> The 'spikiness' of the waveform display in Cool Edit of
> these chords seems to be a very good guide (as you'd expect)
> of how prominent the beats are.  What are some good
> 'spikiness' measures of graphs?

It seems to me the question is more one of regularity than spikiness.
The differences are striking, and can also be viewed in Audacity,
Sound Forge, and WavePad. Audacity is free, and WavePad is shareware,
so it should be possible for most people to view these, though it
would be nice to learn if one of these programs will produce a visual
file (jpg, etc.)

From: Dave Keenan (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
> 
> > If you choose the right 
> > initial phases and relative amplitudes for the three notes you may 
> > actually be able to get the beats to cancel each other out, or 
> > nearly so.
> 
> How is this possible? Each of the beatings occurs at a different 
pitch 
> (that is, for a different pair of nearly coincident partials) so I 
have 
> no idea how this could happen here. I'd love to be educated, 
> particularly with a demonstration!

There are at least two perceptual aspects to beats
(a) cyclic variation of loudness with time
(b) cyclic variation of timbre with time

I'm saying we could eliminate the first, but not the second.

Here's a very contrived example. 

Use a timbre that has equal amplitudes for the fundamental and all 
harmonics up to the sixth, then nothing after that. And one needs to 
be able to set the start-phase of each harmonic independently.

Consider the sync-beating 512:639:766 Hz chord. This will have the 
following beats between the following four pairs of partials.

4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
8 Hz between 6th harmonic of root and 4th harmonic of fifth
4 Hz between 5th harmonic of root and 4th harmonic of major third
4 Hz between 6th harmonic of major third and 5th harmonic of fifth

By carefully arranging the phases of the partials in the original 
timbre (sorry I'm too lazy to work out the details) it should be 
possible to arrange for the three 4 Hz beats to be 120 degrees out of 
phase and thereby cancel each other and leave only the 8 Hz beat.

This isn't terribly satisfactory since with equal amplitude partials 
that 8 Hz beat will still be quite obvious, so the next step is to 
look at a more normal timbre where the higher partials have lower 
amplitude.

In this case it should be possible to make it so that the beats of the 
major and minor third intervals are in phase with each other and add 
to produce a beat which is the same amplitude as the 4 Hz beat in the 
fifth interval but 180 degrees out of phase with it, and thereby 
cancelling it. We still have the 8 Hz beat, but this time at lower 
amplitude.

If one cheated and knocked out the 6th harmonic of the root and the 
4th harmonic of the fifth completely, then one could have no beating 
at all in the triad (in the sense of loudness variation) even though 
every dyad heard alone has obvious beating. Of course we still expect 
to hear a sort of beating of the triad due to the cyclic timbre 
variation.

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: Re: brats test!

--- In [email protected], "wallyesterpaulrus" wrote
> This statement (of George's) makes sense -- of course you can get 
the 
> beatings to be in phase with one another so that they all hit 
maximum 
> amplitude in synch. But they'll each sound equally loud regardless 
of 
> whether they're in phase or not. There's no way to get the beats 
to 
> cancel one another out, as Dave Keenan seemingly suggested. And 
there 
> will be plenty of other partials in the sound anyway so the 
overall 
> amplitude of everything will still remain reasonably constant.

In my terms, you seem to be saying that even with ordinary 
uncontrived beating, the cyclic loundess variation is relatively 
unimportant and the cyclic timbre variation is mostly what we hear. 
That makes sense, and I probably am all wet. But it wouldn't hurt to 
do the experiment.

So now I suppose I have to come up with exact specifications for 
Carl to synthesise.

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > Maybe I have misunderstood this from the beginning considering
> > the subject, in that case I apologize Paul. Maybe Carl will be
> > kind enough to remind me what the brats were.
> 
> A brat is a Beat RATio, usually defined for approximate
> 4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
> beat rate.

Why use this obfuscatory jargon in the first place? How hard can it be 
to type "beat ratio" and to be clear about exactly which intervals 
you're considering the beat ratio of? [Rhetorical questions]

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> You mean the phases staggered?  That's what I can't seem to
> do in Cool Edit.  Dave's spreadsheet looks like it might
> produce sound, but I didn't hear any when playing with it.

No it doesn't produce sound. It only looks at the loudness 
variation, which is indeed small, as Paul suggested. So maybe he's 
right, that most of what we hear as beats is actually timbre 
variation.

> The waveform graph doesn't seem to change too much when
> I play with the phases there, though.

It only does so with the chord you've called A (actually it's closer 
to a concert-pitch C) ;-), namely 512:639:766, the synced equal 
beats version. But you wont get complete cancellation of loudness 
variation without contriving the relative amplitudes and phases of 
the partials in the original timbre.

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> 
> > Actually Ozan, it's the reverse.  B is the non-proportional
> > chord, yet everyone (including you) seems to prefer it. 
> 
> I neither prefer it nor fail to prefer it. I note that the synch
> beating sounds quite different than the non-synch beating, 
producing a
> different musical effect.
>

Yes. Which is how I originally got into this discussion. By saying I 
thought that synced beats had a special sound, and referring to it 
as "second order JI". But I now think that if you're trying to  
approximate first order JI with a temperament, then it might be 
better to avoid "second order JI". Or it might simply not matter 
much either way and might just show that you should try to minimise 
the beat rate of the loudest harmonics.

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: Re: brats test!

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> > http://asmith.id.au/~dkeenan/beats.xls.zip
> > 
> > -- Dave Keenan
> 
> Can you explain the calculations in this spreadsheet and what you're 
> trying to represent with them? The graph makes no sense to me right 
> now as a depiction of any audible quantity; I'd love to be filled in.

It's intended to show an approximation to the envelope of the sound, 
i.e. the variation of loudness with time. But as you say, that may not 
be anything perceptually significant for the phenomenon of beating of 
complex waveforms.

You talk about the beats happening at different "pitches". That's not 
quite right, since pitch is a perceptual quality that may depend on 
more than one partial. If you'd said "different frequencies" then 
you'd be right, but most people I know don't hear the individual sine-
wave partials which mathematically-speaking make up a sound (unless 
perhaps when one is suddenly added or taken away). They instead hear a 
quality I'm calling timbre. Sorry to be so pedantic.

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
> 
> > A single variable that correlates well with the preferences of the 
> > majority here is simply the _error_in_the_fifth_. It's the beat 
rate 
> > of the fifth that dominates, since it involves lower harmonics (2 
> > and 3), which have greater amplitude.
> > 
> > The logical conclusion of this for meantones is, sad to say, equal 
> > temperament.
> 
> How do you arrive at that conclusion?
>

Simple. It's the meantone with the slowest beating fifths.

I don't consider any fifth-generated linear temperament with a fifth 
wider than 700 cents to be a meantone. We are then into the 
Pythagorean region.

-- Dave Keenan

From: Carl Lumma (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > > Maybe the beats in A should have been staggered . . . (they
> > > still wouldn't cancel each other out, but perhaps the effect
> > > would be more natural-sounding.)
> > 
> > You mean the phases staggered?
> 
> Yes.
> 
> > That's what I can't seem to
> > do in Cool Edit.
> 
> If you can stagger the onset times, it's likely the phases will end 
> up staggered too.

No can do.  Well actually, can do, with 3 tracks and mix
down to mono....

> > Dave's spreadsheet looks like it might
> > produce sound, but I didn't hear any when playing with it.
> > The waveform graph doesn't seem to change too much when
> > I play with the phases there, though.

I was unable to get the period of the biggest humps to change
much by changing the phase.  Can you give me a scenario to
look at?

-Carl

From: Carl Lumma (2005-11-04)
Subject: Re: brats test!

> > The 'spikiness' of the waveform display in Cool Edit of
> > these chords seems to be a very good guide (as you'd expect)
> > of how prominent the beats are.  What are some good
> > 'spikiness' measures of graphs?
> 
> It seems to me the question is more one of regularity than
> spikiness.

Not sure what the difference is...

> The differences are striking, and can also be viewed in
> Audacity, Sound Forge, and WavePad. Audacity is free, and
> WavePad is shareware, so it should be possible for most
> people to view these, though it would be nice to learn if
> one of these programs will produce a visual file (jpg, etc.)

There is such a thing as a screen shot...

-Carl

From: Carl Lumma (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > > Maybe I have misunderstood this from the beginning considering
> > > the subject, in that case I apologize Paul. Maybe Carl will be
> > > kind enough to remind me what the brats were.
> > 
> > A brat is a Beat RATio, usually defined for approximate
> > 4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
> > beat rate.
> 
> Why use this obfuscatory jargon in the first place? How hard can
> it be to type "beat ratio" and to be clear about exactly which
> intervals you're considering the beat ratio of? [Rhetorical
> questions]

It wasn't my idea (though I didn't disapprove).

-Carl

From: Carl Lumma (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> > You mean the phases staggered?  That's what I can't seem to
> > do in Cool Edit.  Dave's spreadsheet looks like it might
> > produce sound, but I didn't hear any when playing with it.
> 
> No it doesn't produce sound. It only looks at the loudness 
> variation, which is indeed small, as Paul suggested. So maybe
> he's right, that most of what we hear as beats is actually
> timbre variation.

I didn't entirely follow your method for canceling beats...
did you address Paul's point that two 4 Hz. beats 180deg.
out of phase still won't cancel unless their 'carriers'
were at the same frequency?  Or is that what you mean by
timbre variation?  (Wait, maybe I'm getting it.)  In any
case, what I hear as prominent beats is very clearly seen
in Cool Edit's (or any other wave editor's, as Gene points
out) waveform display as variations in the total amplitude.
Maybe that's because I haven't synthesized the right
timbers and start phases yet.  But for a general-purpose
tuning theory, this is too specialized.  Might be of
interest to folks like Bill Sethares, though.

-Carl

From: Carl Lumma (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

> Yes. Which is how I originally got into this discussion. By saying I 
> thought that synced beats had a special sound, and referring to it 
> as "second order JI". But I now think that if you're trying to  
> approximate first order JI with a temperament, then it might be 
> better to avoid "second order JI". Or it might simply not matter 
> much either way and might just show that you should try to minimise 
> the beat rate of the loudest harmonics.

I think you're right, Dave.  Getting rid of error is enough of a
problem.  And as you point out, it seems to be a good rule for
minimizing beat rates... and with weighted error, this will tend
to favor the 'loudest harmonics', wouldn't it?

-Carl

From: monz (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

Hi Dave,


--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:

> I don't think anyone's claiming "objectivity" for this.
> Carl asked us to say which we preferred, and that's what
> we've been doing.


OK, then my bad ... guess i wasn't paying enough attention
to the earlier posts in this thread. I thought i saw folks
saying "B2 is the best" etc.


> I happen to agree with the majority so far in finding
> that if your aim is to make your temperament sound more
> Just, then synchronised beat ratios may _not_ necessarily
> be the way to go. Or if it is, then it may need careful
> attention to the relative phases of the notes, to minimise
> the beat depth (this idea hasn't been checked by experiment
> yet).


I too have not done significant experiments in audio-testing
various aspects of phase, but all accounts that i've ever
read about digital synthesis agree that phase differences
are audibly insignificant.
 

> A single variable that correlates well with the preferences
> of the majority here is simply the _error_in_the_fifth_.
> It's the beat rate of the fifth that dominates, since it
> involves lower harmonics (2 and 3), which have greater
> amplitude.


The lower harmonics generally do have greater amplitude in
the timbres of acoustic instruments. But in synthesized
timbres that is not necessarily the case ... i suppose
it must be the case in the timbres Carl used.


 
> The logical conclusion of this for meantones is, 
> sad to say, equal temperament.


By "equal temperament", do you mean 12-edo, or just the
EDO meantones in general?



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Dave Keenan (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "monz" <monz@t...> wrote:
> I too have not done significant experiments in audio-testing
> various aspects of phase, but all accounts that i've ever
> read about digital synthesis agree that phase differences
> are audibly insignificant.

In general yes. But the question is, could they become significant in 
the case of synced beats.

> > The logical conclusion of this for meantones is, 
> > sad to say, equal temperament.
> 
> 
> By "equal temperament", do you mean 12-edo, or just the
> EDO meantones in general?

Oops! Sorry. Yes, I meant 12 equal.

-- Dave Keenan

From: Tom Dent (2005-11-04)
Subject: Beats in a chord do not cancel

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> > How is this possible? Each of the beatings occurs at a different 
> pitch 
> > (that is, for a different pair of nearly coincident partials) so I 
> have 
> > no idea how this could happen here. I'd love to be educated, 
> > particularly with a demonstration!
> 
> There are at least two perceptual aspects to beats
> (a) cyclic variation of loudness with time
> (b) cyclic variation of timbre with time
> 
> I'm saying we could eliminate the first, but not the second.
> 
> Here's a very contrived example. 
> 
> Use a timbre that has equal amplitudes for the fundamental and all 
> harmonics up to the sixth, then nothing after that. And one needs to 
> be able to set the start-phase of each harmonic independently.
> 
> Consider the sync-beating 512:639:766 Hz chord. This will have the 
> following beats between the following four pairs of partials.
> 
> 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> 4 Hz between 5th harmonic of root and 4th harmonic of major third
> 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
> 
> By carefully arranging the phases of the partials in the original 
> timbre (sorry I'm too lazy to work out the details) it should be 
> possible to arrange for the three 4 Hz beats to be 120 degrees out of 
> phase and thereby cancel each other and leave only the 8 Hz beat.
> 
> This isn't terribly satisfactory since with equal amplitude partials 
> that 8 Hz beat will still be quite obvious, so the next step is to 
> look at a more normal timbre where the higher partials have lower 
> amplitude.
> 
> In this case it should be possible to make it so that the beats of the 
> major and minor third intervals are in phase with each other and add 
> to produce a beat which is the same amplitude as the 4 Hz beat in the 
> fifth interval but 180 degrees out of phase with it, and thereby 
> cancelling it. We still have the 8 Hz beat, but this time at lower 
> amplitude.
> 
> If one cheated and knocked out the 6th harmonic of the root and the 
> 4th harmonic of the fifth completely, then one could have no beating 
> at all in the triad (in the sense of loudness variation) even though 
> every dyad heard alone has obvious beating. Of course we still expect 
> to hear a sort of beating of the triad due to the cyclic timbre 
> variation.
> 
> -- Dave Keenan


Well, I still think the beating in timbre would be very much
noticeable. In my experience beating only produces a noticeable
oscillation of loudness at the unison and the octave.

> 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> 4 Hz between 5th harmonic of root and 4th harmonic of major third
> 4 Hz between 6th harmonic of major third and 5th harmonic of fifth

The three different 4Hz beats occur at 3 different overtone pitches
and there is no way they can cancel each other out. To put it another
way, in a close position chord *every* beat is a variation in timbre,
since every beat affects the upper partials, not the fundamental.
Beats could in principle only cancel each other out if they were all
at the same pitch (as in a violin section all playing the 'same' note).

~~~T~~~

From: Tom Dent (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

It rather depends on the timbral characteristics of your instrument.
If it happens to have rather weak 3rd, 6th etc. partials then the
beating of the 5ths is not so prominent and you can get good results
by making them narrower.

It might well be that harpsichord makers deliberately made the 3rd
harmonic weaker so that meantone temperaments would be more successful.
~~~T~~~

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
> > 
> > > A single variable that correlates well with the preferences of the 
> > > majority here is simply the _error_in_the_fifth_. It's the beat 
> rate 
> > > of the fifth that dominates, since it involves lower harmonics (2 
> > > and 3), which have greater amplitude.
> > > 
> > > The logical conclusion of this for meantones is, sad to say, equal 
> > > temperament.
> > 
> > How do you arrive at that conclusion?
> >
> 
> Simple. It's the meantone with the slowest beating fifths.
> 
> I don't consider any fifth-generated linear temperament with a fifth 
> wider than 700 cents to be a meantone. We are then into the 
> Pythagorean region.
> 
> -- Dave Keenan
>

From: Brad Lehman (2005-11-04)
Subject: Re: Beats in a chord do not cancel

> In my experience beating only produces a noticeable
> oscillation of loudness at the unison and the octave.

Gotta install more 1/5 or 1/6 comma temperaments on your harpsichord, 
then, and play a bunch of open major 10ths.  The pulsating effect is 
quite noticeable and lovely.  The playing of 12ths also brings it out.

Brad Lehman

From: Brad Lehman (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> It rather depends on the timbral characteristics of your instrument.
> If it happens to have rather weak 3rd, 6th etc. partials then the
> beating of the 5ths is not so prominent and you can get good results
> by making them narrower.
> 
> It might well be that harpsichord makers deliberately made the 3rd
> harmonic weaker so that meantone temperaments would be more 
successful.


Wait.  Why is a carefully-regulated impure 5th, and its beating, some 
bad thing that would need to be hidden away (deliberately or 
otherwise) to make the sound "more successful"?

In 1/4 comma meantone on harpsichords, as some might argue, it's the 
noticeably impure 5ths that give the overall sound some liveliness...
counteracting the deadness and stasis of the pure major 3rds.  
Complete purity (stasis) in triads is not necessarily a top *musical* 
goal for the way music behaves, as forward motion is important too.  
The farther away we get from 1/4 comma, the less "need" there is to 
liven up the sound with non-harmonic grit in the ornamentation, 
because the major 3rds themselves are contributing some motion.

The regular 1/5 and 1/6 comma temperaments have some proportionality 
of impurities in the 3rds vs 5ths.  But, the thing that makes them 
interesting to listen to (as compared with 1/4 comma) is *not* so much 
that any beats synch up sometimes; rather, it's more prominently the 
sense of life in those major 3rds themselves, refusing to sit 
absolutely still.  Don't all of these features contribute somewhat to 
making the overall sound "successful" by various criteria and 
expectations?  The sound gets more complex, and therefore more 
interesting, when both the 5ths and major 3rds have some motion to 
them...whether we're trying to line any of it up, or not.  The ear is 
drawn to the property that *something* is happening there in the sound 
of sustained intervals.

Another important factor is the distance away from the instrument that 
one is listening.  What might seem "successful" in 5ths and 3rds up 
close, might be just the opposite at a distance; and vice versa.  
Given that the instrument and performance manner should be calculated 
to sound best where the king or queen is sitting.......this being a 
musical rudiment of playing the hall correctly......


Brad Lehman

From: Tom Dent (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

Sigh... Did you actually listen to the samples A, B, B1, B2? I was
thinking mainly about the preference that almost everyone expressed,
which turned out to be that the slower the beating of the 5th (though
not going as far as absolute purity), the more pleasant the chord.

Perhaps, though, this was a function of overtone strengths: that the
3rd harmonic was rather stronger than the 5th in the example - perhaps
relatively stronger than in most acoustic instruments.

My idea was that some harpsichord makers might have intended the 5th
harmonic stronger than the 3rd. NOT that they might have wanted to
eliminate the beating of 5ths altogether (which would make the
instrument untunable!). Taking the opinion 'X should be bigger than Y'
to a supposedly 'logical' conclusion 'X should be enormous and Y
should be nonexistent' is a straw man. In 1/4 comma meantone, the
beating 5ths are always going to be noticeable to some extent; the
question is how to stop them *dominating* the aural picture (as they
would if the 3rd harmonic was stronger).

Actually, my spinet has a rather strong 3rd harmonic, which makes it
easy to tune, but some tempered fifths rather *obtrusive* to listen
to. That's why I tend to prefer more nearly (12-)equal tunings on it.

In 1/4 comma meantone, the fifths beat really pretty quickly and in
many cases in the 'intermittence' range which psychoacoustically
causes annoyance. I would want this sort of rapid beating to be almost
submerged below other strong harmonics if the major chord is to be
perceived as a consonance.

Besides, isn't it a strong consensus in the historical tuning industry
that the size of the fifths in a circular temperament is of little
consequence except insofar as they determine the size of thirds? How
could this be true except if the beating of fifths was rather unobtrusive?

I'll let others deal with the 'deadness' of pure intervals... and the
position of the king and queen and how the strengths of harmonics can
reverse themselves in travelling through the air (??). AFAIK most
harpsichord music was written for private enjoyment.


--- In [email protected], "Brad Lehman" <bpl@u...> wrote:
>
> > It rather depends on the timbral characteristics of your instrument.
> > If it happens to have rather weak 3rd, 6th etc. partials then the
> > beating of the 5ths is not so prominent and you can get good results
> > by making them narrower.
> > 
> > It might well be that harpsichord makers deliberately made the 3rd
> > harmonic weaker so that meantone temperaments would be more 
> successful.
> 
> 
> Wait.  Why is a carefully-regulated impure 5th, and its beating, some 
> bad thing that would need to be hidden away (deliberately or 
> otherwise) to make the sound "more successful"?
> 
> In 1/4 comma meantone on harpsichords, as some might argue, it's the 
> noticeably impure 5ths that give the overall sound some liveliness...
> counteracting the deadness and stasis of the pure major 3rds.  
> Complete purity (stasis) in triads is not necessarily a top *musical* 
> goal for the way music behaves, as forward motion is important too.  
> The farther away we get from 1/4 comma, the less "need" there is to 
> liven up the sound with non-harmonic grit in the ornamentation, 
> because the major 3rds themselves are contributing some motion.
> 
> The regular 1/5 and 1/6 comma temperaments have some proportionality 
> of impurities in the 3rds vs 5ths.  But, the thing that makes them 
> interesting to listen to (as compared with 1/4 comma) is *not* so much 
> that any beats synch up sometimes; rather, it's more prominently the 
> sense of life in those major 3rds themselves, refusing to sit 
> absolutely still.  Don't all of these features contribute somewhat to 
> making the overall sound "successful" by various criteria and 
> expectations?  The sound gets more complex, and therefore more 
> interesting, when both the 5ths and major 3rds have some motion to 
> them...whether we're trying to line any of it up, or not.  The ear is 
> drawn to the property that *something* is happening there in the sound 
> of sustained intervals.
> 
> Another important factor is the distance away from the instrument that 
> one is listening.  What might seem "successful" in 5ths and 3rds up 
> close, might be just the opposite at a distance; and vice versa.  
> Given that the instrument and performance manner should be calculated 
> to sound best where the king or queen is sitting.......this being a 
> musical rudiment of playing the hall correctly......
> 
> 
> Brad Lehman
>

From: Tom Dent (2005-11-04)
Subject: Re: Beats in a chord do not cancel

Ah, criticism by means of quote out of context. The context (clear
from Paul's previous messages) being, whether oscillation of
*loudness* or of *timbre* was more relatively important in the
perception of beating.

So, to explain once again, I hear beating at a major third and fifth
(and tenth, and twelfth, etc.) primarily as an oscillation of
*timbre*. Pulsation is a reasonable word - and ironically, it does not
imply variation of loudness. 

The idea that I'm not aware of the existence of audible beats in major
thirds and tenths on a harpsichord is ludicrous.
~~~T~~~

--- In [email protected], "Brad Lehman" <bpl@u...> wrote:
>
> > In my experience beating only produces a noticeable
> > oscillation of loudness at the unison and the octave.
> 
> Gotta install more 1/5 or 1/6 comma temperaments on your harpsichord, 
> then, and play a bunch of open major 10ths.  The pulsating effect is 
> quite noticeable and lovely.  The playing of 12ths also brings it out.
> 
> Brad Lehman
>

From: wallyesterpaulrus (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Dave Keenan" <d.keenan@b...> 
wrote:
> > 
> > > If you choose the right 
> > > initial phases and relative amplitudes for the three notes you 
may 
> > > actually be able to get the beats to cancel each other out, or 
> > > nearly so.
> > 
> > How is this possible? Each of the beatings occurs at a different 
> pitch 
> > (that is, for a different pair of nearly coincident partials) so 
I 
> have 
> > no idea how this could happen here. I'd love to be educated, 
> > particularly with a demonstration!
> 
> There are at least two perceptual aspects to beats
> (a) cyclic variation of loudness with time

Ah, but do you know how to calculate the loudness of a signal 
correctly?

> (b) cyclic variation of timbre with time
> 
> I'm saying we could eliminate the first, but not the second.

You cannot eliminate any of the beatings, so I think I disagree.

> Here's a very contrived example. 
> 
> Use a timbre that has equal amplitudes for the fundamental and all 
> harmonics up to the sixth, then nothing after that. And one needs 
to 
> be able to set the start-phase of each harmonic independently.
> 
> Consider the sync-beating 512:639:766 Hz chord. This will have the 
> following beats between the following four pairs of partials.
> 
> 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> 4 Hz between 5th harmonic of root and 4th harmonic of major third
> 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
> 
> By carefully arranging the phases of the partials in the original 
> timbre (sorry I'm too lazy to work out the details) it should be 
> possible to arrange for the three 4 Hz beats to be 120 degrees out 
of 
> phase and thereby cancel each other and leave only the 8 Hz beat.

Again, your use of the word "cancel" here strikes be as totally 
incorrect. There is no cancellation because all the beats are just as 
audible whether they are in phase or out of phase.

> This isn't terribly satisfactory since with equal amplitude 
partials 
> that 8 Hz beat will still be quite obvious, so the next step is to 
> look at a more normal timbre where the higher partials have lower 
> amplitude.
> 
> In this case it should be possible to make it so that the beats of 
the 
> major and minor third intervals are in phase with each other and 
add 
> to produce a beat which is the same amplitude as the 4 Hz beat in 
the 
> fifth interval but 180 degrees out of phase with it, and thereby 
> cancelling it.

Ditto. No cancellation would take place.

> We still have the 8 Hz beat, but this time at lower 
> amplitude.
> 
> If one cheated and knocked out the 6th harmonic of the root and the 
> 4th harmonic of the fifth completely, then one could have no 
beating 
> at all in the triad (in the sense of loudness variation) even 
though 
> every dyad heard alone has obvious beating.

Well, I'm glad you agree at least about the latter.

(a) You haven't calculated loudness correctly.

(b) Have you listened to check if this supposed lack of loudness 
variation is actually audible as such? And to see if "no beating at 
all in the triad" is at all a fair characterization of the sound? 
(Both should be checked by comparing different phase arrangements.)

> Of course we still expect 
> to hear a sort of beating of the triad due to the cyclic timbre 
> variation.

I don't call it a cyclic timbre variation, I call it beating, plain 
and simple. Although "variation in total loudness" does mean 
something, it's not what contributes to the sensation of beating 
(rather it's variation in loudness at particular pitches that does), 
and in any case you haven't calculated total loudness correctly.

From: wallyesterpaulrus (2005-11-04)
Subject: Re: brats test!

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> > > http://asmith.id.au/~dkeenan/beats.xls.zip
> > > 
> > > -- Dave Keenan
> > 
> > Can you explain the calculations in this spreadsheet and what 
you're 
> > trying to represent with them? The graph makes no sense to me 
right 
> > now as a depiction of any audible quantity; I'd love to be filled 
in.
> 
> It's intended to show an approximation to the envelope of the 
sound, 
> i.e. the variation of loudness with time.

I'd like to know how you think your calculation acheives this 
approximation, and what references/theories you're basing this on.

> But as you say, that may not 
> be anything perceptually significant for the phenomenon of beating 
of 
> complex waveforms.
> 
> You talk about the beats happening at different "pitches". That's 
not 
> quite right, since pitch is a perceptual quality that may depend on 
> more than one partial.

But in order to hear the beating clearly, you focus on a particular 
*pitch* so as to hear out the relevant pair of partials.

> If you'd said "different frequencies" then 
> you'd be right, but most people I know don't hear the individual 
sine-
> wave partials which mathematically-speaking make up a sound (unless 
> perhaps when one is suddenly added or taken away). They instead 
hear a 
> quality I'm calling timbre. Sorry to be so pedantic.

It's OK. Piano tuners are trained to listen for beats *at* the pitch 
of the partials which participate in the beating. Certainly, those 
without such training would be hard pressed to hear all the beatings 
individually. But I still don't think what they do hear is a 
variation in *overall* loudness -- though it seems we may be able to 
test that somewhat with a few more .wav files.

From: wallyesterpaulrus (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Dave Keenan" <d.keenan@b...> 
wrote:
> > 
> > > A single variable that correlates well with the preferences of 
the 
> > > majority here is simply the _error_in_the_fifth_. It's the beat 
> rate 
> > > of the fifth that dominates, since it involves lower harmonics 
(2 
> > > and 3), which have greater amplitude.
> > > 
> > > The logical conclusion of this for meantones is, sad to say, 
equal 
> > > temperament.
> > 
> > How do you arrive at that conclusion?
> >
> 
> Simple. It's the meantone with the slowest beating fifths.
> 
> I don't consider any fifth-generated linear temperament with a 
fifth 
> wider than 700 cents to be a meantone. We are then into the 
> Pythagorean region.

You mean the schismatic region? Or . . . (?)

Anyway, don't you think it would be prudent to see what "the 
preferences of the majority" are when you actually include the equal 
temperament triad, and perhaps a few others that differ from A, B, 
B2, and B3, before making such a conclusion? It seems you're 
extrapolating *way* beyond the realm explored with the current set of 
triads.

From: Brad Lehman (2005-11-04)
Subject: Re: Beats in a chord do not cancel

Well Tom, I missed your previous material where you explained this 
difference, and I'm sorry: I didn't intend any offence here.  The 
beats of 12ths and major 10ths, on harpsichords, simply sound like 
variations of loudness to me--not timbre.

Brad Lehman


--- In [email protected], "Tom Dent" <stringph@g...> wrote:
>
> 
> Ah, criticism by means of quote out of context. The context (clear
> from Paul's previous messages) being, whether oscillation of
> *loudness* or of *timbre* was more relatively important in the
> perception of beating.
> 
> So, to explain once again, I hear beating at a major third and fifth
> (and tenth, and twelfth, etc.) primarily as an oscillation of
> *timbre*. Pulsation is a reasonable word - and ironically, it does 
not
> imply variation of loudness. 
> 
> The idea that I'm not aware of the existence of audible beats in 
major
> thirds and tenths on a harpsichord is ludicrous.
> ~~~T~~~
> 
> --- In [email protected], "Brad Lehman" <bpl@u...> wrote:
> >
> > > In my experience beating only produces a noticeable
> > > oscillation of loudness at the unison and the octave.
> > 
> > Gotta install more 1/5 or 1/6 comma temperaments on your 
harpsichord, 
> > then, and play a bunch of open major 10ths.  The pulsating effect 
is 
> > quite noticeable and lovely.  The playing of 12ths also brings it 
out.
> > 
> > Brad Lehman
> >
>

From: wallyesterpaulrus (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "monz" <monz@t...> wrote:

> I too have not done significant experiments in audio-testing
> various aspects of phase, but all accounts that i've ever
> read about digital synthesis agree that phase differences
> are audibly insignificant.

That's true in general, but this is a particular scenario which is so 
different from the general one that the meaning isn't really even the 
same anymore. Yes, relative phase differences in the (pressure) 
waveforms of different notes are audibly insignificant. But relative 
phases of the far slower amplitude waveforms that we call beats? That's 
a whole other question.

From: Dave Keenan (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Tom Dent" <stringph@g...> wrote:
> It might well be that harpsichord makers deliberately made the 3rd
> harmonic weaker so that meantone temperaments would be more 
successful.

That would require plucking the string at somewhere near 1/3 of its 
length. Is this the case? Seems unlikely.

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "wallyesterpaulrus" > I don't call it a 
cyclic timbre variation, I call it beating, plain 
> and simple. Although "variation in total loudness" does mean 
> something, it's not what contributes to the sensation of beating 
> (rather it's variation in loudness at particular pitches that does), 
> and in any case you haven't calculated total loudness correctly.

OK. You've convinced me.

-- Dave Keenan

From: Brad Lehman (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> Sigh... Did you actually listen to the samples A, B, B1, B2? I was
> thinking mainly about the preference that almost everyone expressed,
> which turned out to be that the slower the beating of the 5th 
(though
> not going as far as absolute purity), the more pleasant the chord.

Of course I listened to them, yesterday, before writing this message:
http://launch.groups.yahoo.com/group/tuning/message/62112

All of them have differently interesting qualities, and I might prefer 
different ones on different days or in different musical contexts.  
Triads in isolation really don't give me anything to go on, in 
formulating any firm preference about pleasantness; they're not the 
music that I care about (with any tensions/resolutions to deal with), 
and there's no melodic factor here at all, let alone any contrast 
against different triads, or any approach from a different harmony.  
Totally cold and without knowing what's going on in them, I liked "A" 
and "B" both a little bit better than "B2" and "B3", as far as 
listening to sustained wobbly triads goes.

And all four of them seemed more regular (i.e. quickly monotonous) 
than I'd expect to hear on any acoustic instrument.   

So?

What good singer would ever be caught holding a note with absolutely 
no variation in it, for more than a few seconds, whether there's 
vibrato in it or not?  :)


Brad Lehman

From: Dave Keenan (2005-11-04)
Subject: Re: brats test!

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> > It's intended to show an approximation to the envelope of the 
> sound, 
> > i.e. the variation of loudness with time.
> 
> I'd like to know how you think your calculation acheives this 
> approximation, and what references/theories you're basing this on.

I square the instantaneous voltage or pressure to get instantaneous 
power and then I filter this with a simple moving average to try to 
eliminate the variations that are too rapid to hear. It's pretty crude 
but surely it's good enough to see where the peaks and troughs are.

-- Dave Keenan

From: Dave Keenan (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
> >
> > --- In [email protected], "wallyesterpaulrus" 
> > <wallyesterpaulrus@y...> wrote:
> > >
> > > --- In [email protected], "Dave Keenan" <d.keenan@b...> 
> wrote:
> > > 
> > > > A single variable that correlates well with the preferences 
of 
> the 
> > > > majority here is simply the _error_in_the_fifth_. It's the 
beat 
> > rate 
> > > > of the fifth that dominates, since it involves lower 
harmonics 
> (2 
> > > > and 3), which have greater amplitude.
> > > > 
> > > > The logical conclusion of this for meantones is, sad to say, 
> equal 
> > > > temperament.
> > > 
> > > How do you arrive at that conclusion?
> > >
> > 
> > Simple. It's the meantone with the slowest beating fifths.
> > 
> > I don't consider any fifth-generated linear temperament with a 
> fifth 
> > wider than 700 cents to be a meantone. We are then into the 
> > Pythagorean region.
> 
> You mean the schismatic region? Or . . . (?)

Yes, or schismic or helmholtz or any other name that its been given 
over the years. I think most people would have known what I meant.

> Anyway, don't you think it would be prudent to see what "the 
> preferences of the majority" are when you actually include the 
equal 
> temperament triad, and perhaps a few others that differ from A, B, 
> B2, and B3, before making such a conclusion? It seems you're 
> extrapolating *way* beyond the realm explored with the current set 
of 
> triads.

Sure. But it's unfortunate that it can hardly be a blind test now.

-- Dave Keenan

From: wallyesterpaulrus (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Tom Dent" <stringph@g...> wrote:

> In 1/4 comma meantone, the fifths beat really pretty quickly and in
> many cases in the 'intermittence' range which psychoacoustically
> causes annoyance.

Really? I haven't found that to be the case with 1/4-comma meantone, 
but only with high-register fifths in temperaments that do more 
damage to the fifths. Can you elaborate on your experience (as well 
as on the psychoacoustics if you will)?

> Besides, isn't it a strong consensus in the historical tuning 
industry
> that the size of the fifths in a circular temperament is of little
> consequence except insofar as they determine the size of thirds?

(a) The thirds in such tunings are formed from chains of three or 
four fifths, so if you change each of a set of adjacent fifths by a 
certain amount, you're changing the relevant thirds by three to four 
times that amount. Conversely, changing a third by a given amount 
only means changing each of the comprising fifths by 1/3 to 1/4 that 
amount. So one has to keep the thirds much more strongly and 
carefully in mind as a goal in such tunings; the individual fifths 
have relatively little leeway, and thus variations in their size 
relatively little impact, just by the nature of the construction.

(b) A 12-tone circular temperament has other constraints on it, such 
as closure of the circle of fifths.

So it's unfair to extrapolate consensuses for 12-tone circulating 
temperaments to tunings in general.

From: wallyesterpaulrus (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "Tom Dent" <stringph@g...> wrote:
> > It might well be that harpsichord makers deliberately made the 3rd
> > harmonic weaker so that meantone temperaments would be more 
> successful.
> 
> That would require plucking the string at somewhere near 1/3 of its 
> length. Is this the case? Seems unlikely.

Why unlikely??

From: wallyesterpaulrus (2005-11-04)
Subject: Re: brats test!

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> > > It's intended to show an approximation to the envelope of the 
> > sound, 
> > > i.e. the variation of loudness with time.
> > 
> > I'd like to know how you think your calculation acheives this 
> > approximation, and what references/theories you're basing this on.
> 
> I square the instantaneous voltage or pressure to get instantaneous 
> power and then I filter this with a simple moving average to try to 
> eliminate the variations that are too rapid to hear.

How rapid?

> It's pretty crude 
> but surely it's good enough to see where the peaks and troughs are.

Doing the correct "convolution" or "filtering by moving average" is one 
aspect of getting this calculation right; another aspect is knowing how 
loudnesses *add* when one adds frequency components either (this 
matters!) within the same critical band or in different critical bands 
(unlike amplitudes, loudnesses do not add in a linear way!) You may be 
able to find the relevant psychoacoustical formulas on the web . . .

From: Dave Keenan (2005-11-04)
Subject: Re: brats test!

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> > > It's intended to show an approximation to the envelope of the 
> > sound, 
> > > i.e. the variation of loudness with time.
> > 
> > I'd like to know how you think your calculation acheives this 
> > approximation, and what references/theories you're basing this 
on.
> 
> I square the instantaneous voltage or pressure to get 
instantaneous 
> power and then I filter this with a simple moving average to try 
to 
> eliminate the variations that are too rapid to hear. It's pretty 
crude 
> but surely it's good enough to see where the peaks and troughs are.

I guess you're gonna say I should have taken the log or otherwise 
converted to decibels in which case I would have seen how utterly 
insignificant the overall loudness variation actually is. Yeah, 
you're right.

-- Dave Keenan
> 
> -- Dave Keenan
>

From: wallyesterpaulrus (2005-11-04)
Subject: Re: Beats in a chord do not cancel

Thanks Tom, you pretty much said what I said!

:)

--- In [email protected], "Tom Dent" <stringph@g...> wrote:
>
> --- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
> >
> > > How is this possible? Each of the beatings occurs at a 
different 
> > pitch 
> > > (that is, for a different pair of nearly coincident partials) 
so I 
> > have 
> > > no idea how this could happen here. I'd love to be educated, 
> > > particularly with a demonstration!
> > 
> > There are at least two perceptual aspects to beats
> > (a) cyclic variation of loudness with time
> > (b) cyclic variation of timbre with time
> > 
> > I'm saying we could eliminate the first, but not the second.
> > 
> > Here's a very contrived example. 
> > 
> > Use a timbre that has equal amplitudes for the fundamental and 
all 
> > harmonics up to the sixth, then nothing after that. And one needs 
to 
> > be able to set the start-phase of each harmonic independently.
> > 
> > Consider the sync-beating 512:639:766 Hz chord. This will have 
the 
> > following beats between the following four pairs of partials.
> > 
> > 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> > 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> > 4 Hz between 5th harmonic of root and 4th harmonic of major third
> > 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
> > 
> > By carefully arranging the phases of the partials in the original 
> > timbre (sorry I'm too lazy to work out the details) it should be 
> > possible to arrange for the three 4 Hz beats to be 120 degrees 
out of 
> > phase and thereby cancel each other and leave only the 8 Hz beat.
> > 
> > This isn't terribly satisfactory since with equal amplitude 
partials 
> > that 8 Hz beat will still be quite obvious, so the next step is 
to 
> > look at a more normal timbre where the higher partials have lower 
> > amplitude.
> > 
> > In this case it should be possible to make it so that the beats 
of the 
> > major and minor third intervals are in phase with each other and 
add 
> > to produce a beat which is the same amplitude as the 4 Hz beat in 
the 
> > fifth interval but 180 degrees out of phase with it, and thereby 
> > cancelling it. We still have the 8 Hz beat, but this time at 
lower 
> > amplitude.
> > 
> > If one cheated and knocked out the 6th harmonic of the root and 
the 
> > 4th harmonic of the fifth completely, then one could have no 
beating 
> > at all in the triad (in the sense of loudness variation) even 
though 
> > every dyad heard alone has obvious beating. Of course we still 
expect 
> > to hear a sort of beating of the triad due to the cyclic timbre 
> > variation.
> > 
> > -- Dave Keenan
> 
> 
> Well, I still think the beating in timbre would be very much
> noticeable. In my experience beating only produces a noticeable
> oscillation of loudness at the unison and the octave.
> 
> > 4 Hz between 3rd harmonic of root and 2nd harmonic of fifth
> > 8 Hz between 6th harmonic of root and 4th harmonic of fifth
> > 4 Hz between 5th harmonic of root and 4th harmonic of major third
> > 4 Hz between 6th harmonic of major third and 5th harmonic of fifth
> 
> The three different 4Hz beats occur at 3 different overtone pitches
> and there is no way they can cancel each other out. To put it 
another
> way, in a close position chord *every* beat is a variation in 
timbre,
> since every beat affects the upper partials, not the fundamental.
> Beats could in principle only cancel each other out if they were all
> at the same pitch (as in a violin section all playing the 'same' 
note).
> 
> ~~~T~~~
>

From: wallyesterpaulrus (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Dave Keenan" <d.keenan@b...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Dave Keenan" <d.keenan@b...> 
wrote:
> > >
> > > --- In [email protected], "wallyesterpaulrus" 
> > > <wallyesterpaulrus@y...> wrote:
> > > >
> > > > --- In [email protected], "Dave Keenan" <d.keenan@b...> 
> > wrote:
> > > > 
> > > > > A single variable that correlates well with the preferences 
> of 
> > the 
> > > > > majority here is simply the _error_in_the_fifth_. It's the 
> beat 
> > > rate 
> > > > > of the fifth that dominates, since it involves lower 
> harmonics 
> > (2 
> > > > > and 3), which have greater amplitude.
> > > > > 
> > > > > The logical conclusion of this for meantones is, sad to 
say, 
> > equal 
> > > > > temperament.
> > > > 
> > > > How do you arrive at that conclusion?
> > > >
> > > 
> > > Simple. It's the meantone with the slowest beating fifths.
> > > 
> > > I don't consider any fifth-generated linear temperament with a 
> > fifth 
> > > wider than 700 cents to be a meantone. We are then into the 
> > > Pythagorean region.
> > 
> > You mean the schismatic region? Or . . . (?)
> 
> Yes, or schismic or helmholtz or any other name that its been given 
> over the years. I think most people would have known what I meant.

Pythagorean to many, if not most, people here would just mean pure 
fifths and triads with major thirds of 408 cents and minor thirds of 
294 cents. If you're talking about using diminished fourths instead 
of major thirds, etc., I would think you should say so explicitly. I 
don't think someone without many years of history talking to you 
would be likely to correctly discern your meaning here, since you 
didn't make any mention of triads, thirds, diminished fourths, etc. 
(Sorry to be so pedantic.)

> > Anyway, don't you think it would be prudent to see what "the 
> > preferences of the majority" are when you actually include the 
> equal 
> > temperament triad, and perhaps a few others that differ from A, 
B, 
> > B2, and B3, before making such a conclusion? It seems you're 
> > extrapolating *way* beyond the realm explored with the current 
set 
> of 
> > triads.
> 
> Sure. But it's unfortunate that it can hardly be a blind test now.

Now? I thought all the triads (A, B, B2, B3) were explicitly 
specified here on this list *before* any of the corresponding 
listening tests. So nothing would be different now. :)

From: Dave Keenan (2005-11-04)
Subject: Re: Beats in a chord do not cancel

--- In [email protected], "Tom Dent" <stringph@g...> wrote:
> Well, I still think the beating in timbre would be very much
> noticeable. In my experience beating only produces a noticeable
> oscillation of loudness at the unison and the octave.

Thanks Tom. You and Paul have convinced me of that now.

-- Dave Keenan

From: monz (2005-11-04)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "monz" <monz@t...> wrote:
> 
> > I too have not done significant experiments in audio-testing
> > various aspects of phase, but all accounts that i've ever
> > read about digital synthesis agree that phase differences
> > are audibly insignificant.
> 
> That's true in general, but this is a particular scenario
> which is so different from the general one that the meaning
> isn't really even the same anymore. Yes, relative phase
> differences in the (pressure) waveforms of different notes
> are audibly insignificant. But relative phases of the far
> slower amplitude waveforms that we call beats? That's 
> a whole other question.


Ah, OK, thanks ... i misunderstood. It's clear now.



-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Petr Parízek (2005-11-05)
Subject: Re: Beats in a chord do not cancel

There is one possible case in which the beats could be cancelled out. But I
can't imagine this happening on an acoustic instrument. It would mean all of
these conditions to be met at the same time:

1. The played chord is a regular part of the subharmonic series (slightly
detuned, of course).

2. One of the intervals is rational.

3. The "almost common" overtone appears in the same intensity in all the
sounding tones.

4. The phase of each tone is carefully chosen in such a way that makes the
required harmonic overtones vanish and leaves only the inharmonic ones.

Petr

From: Petr Parízek (2005-11-05)
Subject: Re: [tuning] Re: Beats in a chord do not cancel

> 1. The played chord is a regular part of the subharmonic series (slightly
> detuned, of course).
>
> 2. One of the intervals is rational.
>
> 3. The "almost common" overtone appears in the same intensity in all the
> sounding tones.
>
> 4. The phase of each tone is carefully chosen in such a way that makes the
> required harmonic overtones vanish and leaves only the inharmonic ones.

I was inexact when I said "rational". Of course, if you wish, all of the
intervals may be rational but only those which fall exactly into the part of
the subharmonic series will work. For example, suppose we have a minor chord
of E4-G4-B4 where the frequency of E4 is 320Hz (i.e. as in the 18. century
or so). If it's a regular part of the subharmonic series, the frequencies
are 320:384:480. If you shift the lowest tone 1Hz higher, then two very
similar sets of overtones start beating. The lowest similar overtone in the
fifth is B5 (i.e. 960:963) and in the case of the minor third it's B6
(1920:1926). Essentially, you can't get rid of the beats in B5 but you can
get rid of the beats in B6 by removing 1920Hz which is the common overtone
of the major third. This is done by adjusting the phase of either G4 or B4
in such a way that the harmonic B6 vanishes and only the inharmonic B6
remains. This may explain another reason for promoting quarter-comma
(eventually third-comma) meantone.

Petr

From: Carl Lumma (2005-11-05)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> Yes, or schismic or helmholtz or any other name that its been
> given over the years. I think most people would have known what
> I meant.

I did.

> Sure. But it's unfortunate that it can hardly be a blind test now.

A new blind test could always be set up, no?

-Carl

From: Carl Lumma (2005-11-05)
Subject: Re: brats test!

> I guess you're gonna say I should have taken the log or otherwise 
> converted to decibels in which case I would have seen how utterly 
> insignificant the overall loudness variation actually is. Yeah, 
> you're right.

Don't be intimidated, Dave.  Mean-squared power is the standard
evaluation of the overall loudness of an audio signal.  It's
completely obvious while looking at any of these wav files in
a waveform editor that the percieved overall loudness variations
match the instantaneous height on the display -- whatever
smoothing these editors do for a 6-second file filling half
the screen works nearly perfectly.

-Carl

From: Carl Lumma (2005-11-05)
Subject: Re: Beats in a chord do not cancel

> --- In [email protected], "Tom Dent" <stringph@g...> wrote:
> > Well, I still think the beating in timbre would be very much
> > noticeable.

Agree.

> > In my experience beating only produces a noticeable
> > oscillation of loudness at the unison and the octave.

Huh??

-Carl

From: Carl Lumma (2005-11-05)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> > Yes, relative phase differences in the (pressure)
> > waveforms of different notes are audibly insignificant.
> > But relative phases of the far slower amplitude
> > waveforms that we call beats? That's a whole other
> > question.
> 
> Ah, OK, thanks ... i misunderstood. It's clear now.

However the answer in this case also seems to be that
phases don't matter all that much.

-Carl

From: Gene Ward Smith (2005-11-06)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> > > A brat is a Beat RATio, usually defined for approximate
> > > 4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
> > > beat rate.
> > 
> > Why use this obfuscatory jargon in the first place? How hard can
> > it be to type "beat ratio" and to be clear about exactly which
> > intervals you're considering the beat ratio of? [Rhetorical
> > questions]
> 
> It wasn't my idea (though I didn't disapprove).

I've said it before, and I'll say it again: it isn't obfuscatory, but
is a precisely defined term introduced exactly because "beat ratio" is
the *general* term. If f is the fifth, and t is the third, then the
brat means *precisely* this ratio:

brat = (6t - 5f)/(4t - 5)

It does not mean some other beat ratio. It does not mean
(5f-6t)/(4t-5). It means exactly *this* ratio. This allows it to be
given, and convey a precise meaning. Otherwise confusion can and will
ensue. Admittedly, it might have been better to base everything on

(4t - 5)/(2f - 3)

and also to state things in terms of the approximate "3", a, and the
approximate "5", b, and so as

(b - 5)/(a - 3)

but I was attempting to follow Wendell.

From: Gene Ward Smith (2005-11-07)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Brad Lehman" <bpl@u...> wrote:

> In 1/4 comma meantone on harpsichords, as some might argue, it's the 
> noticeably impure 5ths that give the overall sound some liveliness...
> counteracting the deadness and stasis of the pure major 3rds.  
> Complete purity (stasis) in triads is not necessarily a top *musical* 
> goal for the way music behaves, as forward motion is important too. 

I don't think this is a linear relationship; I think it does not take
much detuning of the pure major thirds of 1/4 comma meantone to
produce a different sound. Anyway, I like the sound of 31-et, for my part.

From: Gene Ward Smith (2005-11-07)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Tom Dent" <stringph@g...> wrote:
>
> 
> Sigh... Did you actually listen to the samples A, B, B1, B2? I was
> thinking mainly about the preference that almost everyone expressed,
> which turned out to be that the slower the beating of the 5th (though
> not going as far as absolute purity), the more pleasant the chord.

Dave made this point also. Given that the tuning of the major third
was fixed, I think it is pretty well meaningless.

From: Carl Lumma (2005-11-07)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> > Sigh... Did you actually listen to the samples A, B, B1, B2?
> > I was thinking mainly about the preference that almost
> > everyone expressed, which turned out to be that the slower
> > the beating of the 5th (though not going as far as absolute
> > purity), the more pleasant the chord.
> 
> Dave made this point also. Given that the tuning of the major
> third was fixed, I think it is pretty well meaningless.

The major 3rd was not the same in all chords, but this is a
worthwhile point.

-Carl

From: Tom Dent (2005-11-07)
Subject: Re: Beats in a chord do not cancel

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > --- In [email protected], "Tom Dent" <stringph@g...> wrote:
> > > Well, I still think the beating in timbre would be very much
> > > noticeable.
> 
> Agree.
> 
> > > In my experience beating only produces a noticeable
> > > oscillation of loudness at the unison and the octave.
> 
> Huh??
> 
> -Carl

Without knowing what 'Huh??' means, I can only restate in more
pedantically explicit form as follows:

Beats produced by a particular out-of-tune interval have the twin
effects of oscillating loudness and/or timbre.

In my opinion, beats in a mistuned unison (or equison, to be yet more
pedantic) or an octave partake substantially of both effects. But
beats in fifths and thirds sound more like oscillations of timbre than
of loudness.

Of course, this question depends on relative strengths of the
fundamental and overtones. In the limit where the fundamental is weak
and the overtones are strong, beating fifths, thirds etc. *will* have
a quite appreciable effect on overall perceived loudness. But in most
musical situations it is the other way round.

Incidentally, what happens with the .wav generation program if you
simply tell it to stagger the notes of the chord by some fractions of
a second?

~~~T~~~

From: Kraig Grady (2005-11-07)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

Hi Gene!
 Since there are others triads that could be the basis besides the 5 
limit. is it possible to apply this method to other triads?
 For instance in Meta Slendro the basic triad is the 6-7-8. it seems 
this method chord be applied to any proportional triad?
 

>Message: 4         
>   Date: Sun, 06 Nov 2005 22:54:48 -0000
>   From: "Gene Ward Smith" <[email protected]>
>Subject: Re: the simplest pan-proportionally beating 12-tone temperament
>
>--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
>  
>
>>>>A brat is a Beat RATio, usually defined for approximate
>>>>4:5:6 triads as the ratio of the 6:5's beat rate to the 5:4's
>>>>beat rate.
>>>>        
>>>>
>>>Why use this obfuscatory jargon in the first place? How hard can
>>>it be to type "beat ratio" and to be clear about exactly which
>>>intervals you're considering the beat ratio of? [Rhetorical
>>>questions]
>>>      
>>>
>>It wasn't my idea (though I didn't disapprove).
>>    
>>
>
>I've said it before, and I'll say it again: it isn't obfuscatory, but
>is a precisely defined term introduced exactly because "beat ratio" is
>the *general* term. If f is the fifth, and t is the third, then the
>brat means *precisely* this ratio:
>
>brat = (6t - 5f)/(4t - 5)
>
>It does not mean some other beat ratio. It does not mean
>(5f-6t)/(4t-5). It means exactly *this* ratio. This allows it to be
>given, and convey a precise meaning. Otherwise confusion can and will
>ensue. Admittedly, it might have been better to base everything on
>
>(4t - 5)/(2f - 3)
>
>and also to state things in terms of the approximate "3", a, and the
>approximate "5", b, and so as
>
>(b - 5)/(a - 3)
>
>but I was attempting to follow Wendell.
>
>
>
>
>  
>

-- 
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

From: Gene Ward Smith (2005-11-07)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], Kraig Grady <kraiggrady@a...> wrote:
>
> Hi Gene!
>  Since there are others triads that could be the basis besides the 5 
> limit. is it possible to apply this method to other triads?

It certainly is! It is also possible to apply the idea to *tetrads*
and beyond; I've posted some stuff on tuning-math on how you can take
a single-comma temperament and get synchronized beating of tetrads;
chosing the right single-comma temperament can give near-tunings of
rank two ("linear") temperaments also.

From: wallyesterpaulrus (2005-11-08)
Subject: Re: Beats in a chord do not cancel

--- In [email protected], Petr Parízek  wrote:
>
> > 1. The played chord is a regular part of the subharmonic series 
(slightly
> > detuned, of course).
> >
> > 2. One of the intervals is rational.
> >
> > 3. The "almost common" overtone appears in the same intensity in 
all the
> > sounding tones.
> >
> > 4. The phase of each tone is carefully chosen in such a way that 
makes the
> > required harmonic overtones vanish and leaves only the inharmonic 
ones.
> 
> I was inexact when I said "rational". Of course, if you wish, all 
of the
> intervals may be rational but only those which fall exactly into 
the part of
> the subharmonic series will work. For example, suppose we have a 
minor chord
> of E4-G4-B4 where the frequency of E4 is 320Hz (i.e. as in the 18. 
century
> or so). If it's a regular part of the subharmonic series, the 
frequencies
> are 320:384:480. If you shift the lowest tone 1Hz higher, then two 
very
> similar sets of overtones start beating. The lowest similar 
overtone in the
> fifth is B5 (i.e. 960:963) and in the case of the minor third it's 
B6
> (1920:1926). Essentially, you can't get rid of the beats in B5 but 
you can
> get rid of the beats in B6 by removing 1920Hz which is the common 
overtone
> of the major third. This is done by adjusting the phase of either 
G4 or B4
> in such a way that the harmonic B6 vanishes and only the inharmonic 
B6
> remains.

Oh, I think I get what you're saying now. You're considering 
harmonics of the lowest tone "inharmonic" since you've shifted the 
lower tone?

> This may explain another reason for promoting quarter-comma
> (eventually third-comma) meantone.

On acoustic instruments, you'd have little control over the phase, 
and get constructive interference as often as destructive 
interference. So I don't think this effect was relevant in the era 
when meantone was being promoted or the era in which it dominated in 
Western music.

From: wallyesterpaulrus (2005-11-08)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > I guess you're gonna say I should have taken the log or otherwise 
> > converted to decibels in which case I would have seen how utterly 
> > insignificant the overall loudness variation actually is. Yeah, 
> > you're right.
> 
> Don't be intimidated, Dave.  Mean-squared power is the standard
> evaluation of the overall loudness of an audio signal.

In psychoacoustics, far from it, particularly when some components 
fall into the same critical band and others don't. Also, Dave needed 
to use some time-smoothing, which you don't mention at all.

> It's
> completely obvious while looking at any of these wav files in
> a waveform editor that the percieved overall loudness variations
> match the instantaneous height on the display

Are you talking about the case where the beatings were in phase with 
one another? If so, that's no surprise. How about when the phases are 
arranged to keep the overall loudness (by your or Dave's definition) 
constant?

From: wallyesterpaulrus (2005-11-08)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > Yes, relative phase differences in the (pressure)
> > > waveforms of different notes are audibly insignificant.
> > > But relative phases of the far slower amplitude
> > > waveforms that we call beats? That's a whole other
> > > question.
> > 
> > Ah, OK, thanks ... i misunderstood. It's clear now.
> 
> However the answer in this case also seems to be that
> phases don't matter all that much.

I don't recall anyone creating synched-beating sound files to compare 
with different phases in each, so I don't know on what basis you're 
claiming this "answer".

From: Carl Lumma (2005-11-08)
Subject: Re: brats test!

> > Don't be intimidated, Dave.  Mean-squared power is the standard
> > evaluation of the overall loudness of an audio signal.
> 
> In psychoacoustics, far from it, particularly when some
> components fall into the same critical band and others don't.
> Also, Dave needed to use some time-smoothing, which you
> don't mention at all.

Years ago I read a Terhardt paper on this stuff, but I don't
think it's necessary here.  In musical applications, ms power
is good enough.  I did mention smoothing farther down, but
you snipped it.

> > It's completely obvious while looking at any of these wav
> > files in a waveform editor that the percieved overall
> > loudness variations match the instantaneous height on the
> > display
> 
> Are you talking about the case where the beatings were in
> phase with one another? If so, that's no surprise. How about
> when the phases are arranged to keep the overall loudness (by
> your or Dave's definition) constant?

That's a very special case that's not likely to occur in
the real world.

-Carl

From: Carl Lumma (2005-11-08)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> > > > Yes, relative phase differences in the (pressure)
> > > > waveforms of different notes are audibly insignificant.
> > > > But relative phases of the far slower amplitude
> > > > waveforms that we call beats? That's a whole other
> > > > question.
> > > 
> > > Ah, OK, thanks ... i misunderstood. It's clear now.
> > 
> > However the answer in this case also seems to be that
> > phases don't matter all that much.
> 
> I don't recall anyone creating synched-beating sound files to
> compare with different phases in each, so I don't know on what
> basis you're claiming this "answer".

Dave's spreadsheet.  But here's one randomly-phased version
of A.wav...

http://lumma.org/tuning/Aphase.wav

It sounds the same and looks the same as A.wav.

-Carl

From: [email protected] (2005-11-08)
Subject: Re: [tuning] brats test! (was Re: the simplest pan-proportionally beating...)

Aha, this random phased version sounds as consonant as B, assuming that it
is still proportional beating.

Cordially,
Ozan

----- Original Message -----
From: "Carl Lumma" <[email protected]>
To: <[email protected]>
Sent: 08 Kas\ufffdm 2005 Sal\ufffd 23:28
Subject: [tuning] brats test! (was Re: the simplest pan-proportionally
beating...)

> >
> > I don't recall anyone creating synched-beating sound files to
> > compare with different phases in each, so I don't know on what
> > basis you're claiming this "answer".
>
> Dave's spreadsheet.  But here's one randomly-phased version
> of A.wav...
>
> http://lumma.org/tuning/Aphase.wav
>
> It sounds the same and looks the same as A.wav.
>
> -Carl
>
>

From: wallyesterpaulrus (2005-11-10)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > Don't be intimidated, Dave.  Mean-squared power is the standard
> > > evaluation of the overall loudness of an audio signal.
> > 
> > In psychoacoustics, far from it, particularly when some
> > components fall into the same critical band and others don't.
> > Also, Dave needed to use some time-smoothing, which you
> > don't mention at all.
> 
> Years ago I read a Terhardt paper on this stuff, but I don't
> think it's necessary here.  In musical applications, ms power
> is good enough.

Carl, have you ever heard of phons?

> I did mention smoothing farther down, but
> you snipped it.

Sorry. Please restore it.

> > > It's completely obvious while looking at any of these wav
> > > files in a waveform editor that the percieved overall
> > > loudness variations match the instantaneous height on the
> > > display
> > 
> > Are you talking about the case where the beatings were in
> > phase with one another? If so, that's no surprise. How about
> > when the phases are arranged to keep the overall loudness (by
> > your or Dave's definition) constant?
> 
> That's a very special case that's not likely to occur in
> the real world.

But it's exactly the case that Dave was trying to draw attention to 
with all his work, and so far we've neither listened to it nor 
attempted to calculate its loudness (not RMS power convoluted with 
some not-too-long, not-too-short averaging window, which is what Dave 
did) profile. All sorts of phase relationships *do* occur in the real 
world, so we shouldn't be shy about investigating particular ones.

From: wallyesterpaulrus (2005-11-10)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > > > Yes, relative phase differences in the (pressure)
> > > > > waveforms of different notes are audibly insignificant.
> > > > > But relative phases of the far slower amplitude
> > > > > waveforms that we call beats? That's a whole other
> > > > > question.
> > > > 
> > > > Ah, OK, thanks ... i misunderstood. It's clear now.
> > > 
> > > However the answer in this case also seems to be that
> > > phases don't matter all that much.
> > 
> > I don't recall anyone creating synched-beating sound files to
> > compare with different phases in each, so I don't know on what
> > basis you're claiming this "answer".
> 
> Dave's spreadsheet.  But here's one randomly-phased version
> of A.wav...
> 
> http://lumma.org/tuning/Aphase.wav
> 
> It sounds the same and looks the same as A.wav.

I found somne phases that changed the shape of Dave's curve quite a 
bit, but maybe that was just Excel changing the scale of the y-
axis . . . maybe Dave would like to make a specific suggestion here, 
and then would could get everyone to listen to the resulting .wav and 
make comparisons?

From: wallyesterpaulrus (2005-11-10)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > Don't be intimidated, Dave.  Mean-squared power is the standard
> > > evaluation of the overall loudness of an audio signal.
> > 
> > In psychoacoustics, far from it, particularly when some
> > components fall into the same critical band and others don't.
> > Also, Dave needed to use some time-smoothing, which you
> > don't mention at all.
> 
> Years ago I read a Terhardt paper on this stuff, but I don't
> think it's necessary here.  In musical applications, ms power
> is good enough.

How do you do it without the time-smoothing Dave needed to use?

From: Carl Lumma (2005-11-11)
Subject: Re: brats test!

> > Years ago I read a Terhardt paper on this stuff, but I don't
> > think it's necessary here.  In musical applications, ms power
> > is good enough.
> 
> How do you do it without the time-smoothing Dave needed to use?

For applications like replaygain, you just use the whole
file.  For applications like beating in mildly tempered
chords, whatever smoothing Cool Edit is using is obviously
working very well.

-Carl

From: Carl Lumma (2005-11-11)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

> I found somne phases that changed the shape of Dave's curve quite
> a bit, but maybe that was just Excel changing the scale of the
> y-axis . . .

I was able to change the height, trend, and dynamic range of the
curve, but not its period much.

> maybe Dave would like to make a specific suggestion
> here, and then would could get everyone to listen to the
> resulting .wav and make comparisons?

I think I may have just figured out how to get precise phase
differences in Cool Edit... but maybe you could do it easier...

-Carl

From: Carl Lumma (2005-11-11)
Subject: Re: brats test!

> > Years ago I read a Terhardt paper on this stuff, but I don't
> > think it's necessary here.  In musical applications, ms power
> > is good enough.
> 
> Carl, have you ever heard of phons?

Yes, but let's not get distracted.

> > I did mention smoothing farther down, but
> > you snipped it.
> 
> Sorry. Please restore it.

No time...

> > > Are you talking about the case where the beatings were in
> > > phase with one another? If so, that's no surprise. How about
> > > when the phases are arranged to keep the overall loudness
> > > (by your or Dave's definition) constant?
> > 
> > That's a very special case that's not likely to occur in
> > the real world.
> 
> But it's exactly the case that Dave was trying to draw attention
> to with all his work,

...which you yourself said couldn't cause the beats to cancel.

> and so far we've neither listened to it nor attempted to
> calculate its loudness (not RMS power convoluted with 
> some not-too-long, not-too-short averaging window, which
> is what Dave did) profile.

I have no trouble believing the overall amplitude can be
flattened, but like you I think the "timbre change" would
be significant.  But even if it's not...

> All sorts of phase relationships *do* occur in
> the real world, so we shouldn't be shy about investigating
> particular ones.

...such a particular condition would almost never happen
without special-purpose hardware.  It'd be interesting
to hear it, and I'd be happy to try and synthesize whatever
test chords are proposed, but I'm not inclined to bend over
backward for any of this.  I'd be much more interested in
how to calculate the heard beat rate of a chord....

-Carl

From: wallyesterpaulrus (2005-11-14)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > think it's necessary here.  In musical applications, ms power
> > > is good enough.
> > 
> > How do you do it without the time-smoothing Dave needed to use?
> 
> For applications like replaygain, you just use the whole
> file. 

Well that wouldn't tell you anything useful, as a file with wildly 
varying amplitude and one with constant amplitude could ned up with 
exactly the same number, right?

> For applications like beating in mildly tempered
> chords, whatever smoothing Cool Edit is using is obviously
> working very well.

You said ms power, but in reality you don't know what it is --
 "whatever smoothing Cool Edit is doing" -- that you say is "good 
enough"? Or are we misunderstanding one another?

From: wallyesterpaulrus (2005-11-14)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > I found somne phases that changed the shape of Dave's curve quite
> > a bit, but maybe that was just Excel changing the scale of the
> > y-axis . . .
> 
> I was able to change the height, trend, and dynamic range of the
> curve, but not its period much.

Of course not -- each of the beat rates remain the same regardless of 
phase, and they're all equal rates!

> > maybe Dave would like to make a specific suggestion
> > here, and then would could get everyone to listen to the
> > resulting .wav and make comparisons?
> 
> I think I may have just figured out how to get precise phase
> differences in Cool Edit... but maybe you could do it easier...

Dave?

From: wallyesterpaulrus (2005-11-14)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > think it's necessary here.  In musical applications, ms power
> > > is good enough.
> > 
> > Carl, have you ever heard of phons?
> 
> Yes, but let's not get distracted.

OK . . .

> > > I did mention smoothing farther down, but
> > > you snipped it.
> > 
> > Sorry. Please restore it.
> 
> No time...

OK again . . .

> > > > Are you talking about the case where the beatings were in
> > > > phase with one another? If so, that's no surprise. How about
> > > > when the phases are arranged to keep the overall loudness
> > > > (by your or Dave's definition) constant?
> > > 
> > > That's a very special case that's not likely to occur in
> > > the real world.
> > 
> > But it's exactly the case that Dave was trying to draw attention
> > to with all his work,
> 
> ...which you yourself said couldn't cause the beats to cancel.
 
And he agreed. But the idea that total loudness could remain constant 
remains intact.

> > and so far we've neither listened to it nor attempted to
> > calculate its loudness (not RMS power convoluted with 
> > some not-too-long, not-too-short averaging window, which
> > is what Dave did) profile.
> 
> I have no trouble believing the overall amplitude can be
> flattened, but like you I think the "timbre change" would
> be significant.  But even if it's not...
>
> > All sorts of phase relationships *do* occur in
> > the real world, so we shouldn't be shy about investigating
> > particular ones.
> 
> ...such a particular condition would almost never happen
> without special-purpose hardware.  It'd be interesting
> to hear it, and I'd be happy to try and synthesize whatever
> test chords are proposed, but I'm not inclined to bend over
> backward for any of this.  I'd be much more interested in
> how to calculate the heard beat rate of a chord....

You never hear more than one beat rate in a chord? Try some tetrads 
with two fifths . . .

From: Carl Lumma (2005-11-15)
Subject: Re: brats test!

> > > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > > think it's necessary here.  In musical applications, ms power
> > > > is good enough.
> > > 
> > > How do you do it without the time-smoothing Dave needed to use?
> > 
> > For applications like replaygain, you just use the whole
> > file. 
> 
> Well that wouldn't tell you anything useful, as a file with wildly 
> varying amplitude and one with constant amplitude could ned up with 
> exactly the same number, right?

I've applied this kind of gain to 20GB of music of all kinds,
and I can tell you it works fantastically well, when the amount
of gain is limited to prevent clipping.

> > For applications like beating in mildly tempered
> > chords, whatever smoothing Cool Edit is using is obviously
> > working very well.
> 
> You said ms power, but in reality you don't know what it is --
>  "whatever smoothing Cool Edit is doing" -- that you say is "good 
> enough"? Or are we misunderstanding one another?

Dave said he used ms power.  Cool Edit plots the values of the
samples on a linear scale between zero and the max value
possible at the given bit depth, then connects the dots using
some bezier-looking thing.  That's if you can see every sample.
When time smoothing is needed, it divides the number of columns
of x-axis pixels by the number of samples to display, finds the
maximum abs sample value in each resulting bin, then colors
pixels between + and - that value on the column.

-Carl

From: Carl Lumma (2005-11-15)
Subject: Re: brats test!

> > > But it's exactly the case that Dave was trying to draw
> > > attention to with all his work,
> > 
> > ...which you yourself said couldn't cause the beats to cancel.
>  
> And he agreed. But the idea that total loudness could remain
> constant remains intact.

Sure.  It's a neat idea.  And something like a Hammond organ
has all the wheels running whether the keys are down are not,
so these phases might be set in the factory, as Dave pointed
out.

> > > All sorts of phase relationships *do* occur in
> > > the real world, so we shouldn't be shy about investigating
> > > particular ones.
> > 
> > ...such a particular condition would almost never happen
> > without special-purpose hardware.  It'd be interesting
> > to hear it, and I'd be happy to try and synthesize whatever
> > test chords are proposed, but I'm not inclined to bend over
> > backward for any of this.  I'd be much more interested in
> > how to calculate the heard beat rate of a chord....
> 
> You never hear more than one beat rate in a chord?

No no, I didn't say that.  In fact beat rates on my piano
are nothing like those in these synthesized examples.
So I wouldn't want to rule out brats just yet.  But this
is the third time I've synthesized comparisons (using
different methods) and they've never held up to Robert's
claims.  But piano acoustics are a strange bunch....

-Carl

From: wallyesterpaulrus (2005-11-15)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > > > Years ago I read a Terhardt paper on this stuff, but I don't
> > > > > think it's necessary here.  In musical applications, ms 
power
> > > > > is good enough.
> > > > 
> > > > How do you do it without the time-smoothing Dave needed to 
use?
> > > 
> > > For applications like replaygain, you just use the whole
> > > file. 
> > 
> > Well that wouldn't tell you anything useful, as a file with 
wildly 
> > varying amplitude and one with constant amplitude could ned up 
with 
> > exactly the same number, right?
> 
> I've applied this kind of gain

This kind of gain?

> to 20GB of music of all kinds,
> and I can tell you it works fantastically well, when the amount
> of gain is limited to prevent clipping.

I can't tell if you've completely changed the subject, or what. What 
does this have to do with calculating loudness as a function of time?

> > > For applications like beating in mildly tempered
> > > chords, whatever smoothing Cool Edit is using is obviously
> > > working very well.
> > 
> > You said ms power, but in reality you don't know what it is --
> >  "whatever smoothing Cool Edit is doing" -- that you say is "good 
> > enough"? Or are we misunderstanding one another?
> 
> Dave said he used ms power.

Read his posts again. He had to use time-averaging (or smoothing) 
over some (fairly arbitrarily chosen) time window.

> Cool Edit plots the values of the
> samples on a linear scale between zero and the max value
> possible at the given bit depth, then connects the dots using
> some bezier-looking thing.  That's if you can see every sample.
> When time smoothing is needed, it divides the number of columns
> of x-axis pixels by the number of samples to display, finds the
> maximum abs sample value in each resulting bin, then colors
> pixels between + and - that value on the column.

Go on. So far none of this sounds like an ms power calculation of any 
kind.

From: wallyesterpaulrus (2005-11-15)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:

> > > ...such a particular condition would almost never happen
> > > without special-purpose hardware.  It'd be interesting
> > > to hear it, and I'd be happy to try and synthesize whatever
> > > test chords are proposed, but I'm not inclined to bend over
> > > backward for any of this.  I'd be much more interested in
> > > how to calculate the heard beat rate of a chord....
> > 
> > You never hear more than one beat rate in a chord?
> 
> No no, I didn't say that. 

Why do you think calculating beat rates in a chord presents any 
special problems, then?

> In fact beat rates on my piano
> are nothing like those in these synthesized examples.

How did you tune it?

> So I wouldn't want to rule out brats just yet.  But this
> is the third time I've synthesized comparisons (using
> different methods) and they've never held up to Robert's
> claims.  But piano acoustics are a strange bunch....

From: Carl Lumma (2005-11-15)
Subject: Re: brats test!

> > > > > > In musical applications, ms power
> > > > > > is good enough.
> > > > > 
> > > > > How do you do it without the time-smoothing?
> > > >
> > > > For applications like replaygain, you just use the
> > > > whole file. 
> > > 
> > > Well that wouldn't tell you anything useful,
> >
> > I've applied this kind of gain
> 
> This kind of gain?

Replaygain.  It's a constant amount of amplification
that brings the ms power of the file to a predetermined
level. 

> > Dave said he used ms power.
> 
> Read his posts again.

"I square the instantaneous voltage or pressure to get
instantaneous power and then I filter this with a simple moving
average"

Sounds like ms to me (ms voltage, to be more accurate).

> He had to use time-averaging (or smoothing) over
> some (fairly arbitrarily chosen) time window.

So?

> > Cool Edit plots the values of the
> > samples on a linear scale between zero and the max value
> > possible at the given bit depth, then connects the dots using
> > some bezier-looking thing.  That's if you can see every sample.
> > When time smoothing is needed, it divides the number of columns
> > of x-axis pixels by the number of samples to display, finds the
> > maximum abs sample value in each resulting bin, then colors
> > pixels between + and - that value on the column.
> 
> Go on.

Nothing more to tell.

> So far none of this sounds like an ms power calculation of
> any kind.

It isn't.

-Carl

From: Carl Lumma (2005-11-15)
Subject: Re: brats test!

--- In [email protected], "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
>
> --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> 
> > > > ...such a particular condition would almost never happen
> > > > without special-purpose hardware.  It'd be interesting
> > > > to hear it, and I'd be happy to try and synthesize whatever
> > > > test chords are proposed, but I'm not inclined to bend over
> > > > backward for any of this.  I'd be much more interested in
> > > > how to calculate the heard beat rate of a chord....
> > > 
> > > You never hear more than one beat rate in a chord?
> > 
> > No no, I didn't say that. 
> 
> Why do you think calculating beat rates in a chord presents any 
> special problems, then?

I'm completely at a loss for what you're asking here.

> > In fact beat rates on my piano
> > are nothing like those in these synthesized examples.
> 
> How did you tune it?

It's far out of tune at the moment.  But I mean, beating
chords on a piano don't seem to behave like synthesized
ones.

-Carl

From: Dave Keenan (2005-11-16)
Subject: brats test! (was Re: the simplest pan-proportionally beating...)

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > I found somne phases that changed the shape of Dave's curve quite
> > a bit, but maybe that was just Excel changing the scale of the
> > y-axis . . .

No. I'm pretty sure I fixed the Y axis.

> > maybe Dave would like to make a specific suggestion
> > here, and then would could get everyone to listen to the
> > resulting .wav and make comparisons?
> 
> I think I may have just figured out how to get precise phase
> differences in Cool Edit... but maybe you could do it easier...

I think it is so unlikely to produce significant audible differences 
that I'm not willing to spend the time, or ask others to spend 
theirs.

I was very interested in what Petr Parizek pointed out in
http://launch.groups.yahoo.com/group/tuning/message/62226
and
http://launch.groups.yahoo.com/group/tuning/message/62227
although I don't think it has anything to do with the historical 
popularity of quarter-comma or third-comma meantone.

i.e. with a utonal chord such as a minor triad the beating 
occurs "in one place". i.e. around the frequency of the guidetone 
(and its multiples). But I soon convinced myself that one can't 
cancel all beats in a mistuned chord there either.

-- Dave Keenan

From: wallyesterpaulrus (2005-11-16)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > > > > > In musical applications, ms power
> > > > > > > is good enough.
> > > > > > 
> > > > > > How do you do it without the time-smoothing?
> > > > >
> > > > > For applications like replaygain, you just use the
> > > > > whole file. 
> > > > 
> > > > Well that wouldn't tell you anything useful,
> > >
> > > I've applied this kind of gain
> > 
> > This kind of gain?
> 
> Replaygain.  It's a constant amount of amplification
> that brings the ms power of the file to a predetermined
> level. 

I can't see what this has to do with this discussion.

> > > Dave said he used ms power.
> > 
> > Read his posts again.
> 
> "I square the instantaneous voltage or pressure to get
> instantaneous power and then I filter this with a simple moving
> average"
> 
> Sounds like ms to me (ms voltage, to be more accurate).
 
My point was the "moving average" Dave refers to.

> > He had to use time-averaging (or smoothing) over
> > some (fairly arbitrarily chosen) time window.
> 
> So?

I thought we were talking about how to calculate loudness as a 
function of time.

> > > Cool Edit plots the values of the
> > > samples on a linear scale between zero and the max value
> > > possible at the given bit depth, then connects the dots using
> > > some bezier-looking thing.  That's if you can see every sample.
> > > When time smoothing is needed, it divides the number of columns
> > > of x-axis pixels by the number of samples to display, finds the
> > > maximum abs sample value in each resulting bin, then colors
> > > pixels between + and - that value on the column.
> > 
> > Go on.
> 
> Nothing more to tell.
> 
> > So far none of this sounds like an ms power calculation of
> > any kind.
> 
> It isn't.

I guess I lost your train of thought, then.

From: wallyesterpaulrus (2005-11-16)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> --- In [email protected], "wallyesterpaulrus" 
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In [email protected], "Carl Lumma" <clumma@y...> wrote:
> > 
> > > > > ...such a particular condition would almost never happen
> > > > > without special-purpose hardware.  It'd be interesting
> > > > > to hear it, and I'd be happy to try and synthesize whatever
> > > > > test chords are proposed, but I'm not inclined to bend over
> > > > > backward for any of this.  I'd be much more interested in
> > > > > how to calculate the heard beat rate of a chord....
> > > > 
> > > > You never hear more than one beat rate in a chord?
> > > 
> > > No no, I didn't say that. 
> > 
> > Why do you think calculating beat rates in a chord presents any 
> > special problems, then?
> 
> I'm completely at a loss for what you're asking here.

Carl, *you* were the one who wrote, "I'd be much more interested in 
how to calculate the heard beat rate of a chord...." (but snipped it 
above).

> > > In fact beat rates on my piano
> > > are nothing like those in these synthesized examples.
> > 
> > How did you tune it?
> 
> It's far out of tune at the moment.  But I mean, beating
> chords on a piano don't seem to behave like synthesized
> ones.

Are you talking about high-quality synthesized piano emulations, or 
some other kind of synth sound?

From: Carl Lumma (2005-11-16)
Subject: Re: brats test!

> > > > > > I'd be much more interested in
> > > > > > how to calculate the heard beat rate of a chord....
> > > > > 
> > > > > You never hear more than one beat rate in a chord?
> > > > 
> > > > No no, I didn't say that. 
> > > 
> > > Why do you think calculating beat rates in a chord presents
> > > any special problems, then?
> > 
> > I'm completely at a loss for what you're asking here.
> 
> Carl, *you* were the one who wrote, "I'd be much more interested
> in how to calculate the heard beat rate of a chord...." (but
> snipped it above).

I didn't snip it, it's right there.  I didn't say it represented
a special problem, but Dave said he didn't know how to do it,
and I don't know how to do it.  But I have a few ideas.

> > > > In fact beat rates on my piano
> > > > are nothing like those in these synthesized examples.
> > > 
> > > How did you tune it?
> > 
> > It's far out of tune at the moment.  But I mean, beating
> > chords on a piano don't seem to behave like synthesized
> > ones.
> 
> Are you talking about high-quality synthesized piano
> emulations, or some other kind of synth sound?

I'm talking about the kind of synth sounds I've used to
test brats.

-Carl

From: wallyesterpaulrus (2005-11-17)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > > > > > > I'd be much more interested in
> > > > > > > how to calculate the heard beat rate of a chord....
> > > > > > 
> > > > > > You never hear more than one beat rate in a chord?
> > > > > 
> > > > > No no, I didn't say that. 
> > > > 
> > > > Why do you think calculating beat rates in a chord presents
> > > > any special problems, then?
> > > 
> > > I'm completely at a loss for what you're asking here.
> > 
> > Carl, *you* were the one who wrote, "I'd be much more interested
> > in how to calculate the heard beat rate of a chord...." (but
> > snipped it above).
> 
> I didn't snip it, it's right there.  I didn't say it represented
> a special problem, but Dave said he didn't know how to do it,
> and I don't know how to do it.  But I have a few ideas.

If you hear more than one beat rate in a chord, it's not only not a 
special problem, it's not a problem at all. You just calculate the 
beat rates for every interval.

> > > > > In fact beat rates on my piano
> > > > > are nothing like those in these synthesized examples.
> > > > 
> > > > How did you tune it?
> > > 
> > > It's far out of tune at the moment.  But I mean, beating
> > > chords on a piano don't seem to behave like synthesized
> > > ones.
> > 
> > Are you talking about high-quality synthesized piano
> > emulations, or some other kind of synth sound?
> 
> I'm talking about the kind of synth sounds I've used to
> test brats.

Are you keeping that secret or something?

From: Carl Lumma (2005-11-17)
Subject: Re: brats test!

> If you hear more than one beat rate in a chord, it's not only not a 
> special problem, it's not a problem at all. You just calculate the 
> beat rates for every interval.

In the chords I've been listening to, there is one primary
beat.

-Carl

From: Carl Lumma (2005-11-17)
Subject: Re: brats test!

> > I'm talking about the kind of synth sounds I've used to
> > test brats.
> 
> Are you keeping that secret or something?

I've been posting files...

-Carl

From: wallyesterpaulrus (2005-11-17)
Subject: Re: brats test!

--- In [email protected], "Carl Lumma" <clumma@y...> wrote:
>
> > If you hear more than one beat rate in a chord, it's not only not a 
> > special problem, it's not a problem at all. You just calculate the 
> > beat rates for every interval.
> 
> In the chords I've been listening to, there is one primary
> beat.

Probably just the loudest of the interval-beats, no?

From: George D. Secor (2005-11-17)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "George D. Secor" <gdsecor@y...> 
wrote:
[to Gene Ward Smith:]
> ... I still don't think either one of us has answered Aaron's 
> original question:
> http://groups.yahoo.com/group/tuning/message/61731
> which I repeat here:
> 
> <<  what is the simplest possible 12-note temperament where all 24 
> major and minor triads have rationally proportional beating? 
> here 'simplest' means that the brats (beat ratios for the un-
> initiated) are the lowest numbers in the numerator and denominator 
> that they can be..... >>
> 
> I also assume the following additional requirements:
> 
> That it be a *circulating* temperament, and
> That it should, at the very least, be reasonably useful from a 
> musical standpoint -- or should I go beyond that by insisting that 
it 
> should sound good (just so we don't overlook the most important 
point 
> of this exercise)?
> 
> --George

I've come to the conclusion that it's almost impossible to construct 
a *musically useful* 12-note temperament where *all 24* major and 
minor triads have rationally proportional beating that's reasonably 
simple, even if you allow close-to-rational brats (as seems to be 
necessary for the minor triads with tempered fifths).

Here are the two best temperaments I've been able to come up with so 
far:

1) A well-temperament containing a chain of five ~1/5-(Didymus-)comma 
fifths:

! secor_WT08.scl
!
George Secor's 12-tone well-temperament, proportional beating 
(attempt #8)
 12
!
 90.22500
 195.30322
 294.13500
 390.60644
 498.04500
 588.05203
 697.65161
 792.18000
 892.95483
 996.09000
 1088.25804
 2/1

Here are the brats and total absolute error for each triad:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.5000 34.41
Bb... ------- ------- 1.5000 25.80 
F.... ------- ------- 1.5000 17.19
C....  1.6667  5.0000 3.0000 17.19
G....  1.6667  5.0000 3.0000 17.19
D....  2.5000  6.2500 2.5000 21.48
A....  4.2621  8.8931 2.0866 30.52
E....  5.9435 11.4153 1.9206 39.13
B.... 15.1830 25.2745 1.6647 43.45
F#... 167.150 248.224 1.4850 43.45
C#... ------- ------- 1.5000 43.01
G#... ------- ------- 1.5000 43.01

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.0000 43.45
Bb... ------- ------- 1.0000 43.01
F.... ------- ------- 1.0000 43.01
C....  7.9456  9.9456 1.2517 43.01
G....  5.9653  7.9653 1.3353 34.41
D....  3.9802  5.9802 1.5025 25.80
A....  1.9901  3.9901 2.0050 17.19
E....  1.9901  3.9901 2.0050 17.19
B....  5.9409  7.9409 1.3367 17.19
F#... 100.222 98.2225 1.9800 21.91
C#... ------- ------- 1.0000 30.52
G#... ------- ------- 1.0000 39.13

I count 22 proportionally beating triads (or 24 if you're willing to 
consider brats of 1.92 and 2.09 a close enough approximation to 2).

2) A temperament (extra)ordinaire containing a chain of six ~1/5-
(Didymus-)comma fifths:

! secor_TEO8.scl
!
George Secor's 12-tone temperament (extra)ordinaire, proportional 
beating (attempt #8)
 12
!
 87.84337
 195.30322
 294.39757
 390.60644
 500.19803
 585.90965
 697.65161
 789.76651
 892.95483
 998.24303
 1088.25804
 2/1

And here are the brats and total absolute error for each triad in 
this one:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 15.0000 20.0000 1.3333 33.88
Bb... ------- ------- 1.5000 21.49 
F....  5.0000 10.0000 2.0000 17.19
C....  1.6667  5.0000 3.0000 17.19
G....  1.6667  5.0000 3.0000 17.19
D....  1.6667  5.0000 3.0000 17.19
A....  3.3333  7.5000 2.2500 25.76
E....  5.0000 10.0000 2.0000 34.30
B....  7.7321 14.0982 1.8233 48.26
F#... 2052.42 3081.13 1.5012 52.08
C#... 1732.84 2061.75 1.5018 52.15
G#... 14.9892 19.9838 1.3332 47.84

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 27.3363 25.3363 0.9268 52.04
Bb... ------- ------- 1.0000 52.08
F.... 22.0531 24.0531 1.0907 52.15
C....  7.8249  9.8249 1.2556 42.49
G....  4.9728  6.9728 1.4022 30.10
D....  2.9851  4.9851 1.6700 21.49
A....  1.9901  3.9901 2.0050 17.19
E....  1.9901  3.9901 2.0050 17.19
B....  1.9901  3.9901 2.0050 17.19
F#... 803.568 805.568 1.0025 17.19
C#... 803.590 805.590 1.0025 25.76
G#... 14.7441 12.7441 0.8644 39.65

For this one I can claim no more than 21 proportionally beating 
triads, counting a brat of 0.8644 as 6/7 (or 23, if you're willing to 
consider brats of 0.93 and 1.09 as approximately 1).

I should emphasize that my first priority was to follow sound 
principles in designing a circulating temperament, with proportional 
beating being a secondary consideration.

I was wondering if Gene could *rationalize* each of these, as he did 
with my previous "latest" temperament (extra)ordinaire:
http://groups.yahoo.com/group/tuning/message/62013
for which I should express my belated thanks!  For this one I can 
claim proportional beating in 11 (or possibly 12) major triads, but 
only 6 minor triads.  (I should add that the major-triad brats are 
simpler than in either of the newer temperaments given above.)

BTW, Gene, I now consider the ratios you gave as the official 
description of the final version of the original temperament (extra)
ordinaire about which I enthused back in August:
http://groups.yahoo.com/group/tuning/message/59999
with the only significant difference between the original and final 
versions being a brat change from 7/3 to 2 for the A major triad.

--George

From: Gene Ward Smith (2005-11-18)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "George D. Secor" <gdsecor@y...> wrote:

> I was wondering if Gene could *rationalize* each of these, as he did 
> with my previous "latest" temperament (extra)ordinaire:

Here they are:

! secorwt08.scl
George Secor well-temperament, rationalized version
12
!
256/243
124661/111375
32/27
502429/400950
4/3
1024/729
499946/334125
128/81
671389/400950
16/9
11392/6075
2

! secorte08.scl
George Secor extraordinare temperament, rationalized version
12
!
5075/4824
75/67
28591/24120
2015/1608
805/603
5075/3618
401/268
5075/3216
1010/603
3220/1809
15/8
2

From: Carl Lumma (2005-11-18)
Subject: Re: brats test!

> > In the chords I've been listening to, there is one primary
> > beat.
> 
> Probably just the loudest of the interval-beats, no?

I think so.

-Carl

From: George D. Secor (2005-11-18)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@y...> wrote:
> 
> > I was wondering if Gene could *rationalize* each of these, as he 
did 
> > with my previous "latest" temperament (extra)ordinaire:
> 
> Here they are:
> ...

Thanks, Gene -- that was quick!

These rational versions are quite interesting and (to me) a bit 
surprising -- I'll reply in more detail later (when I have more time), 
on tuning-math.

--George

From: George D. Secor (2006-01-03)
Subject: New Year's Gift for Aaron (was: the simplest pan-proportionally beating 12-t..)

--- In [email protected], Aaron Krister Johnson <aaron@a...> 
wrote:
> 
> it seems that all this exploration into equal-beating well 
temperaments has 
> given me a great question:
> 
> what is the simplest possible 12-note temperament where all 24 
major and minor 
> triads have rationally proportional beating? here 'simplest' means 
that the 
> brats (beat ratios for the un-initiated) are the lowest numbers in 
the 
> numerator and denominator that they can be.....
> 
> anyone? this is ripe for george or gene to explore.
> 
> -aaron.

Aaron, it turned out to be something of a challenge to find a tuning 
with reasonably simple brats for the 12 major triads alone.  One 
example that was brought to my attention is Paul Bailey's well-
temperament, for which Gene gave ratios:
http://groups.yahoo.com/group/tuning-math/message/13670

On the first day of this new year, while using Gene's "Christmas 
present for George" (on tuning-math), I stumbled across the following 
solution (a high-contrast well-temperament) with major-triad brats 
(starting on C) of:
 4, 4, 3, 2, 1.5, 16/9, 1.5, 1.5, 1.5, 1.5, 1.5, 1.5
and minor-triad brats (starting on C) of :
 1.125, 1.5, 15/7, 4.2, 1, 1.5, 1, 1, 1, 1, 1, 1

For the actual ratios (in .scl format) see:
http://groups.yahoo.com/group/tuning-math/message/13733

There's also another (2nd-best, lower-contrast) well-temperament at 
the above link that may be more useful.

Best,

--George

From: George D. Secor (2006-04-26)
Subject: 1/7-comma proportional-beating well-temperament

This is in reply to
http://groups.yahoo.com/group/MakeMicroMusic/message/13088
which is quoted here:

--- In [email protected], "Aaron Krister Johnson" 
<aaron@...> wrote:
>
> --- In [email protected], Carl Lumma <ekin@> wrote:
> 
> > Is there something microtonal about it?
> 
> in truth, i believe i had the korg X5DR set to neidhardt 1. so in a
> *very mild* sense, yes.
> 
> failing that, you could say that each chromatic step is close to 
18/17
> apart!
> 
> -aaron

http://groups.yahoo.com/group/MakeMicroMusic/message/13080

Aaron, this is really very nice.  I especially liked variation 3.

But should I infer from your using Neidhard 1 that you haven't 
settled on any particular modern (low-contrast) well-temperament to 
replace 12-ET?

At one point you were experimenting with a 1/7-comma circulating 
temperament, observing that it has "some very sweet builtin beat 
ratios":
http://groups.yahoo.com/group/tuning/message/60068

In checking this out, I found that 1/7-comma major brats are 
approximately 2, and the minor brats ~4/3, which is very sweet, 
indeed!

About a month later I came up with a proportional-beating 1/7-comma 
well-temperament:
http://groups.yahoo.com/group/tuning/message/60252
with a correction at:
http://groups.yahoo.com/group/tuning/message/60264
but it appears that you didn't stick around this list long enough to 
see it.

A couple months after that Gene Ward Smith transformed it into a 
rational tuning (tuning-math #13615), which I transposed to arrive at 
the following modern well-temperament:

! Secor1_7WT.scl
!
George Secor's 1/7-comma well-temperament (Gene Ward Smith rational 
version)
12
!
6264/5929
33232/29645
105592/88935
5322/4235
39597/29645
8352/5929
31706/21175
9396/5929
9952/5929
52796/29645
55782/29645
2/1

All of the major brats (in the 3rd column of numbers) except one (on 
B) are simple ratios:

Major -----Beat Ratios------ Total absolute
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.5000 30.69
Bb... ------- ------- 1.5000 24.52 
F....  5.0000 10.0000 2.0000 24.56
C....  5.0000 10.0000 2.0000 24.59
G....  5.0000 10.0000 2.0000 24.63
D....  5.0000 10.0000 2.0000 24.41
A....  6.6667 12.5000 1.8750 30.54
E....  8.3333 15.0000 1.8000 36.68
B....  8.7211 15.5817 1.7867 39.31
F#... ------- ------- 1.5000 39.31
C#... ------- ------- 1.5000 39.31
G#... 15.0000 25.0000 1.6667 36.86

Most of the minor brats are very close approximations of simple 
ratios:

Minor -----Beat Ratios------ Total absolute
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.0000 39.31
Bb... ------- ------- 1.0000 39.31
F.... 10.7109 12.7109 1.1867 39.31
C....  9.9048 11.9048 1.2019 36.86
G....  7.9089  9.9089 1.2529 30.69
D....  5.9943  7.9943 1.3336 24.52
A....  5.9924  7.9924 1.3338 24.56
E....  5.9899  7.9899 1.3339 24.59
B....  5.7516  7.7516 1.3477 24.63
F#... ------- ------- 1.0000 24.41
C#... ------- ------- 1.0000 30.54
G#... 17.7124 19.7124 1.1129 36.68

As pretty as the numbers are, it's the sound the counts.  Please try 
it and let me know what you think.

Best,

--George

From: Ozan Yarman (2006-04-29)
Subject: Re: [tuning] 1/7-comma proportional-beating well-temperament

With all due respect to your efforts, could you not do a little better dear
George? I had the impression that the previous temperament extra-ordinaire
had it all.

One more thing if I may... could you send me all your related 12-tone Scala
files in a zip folder? I got them all messed up here in my hardrive.

Also, it might be a good idea to build a public scale archive for general
use, much like in wikipedia?

Cordially,
Oz.


----- Original Message -----
From: "George D. Secor" <[email protected]>
To: <[email protected]>
Sent: 26 Nisan 2006 \ufffdar\ufffdamba 23:39
Subject: [tuning] 1/7-comma proportional-beating well-temperament


> This is in reply to
> http://groups.yahoo.com/group/MakeMicroMusic/message/13088
> which is quoted here:
>
> --- In [email protected], "Aaron Krister Johnson"
> <aaron@...> wrote:
> >
> > --- In [email protected], Carl Lumma <ekin@> wrote:
> >
> > > Is there something microtonal about it?
> >
> > in truth, i believe i had the korg X5DR set to neidhardt 1. so in a
> > *very mild* sense, yes.
> >
> > failing that, you could say that each chromatic step is close to
> 18/17
> > apart!
> >
> > -aaron
>
> http://groups.yahoo.com/group/MakeMicroMusic/message/13080
>
> Aaron, this is really very nice.  I especially liked variation 3.
>
> But should I infer from your using Neidhard 1 that you haven't
> settled on any particular modern (low-contrast) well-temperament to
> replace 12-ET?
>
> At one point you were experimenting with a 1/7-comma circulating
> temperament, observing that it has "some very sweet builtin beat
> ratios":
> http://groups.yahoo.com/group/tuning/message/60068
>
> In checking this out, I found that 1/7-comma major brats are
> approximately 2, and the minor brats ~4/3, which is very sweet,
> indeed!
>
> About a month later I came up with a proportional-beating 1/7-comma
> well-temperament:
> http://groups.yahoo.com/group/tuning/message/60252
> with a correction at:
> http://groups.yahoo.com/group/tuning/message/60264
> but it appears that you didn't stick around this list long enough to
> see it.
>
> A couple months after that Gene Ward Smith transformed it into a
> rational tuning (tuning-math #13615), which I transposed to arrive at
> the following modern well-temperament:
>
> ! Secor1_7WT.scl
> !
> George Secor's 1/7-comma well-temperament (Gene Ward Smith rational
> version)
> 12
> !
> 6264/5929
> 33232/29645
> 105592/88935
> 5322/4235
> 39597/29645
> 8352/5929
> 31706/21175
> 9396/5929
> 9952/5929
> 52796/29645
> 55782/29645
> 2/1
>
> All of the major brats (in the 3rd column of numbers) except one (on
> B) are simple ratios:
>
> Major -----Beat Ratios------ Total absolute
> Triad M3/5th. m3/5th. m3/M3  error (cents)
> ----- ------- ------- ------ -------------
> Eb... ------- ------- 1.5000 30.69
> Bb... ------- ------- 1.5000 24.52
> F....  5.0000 10.0000 2.0000 24.56
> C....  5.0000 10.0000 2.0000 24.59
> G....  5.0000 10.0000 2.0000 24.63
> D....  5.0000 10.0000 2.0000 24.41
> A....  6.6667 12.5000 1.8750 30.54
> E....  8.3333 15.0000 1.8000 36.68
> B....  8.7211 15.5817 1.7867 39.31
> F#... ------- ------- 1.5000 39.31
> C#... ------- ------- 1.5000 39.31
> G#... 15.0000 25.0000 1.6667 36.86
>
> Most of the minor brats are very close approximations of simple
> ratios:
>
> Minor -----Beat Ratios------ Total absolute
> Triad M3/5th. m3/5th. m3/M3  error (cents)
> ----- ------- ------- ------ -------------
> Eb... ------- ------- 1.0000 39.31
> Bb... ------- ------- 1.0000 39.31
> F.... 10.7109 12.7109 1.1867 39.31
> C....  9.9048 11.9048 1.2019 36.86
> G....  7.9089  9.9089 1.2529 30.69
> D....  5.9943  7.9943 1.3336 24.52
> A....  5.9924  7.9924 1.3338 24.56
> E....  5.9899  7.9899 1.3339 24.59
> B....  5.7516  7.7516 1.3477 24.63
> F#... ------- ------- 1.0000 24.41
> C#... ------- ------- 1.0000 30.54
> G#... 17.7124 19.7124 1.1129 36.68
>
> As pretty as the numbers are, it's the sound the counts.  Please try
> it and let me know what you think.
>
> Best,
>
> --George
>
>
>

From: George D. Secor (2006-05-01)
Subject: 1/7-comma proportional-beating WT - and all the others as well!

--- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
>
> With all due respect to your efforts, could you not do a little 
better dear
> George? I had the impression that the previous temperament extra-
ordinaire
> had it all.

Oh, yes, I quite agree that it did (and still does) have it all!

However, "all" is evidently too much for some people, as Paul Bailey 
wrote me (off-list) that although he was initially impressed by it, 
he wouldn't be forwarding my 1/7-comma temperament to "a few other 
piano tuners" because  he remembered that:

<< I've learned from other tuners [that] 18 cents is about the 
biggest [major] third that's acceptable to 'academic' pianists. >>

My 1/7th-comma WT has two major thirds 19.65 cents wide.  I sent the 
following to Paul today for comment (with widest major 3rds +18.49 
cents, but many of the brats not as simple):

! Secor1_7MCRWT.scl
!
George Secor's 1/7-comma minimum-contrast rational well-temperament
12
!
 14285/13512
 5049/4504
 428221/360320
 1415/1126
 752/563
 25407/18016
 843/563
 1785/1126
 945/563
 80249/45040
 8475/4504
 2/1

> One more thing if I may... could you send me all your related 12-
tone Scala
> files in a zip folder? I got them all messed up here in my hardrive.

It's no problem to list them all here (in ascending order of key 
contrast).  They all have proportional-beating triads, both major and 
minor.  If a temperament is too tame (or too wild) for your purposes, 
then just try one farther down (or up) the list.  You'll notice that 
Gene Ward Smith is credited with determining the ratios for some of 
these:


! Secor1_7WT.scl
!
George Secor's 1/7-comma well-temperament (ratios supplied by G. W. 
Smith)

12
!
6264/5929
33232/29645
105592/88935
5322/4235
39597/29645
8352/5929
31706/21175
9396/5929
9952/5929
52796/29645
55782/29645
2/1


! Secor2_11WT.scl
!
George Secor's rational 2/11-comma well-temperament
12
!
560/531
2643/2360
70/59
74/59
315/236
2240/1593
883/590
280/177
890/531
105/59
443/236
2/1


! Secor1_5WT.scl
!
George Secor's 1/5-comma well-temperament (ratios supplied by G. W. 
Smith)
 12
!
 256/243
 75/67
 32/27
 2015/1608
 4/3
 9101/6480
 401/268
 128/81
 1010/603
 16/9
 15/8
 2/1


! Secor5_23WT.scl
!
George Secor's rational 5/23-comma proportional-beating well-
temperament
 12
!
 256/243
 966664/863865
 32/27
 120176/95985
 4/3
 1024/729
 143588/95985
 128/81
 160624/95985
 16/9
 4096/2187
 2/1


! Secor1_5TX.scl
!
George Secor's 1/5-comma temperament extraordinaire (ratios supplied 
by G. W. Smith)
 12
!
 5075/4824
 75/67
 28591/24120
 2015/1608
 805/603
 5075/3618
 401/268
 5075/3216
 1010/603
 3220/1809
 15/8
 2/1


! Secor5_23TX.scl
!
George Secor's rational 5/23-comma temperament extraordinaire
 12
!
 390/371
 3321/2968
 4397/3710
 929/742
 2476/1855
 8325/5936
 555/371
 9365/5936
 621/371
 9904/5565
 5559/2968
 2/1


! Secor1_4TX.scl
!
George Secor's rational 1/4-comma temperament extraordinaire
 12
!
 4873/4644
 481/430
 61837/52245
 484/387
 7747/5805
 8132/5805
 193/129
 1219/774
 1942/1161
 30988/17415
 3621/1935
 2/1


The latest version of the original temperament (extra)ordinaire that 
you like so well is the 5/23-comma tuning (Secor5_23TX.scl).

In addition to the above, I also have two well-temperaments in which 
all 24 major and minor triads have relatively simple brats; see:

http://groups.yahoo.com/group/tuning-math/message/13733

> Also, it might be a good idea to build a public scale archive for 
general
> use, much like in wikipedia?

What's wrong with the Scala archive -- if Manuel will add these?

Best,

--George

From: Kraig Grady (2006-05-02)
Subject: 1/7-comma proportional-beating WT - and all the others as well!

Hello George!
I hate to bring up a tangential thought to your post so early, but it 
was something that came up during the Johnston Symposium both from Monzo 
and myself.
 This was the idea of notating ratios in terms of factors instead of the 
resulting high number we sometimes use.
 The advantage is that, at least, i have to figure it out this way 
anyway when running across a ratio, i hadn't used before or very frequently.
 This way one would also know the implied basic function of the ratio.
 admittedly we don't always use intervals in this way. Thought you might 
have some thoughts on this as a possibility.

 I have BTW quite liked many of these scales you have come up along 
these lines and look forward to trying this one out in the future when i 
have a little less on my table.
 It seems thirds this large might be useful to Margo Schulter

> Message: 8         
>    Date: Mon, 01 May 2006 20:18:33 -0000
>    From: "George D. Secor" <[email protected]>
> Subject: 1/7-comma proportional-beating WT - and all the others as well!
>
> --- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
>   
>> With all due respect to your efforts, could you not do a little 
>>     
> better dear
>   
>> George? I had the impression that the previous temperament extra-
>>     
> ordinaire
>   
>> had it all.
>>     
>
> Oh, yes, I quite agree that it did (and still does) have it all!
>
> However, "all" is evidently too much for some people, as Paul Bailey 
> wrote me (off-list) that although he was initially impressed by it, 
> he wouldn't be forwarding my 1/7-comma temperament to "a few other 
> piano tuners" because  he remembered that:
>
> << I've learned from other tuners [that] 18 cents is about the 
> biggest [major] third that's acceptable to 'academic' pianists. >>
>
> My 1/7th-comma WT has two major thirds 19.65 cents wide.  I sent the 
> following to Paul today for comment (with widest major 3rds +18.49 
> cents, but many of the brats not as simple):
>
> ! Secor1_7MCRWT.scl
> !
> George Secor's 1/7-comma minimum-contrast rational well-temperament
> 12
> !
>  14285/13512
>  5049/4504
>  428221/360320
>  1415/1126
>  752/563
>  25407/18016
>  843/563
>  1785/1126
>  945/563
>  80249/45040
>  8475/4504
>  2/1
>
>   
>> One more thing if I may... could you send me all your related 12-
>>     
> tone Scala
>   
>> files in a zip folder? I got them all messed up here in my hardrive.
>>     
>
> It's no problem to list them all here (in ascending order of key 
> contrast).  They all have proportional-beating triads, both major and 
> minor.  If a temperament is too tame (or too wild) for your purposes, 
> then just try one farther down (or up) the list.  You'll notice that 
> Gene Ward Smith is credited with determining the ratios for some of 
> these:
>
>
> ! Secor1_7WT.scl
> !
> George Secor's 1/7-comma well-temperament (ratios supplied by G. W. 
> Smith)
>
> 12
> !
> 6264/5929
> 33232/29645
> 105592/88935
> 5322/4235
> 39597/29645
> 8352/5929
> 31706/21175
> 9396/5929
> 9952/5929
> 52796/29645
> 55782/29645
> 2/1
>
>
> ! Secor2_11WT.scl
> !
> George Secor's rational 2/11-comma well-temperament
> 12
> !
> 560/531
> 2643/2360
> 70/59
> 74/59
> 315/236
> 2240/1593
> 883/590
> 280/177
> 890/531
> 105/59
> 443/236
> 2/1
>
>
> ! Secor1_5WT.scl
> !
> George Secor's 1/5-comma well-temperament (ratios supplied by G. W. 
> Smith)
>  12
> !
>  256/243
>  75/67
>  32/27
>  2015/1608
>  4/3
>  9101/6480
>  401/268
>  128/81
>  1010/603
>  16/9
>  15/8
>  2/1
>
>
> ! Secor5_23WT.scl
> !
> George Secor's rational 5/23-comma proportional-beating well-
> temperament
>  12
> !
>  256/243
>  966664/863865
>  32/27
>  120176/95985
>  4/3
>  1024/729
>  143588/95985
>  128/81
>  160624/95985
>  16/9
>  4096/2187
>  2/1
>
>
> ! Secor1_5TX.scl
> !
> George Secor's 1/5-comma temperament extraordinaire (ratios supplied 
> by G. W. Smith)
>  12
> !
>  5075/4824
>  75/67
>  28591/24120
>  2015/1608
>  805/603
>  5075/3618
>  401/268
>  5075/3216
>  1010/603
>  3220/1809
>  15/8
>  2/1
>
>
> ! Secor5_23TX.scl
> !
> George Secor's rational 5/23-comma temperament extraordinaire
>  12
> !
>  390/371
>  3321/2968
>  4397/3710
>  929/742
>  2476/1855
>  8325/5936
>  555/371
>  9365/5936
>  621/371
>  9904/5565
>  5559/2968
>  2/1
>
>
> ! Secor1_4TX.scl
> !
> George Secor's rational 1/4-comma temperament extraordinaire
>  12
> !
>  4873/4644
>  481/430
>  61837/52245
>  484/387
>  7747/5805
>  8132/5805
>  193/129
>  1219/774
>  1942/1161
>  30988/17415
>  3621/1935
>  2/1
>
>
> The latest version of the original temperament (extra)ordinaire that 
> you like so well is the 5/23-comma tuning (Secor5_23TX.scl).
>
> In addition to the above, I also have two well-temperaments in which 
> all 24 major and minor triads have relatively simple brats; see:
>
> http://groups.yahoo.com/group/tuning-math/message/13733
>
>   
>> Also, it might be a good idea to build a public scale archive for 
>>     
> general
>   
>> use, much like in wikipedia?
>>     
>
> What's wrong with the Scala archive -- if Manuel will add these?
>
> Best,
>
> --George
>
>
>
>
>
>
> __
>  
> ------------------------------------------------------------------------
>
>
>
>
>
>   

-- 
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

From: George D. Secor (2006-05-02)
Subject: Re: 1/7-comma proportional-beating WT - and all the others as well!

--- In [email protected], Kraig Grady <kraiggrady@...> wrote:
>
> Hello George!
> I hate to bring up a tangential thought to your post so early, but 
it 
> was something that came up during the Johnston Symposium both from 
Monzo 
> and myself.
>  This was the idea of notating ratios in terms of factors instead 
of the 
> resulting high number we sometimes use.
>  The advantage is that, at least, i have to figure it out this way 
> anyway when running across a ratio, i hadn't used before or very 
frequently.
>  This way one would also know the implied basic function of the 
ratio.
>  admittedly we don't always use intervals in this way. Thought you 
might 
> have some thoughts on this as a possibility.

Hi Kraig,

I am in complete agreement with you on this point, so much so, in 
fact, that for the past few years I have made it a general practice 
in my tuning spreadsheets to express the numbers of ratios as the 
calculated products of their factors whenever it's meaningful to do 
so.

>  I have BTW quite liked many of these scales you have come up along 
> these lines and look forward to trying this one out in the future 
when i 
> have a little less on my table.

OOPS!  Please disregard this latest one!!!  Although the paint's not 
dry yet, I've already come to the conclusion that I can do better 
(and expect that I'll eventually replace my 1/7-comma and 2/11-comma 
well-temperaments with something else).

BTW, getting back to factoring ratios, I hope you won't try to factor 
the ones with the largest numbers in the following one:

> > ! Secor1_5WT.scl
> > !> > George Secor's 1/5-comma well-temperament (ratios supplied 
by G. W. 
> > Smith)
> >  12
> > !
> >  256/243
> >  75/67
> >  32/27
> >  2015/1608
> >  4/3
> >  9101/6480
> >  401/268
> >  128/81
> >  1010/603
> >  16/9
> >  15/8
> >  2/1

Most of the numbers are so small that you're apt to think it's JI.  
However, if you calculate the number of cents in each tone, you'll 
find that all of the fifths in a chain from C to B are 4.3 cents 
(rounded to 1 decimal place), so essentially this really is a *1/5-
comma temperament*.

--George

From: Ozan Yarman (2006-05-04)
Subject: Re: [tuning] 1/7-comma proportional-beating WT - and all the others as well!

Dear George,



> --- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > With all due respect to your efforts, could you not do a little
> better dear
> > George? I had the impression that the previous temperament extra-
> ordinaire
> > had it all.
>
> Oh, yes, I quite agree that it did (and still does) have it all!
>


Glad to hear it!


>
> However, "all" is evidently too much for some people, as Paul Bailey
> wrote me (off-list) that although he was initially impressed by it,
> he wouldn't be forwarding my 1/7-comma temperament to "a few other
> piano tuners" because  he remembered that:
>
> << I've learned from other tuners [that] 18 cents is about the
> biggest [major] third that's acceptable to 'academic' pianists. >>
>
> My 1/7th-comma WT has two major thirds 19.65 cents wide.  I sent the
> following to Paul today for comment (with widest major 3rds +18.49
> cents, but many of the brats not as simple):
>


Nonsense. They just don't wish to exploit the possibilities.


> ! Secor1_7MCRWT.scl
> !
> George Secor's 1/7-comma minimum-contrast rational well-temperament
> 12
> !
>  14285/13512
>  5049/4504
>  428221/360320
>  1415/1126
>  752/563
>  25407/18016
>  843/563
>  1785/1126
>  945/563
>  80249/45040
>  8475/4504
>  2/1
>


Urm. I did not like this one too much I'm afraid.


> > One more thing if I may... could you send me all your related 12-
> tone Scala
> > files in a zip folder? I got them all messed up here in my hardrive.
>
> It's no problem to list them all here (in ascending order of key
> contrast).  They all have proportional-beating triads, both major and
> minor.  If a temperament is too tame (or too wild) for your purposes,
> then just try one farther down (or up) the list.  You'll notice that
> Gene Ward Smith is credited with determining the ratios for some of
> these:
>


They are all swell, but the copy-paste procedure is a tad tiresome. I would
be truly glad if you could send not only the SCL files, but also the excel
documents detailing the BRATs for further analysis.


>
> ! Secor1_7WT.scl
> !
> George Secor's 1/7-comma well-temperament (ratios supplied by G. W.
> Smith)
>
> 12
> !
> 6264/5929
> 33232/29645
> 105592/88935
> 5322/4235
> 39597/29645
> 8352/5929
> 31706/21175
> 9396/5929
> 9952/5929
> 52796/29645
> 55782/29645
> 2/1
>


This one is wild and brilliant.


>
> ! Secor2_11WT.scl
> !
> George Secor's rational 2/11-comma well-temperament
> 12
> !
> 560/531
> 2643/2360
> 70/59
> 74/59
> 315/236
> 2240/1593
> 883/590
> 280/177
> 890/531
> 105/59
> 443/236
> 2/1
>


More tamed, but still sparkly.


>
> ! Secor1_5WT.scl
> !
> George Secor's 1/5-comma well-temperament (ratios supplied by G. W.
> Smith)
>  12
> !
>  256/243
>  75/67
>  32/27
>  2015/1608
>  4/3
>  9101/6480
>  401/268
>  128/81
>  1010/603
>  16/9
>  15/8
>  2/1
>


Quite balanced, save a few beating glitches.


>
> ! Secor5_23WT.scl
> !
> George Secor's rational 5/23-comma proportional-beating well-
> temperament
>  12
> !
>  256/243
>  966664/863865
>  32/27
>  120176/95985
>  4/3
>  1024/729
>  143588/95985
>  128/81
>  160624/95985
>  16/9
>  4096/2187
>  2/1
>

Not quite to my liking.

>
> ! Secor1_5TX.scl
> !
> George Secor's 1/5-comma temperament extraordinaire (ratios supplied
> by G. W. Smith)
>  12
> !
>  5075/4824
>  75/67
>  28591/24120
>  2015/1608
>  805/603
>  5075/3618
>  401/268
>  5075/3216
>  1010/603
>  3220/1809
>  15/8
>  2/1


Intriguing.

>
>
> ! Secor5_23TX.scl
> !
> George Secor's rational 5/23-comma temperament extraordinaire
>  12
> !
>  390/371
>  3321/2968
>  4397/3710
>  929/742
>  2476/1855
>  8325/5936
>  555/371
>  9365/5936
>  621/371
>  9904/5565
>  5559/2968
>  2/1
>


This one is simply terrific, though lacking the brilliance of the previous
listed, which can be compensated by sheer volume of sound I believe.

I made a hasty recording using this temperament on FTS:

http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3

My hands were too crude for the plastic keyboard of Yamaha P-200. My
apologies beforehand for the reviling pianism.


>
> ! Secor1_4TX.scl
> !
> George Secor's rational 1/4-comma temperament extraordinaire
>  12
> !
>  4873/4644
>  481/430
>  61837/52245
>  484/387
>  7747/5805
>  8132/5805
>  193/129
>  1219/774
>  1942/1161
>  30988/17415
>  3621/1935
>  2/1
>
>


This one is extra rugged.


> The latest version of the original temperament (extra)ordinaire that
> you like so well is the 5/23-comma tuning (Secor5_23TX.scl).
>
> In addition to the above, I also have two well-temperaments in which
> all 24 major and minor triads have relatively simple brats; see:
>
> http://groups.yahoo.com/group/tuning-math/message/13733
>


Are they not extraordinary?


> > Also, it might be a good idea to build a public scale archive for
> general
> > use, much like in wikipedia?
>
> What's wrong with the Scala archive -- if Manuel will add these?
>


The issue is, this really should not be a job cast over the shoulders of one
person, however hard-working. There are simply too many scales to track and
catalogue. Minute changes and corrections may be forgotten or omitted
altogether due to confusion. Scale archiving could be made a public effort
within the tuning list. In this way, we could have a Wikipedia-like database
and supply extra information for each scale. I for one would like to
describe my 79 MOS 159-tET in detail in several pages. CVs are also
important! It should be possible to view scales not only by the name of
their creator, but date of creation, cardinality, equalness of intervals,
etc...


> Best,
>
> --George
>
>
>

Cordially,
Oz.

From: Carl Lumma (2006-05-04)
Subject: Re: 1/7-comma proportional-beating WT - and all the others as well!

> > ! Secor5_23TX.scl
> > !
> > George Secor's rational 5/23-comma temperament extraordinaire
> >  12
> > !
> >  390/371
> >  3321/2968
> >  4397/3710
> >  929/742
> >  2476/1855
> >  8325/5936
> >  555/371
> >  9365/5936
> >  621/371
> >  9904/5565
> >  5559/2968
> >  2/1
> 
> This one is simply terrific, though lacking the brilliance of
> the previous listed, which can be compensated by sheer volume
> of sound I believe.
> 
> I made a hasty recording using this temperament on FTS:
> 
> http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
> 
> My hands were too crude for the plastic keyboard of Yamaha P-200.
> My apologies beforehand for the reviling pianism.

Not at all.  This is quite nice.  Though for people like me,
and I think we're in the minority around here, sounds like
those at 1:02, though bareable and sometimes even flavorful,
are in the end not worth the improved consonance of the 'near'
keys relative to 12-equal.

-Carl

From: George D. Secor (2006-05-04)
Subject: Re: 1/7-comma proportional-beating WT - and all the others as well!

--- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
> ...
> [GS:]
> > My 1/7th-comma WT has two major thirds 19.65 cents wide.  I sent 
the
> > following to Paul today for comment (with widest major 3rds +18.49
> > cents, but many of the brats not as simple):
> > 
> > ! Secor1_7MCRWT.scl
> > !
> > George Secor's 1/7-comma minimum-contrast rational well-
temperament
> > ...
> 
> Urm. I did not like this one too much I'm afraid.

I don't either -- I said I was scrapping it, because I'm working on 
something that's much better, which I believe will be acceptable 
to 'academic' pianists.  (I expect even Carl will like it. :-)

> > > One more thing if I may... could you send me all your related 
12-tone Scala
> > > files in a zip folder? I got them all messed up here in my 
hardrive.
> >
> > It's no problem to list them all here (in ascending order of key
> > contrast).  They all have proportional-beating triads, both major 
and
> > minor.  If a temperament is too tame (or too wild) for your 
purposes,
> > then just try one farther down (or up) the list.  You'll notice 
that
> > Gene Ward Smith is credited with determining the ratios for some 
of
> > these:
> 
> They are all swell, but the copy-paste procedure is a tad tiresome. 
I would
> be truly glad if you could send not only the SCL files, but also 
the excel
> documents detailing the BRATs for further analysis.

Okay, I'll have to send those directly to you (soon, when I have a 
little more time), since they would take up too much space in the 
files section.

> > ! Secor1_7WT.scl
> > !
> > George Secor's 1/7-comma well-temperament (ratios supplied by G. 
W.
> > Smith)
> > ...
> 
> This one is wild and brilliant.

Hmmm, that's interesting, because I think I can improve on this one 
as well.

> >
> > ! Secor2_11WT.scl
> > !
> > George Secor's rational 2/11-comma well-temperament
> > 12
> > !
> > 560/531
> > 2643/2360
> > 70/59
> > 74/59
> > 315/236
> > 2240/1593
> > 883/590
> > 280/177
> > 890/531
> > 105/59
> > 443/236
> > 2/1
> 
> More tamed, ...

I don't quite understand why you would think so, because it has a 
greater key contrast the the 1/7-comma temperament.

> >
> > ! Secor1_5WT.scl
> > !
> > George Secor's 1/5-comma well-temperament (ratios supplied by G. 
W.
> > Smith)
> >  ...
> 
> Quite balanced, save a few beating glitches.

Okay.

> >
> > ! Secor5_23WT.scl
> > !
> > George Secor's rational 5/23-comma proportional-beating well-
> > temperament
> >  ...
> 
> Not quite to my liking.

I guess you like the consonance spread out to more keys, as in the 
1/5-comma WT that precedes it.  The 5/23-comma WT one has 7 just 
fifths, hence is relatively easy to tune.  Also, it contains a lot of 
simple brats, hence is one of my favorites.

> >
> > ! Secor1_5TX.scl
> > !
> > George Secor's 1/5-comma temperament extraordinaire (ratios 
supplied
> > by G. W. Smith)
> > ...
> 
> Intriguing.

It has a lot of exact brats (both major & minor), and most of the 
others approximate relatively simple numbers.

> >
> > ! Secor5_23TX.scl
> > !
> > George Secor's rational 5/23-comma temperament extraordinaire
> >  ...
> 
> This one is simply terrific, though lacking the brilliance of the 
previous
> listed, which can be compensated by sheer volume of sound I believe.

As I indicated, this is the latest version of the original 
temperament (extra)ordinaire that you liked so well.

> I made a hasty recording using this temperament on FTS:
> 
> http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
> 
> My hands were too crude for the plastic keyboard of Yamaha P-200. My
> apologies beforehand for the reviling pianism.

Sorry, I won't be able to listen to this until later.

> >
> > ! Secor1_4TX.scl
> > !
> > George Secor's rational 1/4-comma temperament extraordinaire
> >  ...
> 
> This one is extra rugged.

You betcha!  Its maximum contrast is not for the faint of heart.  Did 
you notice how consonant the A major triad is?

> ...
> > > Also, it might be a good idea to build a public scale archive 
for general
> > > use, much like in wikipedia?
> >
> > What's wrong with the Scala archive -- if Manuel will add these?
> 
> The issue is, this really should not be a job cast over the 
shoulders of one
> person, however hard-working. There are simply too many scales to 
track and
> catalogue. Minute changes and corrections may be forgotten or 
omitted
> altogether due to confusion. Scale archiving could be made a public 
effort
> within the tuning list. In this way, we could have a Wikipedia-like 
database
> and supply extra information for each scale. I for one would like to
> describe my 79 MOS 159-tET in detail in several pages. CVs are also
> important! It should be possible to view scales not only by the 
name of
> their creator, but date of creation, cardinality, equalness of 
intervals,
> etc...

Are you volunteering to chair a committee?  ;-)

Best,

--George

From: George D. Secor (2006-05-04)
Subject: Re: 1/7-comma proportional-beating WT - and all the others as well!

--- In [email protected], "Carl Lumma" <clumma@...> wrote:
> [GS:]
> > > ! Secor5_23TX.scl
> > > !
> > > George Secor's rational 5/23-comma temperament extraordinaire
> > >  ...
> > [Oz:]
> > I made a hasty recording using this temperament on FTS:
> > 
> > http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
> > 
> > My hands were too crude for the plastic keyboard of Yamaha P-200.
> > My apologies beforehand for the reviling pianism.
> 
> Not at all.  This is quite nice.  Though for people like me,
> and I think we're in the minority around here, sounds like
> those at 1:02, though bareable and sometimes even flavorful,

I'm answering this without having heard the recording.  Your comment 
seems to indicate that I accomplished my objectives with this 
temperament (extra)ordinaire, which were:

1) to have no wolf fifths (in this I was successful, since all fifths 
are tempered by less than 4.7c), and
2) to have no triads (or thirds) that are tempered so much as to be 
judged 'unbearable' (open to debate, since it's a matter of opinion 
whether a major 3rd tempered 27c wide is bearable or not).

I'd be interested in your reaction to the "extra-rugged" 1/4-comma 
temperament, if Oz would be so kind as to make a recording of that 
one.

> are in the end not worth the improved consonance of the 'near'
> keys relative to 12-equal.

It depends on what you're using the temperament for.  It was intended 
principally for playing older music with no more than 3 sharps or 2 
flats in the key signature (i.e., the meantone keys), while allowing 
brief modulations into other keys without encountering any wolves.

--George

From: Carl Lumma (2006-05-06)
Subject: Re: 1/7-comma proportional-beating WT - and all the others as well!

> > Urm. I did not like this one too much I'm afraid.
> 
> I don't either -- I said I was scrapping it, because I'm working on 
> something that's much better, which I believe will be acceptable 
> to 'academic' pianists.  (I expect even Carl will like it. :-)

I'll watch this space!

-C.

From: Carl Lumma (2006-05-06)
Subject: Re: 1/7-comma proportional-beating WT - and all the others as well!

> > Not at all.  This is quite nice.  Though for people like me,
> > and I think we're in the minority around here, sounds like
> > those at 1:02, though bareable and sometimes even flavorful,
> > are in the end not worth the improved consonance of the 'near'
> > keys relative to 12-equal.
> 
> It depends on what you're using the temperament for.  It was
> intended principally for playing older music with no more than
> 3 sharps or 2 flats in the key signature (i.e., the meantone
> keys), while allowing brief modulations into other keys without
> encountering any wolves.

Of course.  Meantone is great for Byrd, etc.  But for Bach
and later, 27-wide 3rds just don't work for me *on the piano*.
I can stand them, but I don't prefer them.

-Carl

From: George D. Secor (2006-05-08)
Subject: Proportional-beating Victorian WT (was: 1/7-comma ...)

--- In [email protected], "Carl Lumma" <clumma@...> wrote:
>
> > > Urm. I did not like this one too much I'm afraid.
> > 
> > I don't either -- I said I was scrapping it, because I'm working 
on 
> > something that's much better, which I believe will be acceptable 
> > to 'academic' pianists.  (I expect even Carl will like it. :-)
> 
> I'll watch this space!
> 
> -C.

Okay, Carl, here's my all-purpose 12-ET replacement!

Offlist, Paul Bailey directed my attention to the Broadwood Victorian 
well-temperament (which was devised by none other than Alexander J. 
Ellis), with which I'm highly impressed, to say the least!  I 
concluded that the only way that it might be improved was to make 
some of the triads proportional-beating, which I did this past 
weekend:

! Secor-VRWT.scl
!
George Secor's Victorian rational well-temperament (based on Ellis #2)
12
!
 545/516
 6071/5418
 707447/595980
 755/602
 30118/22575
 30535/21672
 451/301
 34315/21672
 1514/903
 2755/1548
 20365/10836
 2/1

Here are the brats and total absolute error for each triad:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 35.0000 55.0000 1.5714 35.40
Bb... 25.0000 40.0000 1.6000 27.14 
F.... 15.0000 25.0000 1.6667 20.64
C....  5.0000 10.0000 2.0000 15.32
G....  2.0000  5.5000 2.7500 21.89
D....  4.0000  8.5000 2.1250 29.04
A....  5.0000 10.0000 2.0000 36.50
E.... 17.0000 28.0000 1.6471 37.88
B.... 43.0553 67.0829 1.5581 37.83
F#... 35.6667 56.0000 1.5701 37.91
C#... 30.0780 47.6170 1.5831 38.23
G#... 41.2901 64.4352 1.5605 37.62

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 44.5383 46.5383 1.0449 37.83
Bb... 42.3038 44.3038 1.0473 37.91
F.... 34.7415 36.7415 1.0576 38.23
C.... 17.5177 19.5177 1.1142 37.62
G....  5.0786  7.0786 1.3938 35.40
D....  4.3200  6.3200 1.4630 27.14
A....  2.5000  4.5000 1.8000 20.64
E....  7.0000  9.0000 1.2857 15.32
B.... 28.8000 30.8000 1.0694 21.89
F#... 32.0000 34.0000 1.0625 29.04
C#... 34.0000 36.0000 1.0588 36.50
G#... 49.3547 51.3547 1.0405 37.88

I judged that the most important thing was to have whole numbers for 
as many M3:5th ratios in the major triads as possible (beginning with 
the 4 or 5 most consonant, considering that the rest may amount to 
not much more than "eye candy"), and in the process I managed to get 
some fairly simple brat numbers in the minor triads.

The brats aren't as simple as in the 1/7-comma WT that I posted 
previously (as Secor1_7WT.scl), but the most and least consonant 
triads (and arguably the harmonic balance) are all better, 
respectively, in this latest one.

Comments, anyone?

--George

From: Carl Lumma (2006-05-08)
Subject: Re: Proportional-beating Victorian WT (was: 1/7-comma ...)

> Okay, Carl, here's my all-purpose 12-ET replacement!
> 
> Offlist, Paul Bailey directed my attention to the Broadwood
> Victorian well-temperament (which was devised by none other
> than Alexander J. Ellis), with which I'm highly impressed,
> to say the least!

Do you have Jorgensen?  Looks like something that ought to
be on Wikisource.  (It's over $400 on Amazon.)

Could you post the Broadwood tuning for comparison?
A Jorgensen TOC I found on the web shows two of
them; "usual" and "best".

> I concluded that the only way that it might be improved was
> to make some of the triads proportional-beating, which I did
> this past weekend:
> 
> ! Secor-VRWT.scl
> !
> George Secor's Victorian rational well-temperament
> 12
> !
>  545/516
>  6071/5418
>  707447/595980
>  755/602
>  30118/22575
>  30535/21672
>  451/301
>  34315/21672
>  1514/903
>  2755/1548
>  20365/10836
>  2/1
> 
//
> The brats aren't as simple as in the 1/7-comma WT that I posted 
> previously (as Secor1_7WT.scl), but the most and least consonant 
> triads (and arguably the harmonic balance) are all better, 
> respectively, in this latest one.
> 
> Comments, anyone?

With a worst major 3rd of 405 cents, this is something I'll
want to play.  Having just moved, however, I don't have my
keyboard set up at the moment...

-Carl

From: George D. Secor (2006-05-08)
Subject: Re: Proportional-beating Victorian WT (was: 1/7-comma ...)

--- In [email protected], "Carl Lumma" <clumma@...> wrote:
> 
> Could you post the Broadwood tuning for comparison?
> A Jorgensen TOC I found on the web shows two of
> them; "usual" and "best".

This is the "usual", which I like better than the "best". ;-)

! Broadwood.scl
!
Broadwood's Usual (Ellis #2) Victorian Well-temperament
 12
!
 95.00000
 197.00000
 297.00000
 392.00000
 499.00000
 594.00000
 700.00000
 796.00000
 894.00000
 998.00000
 1093.00000
 2/1

--George

From: George D. Secor (2006-05-09)
Subject: Rational temperament brat & beat calculator (was: 1/7-comma ...)

--- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
> ...
> > > One more thing if I may... could you send me all your related 
12-tone Scala
> > > files in a zip folder? I got them all messed up here in my 
hardrive.
> >
> > It's no problem to list them all here (in ascending order of key
> > contrast).  They all have proportional-beating triads, both major 
and
> > minor.  If a temperament is too tame (or too wild) for your 
purposes,
> > then just try one farther down (or up) the list.  You'll notice 
that
> > Gene Ward Smith is credited with determining the ratios for some 
of
> > these:
> 
> They are all swell, but the copy-paste procedure is a tad tiresome. 
I would
> be truly glad if you could send not only the SCL files, but also 
the excel
> documents detailing the BRATs for further analysis.

I'll be sending these to you shortly.  Anyone else wanting to analyze 
the brats in a rational temperament can download the file in the 
following link and input the numbers themselves from any of the 
rational-temperament .scl file listings that Gene and I have posted 
(into the cyan-colored cells):

http://groups.yahoo.com/group/tuning-math/files/secor/RT-wksht.xls

(You need to be a member of the tuning-math group to access the file.)

This is the actual worksheet that I've been using to construct 
rational temperaments.  The beat ratios are in columns O thru Q (page 
downward to see the minor triads); the formal "brat" numbers that 
Gene & I have been using are in column Q.  The beat rates of the 
individual intervals in each triad are in columns S thru U, and the 
frequencies of each tone are in columns W thru Y.  If you wish, you 
can change the reference frequency (of A) in cell K2, but that will 
not affect the numbers in any columns to the right of S.

Should you want to see the brats for a temperament where the pitches 
are given in cents, you can input those numbers directly into cells 
E8-10,12-19.  (This will overwrite the formulas in those cells, but 
they can be restored easily by copying cell E11.)

--George

From: Aaron Krister Johnson (2006-05-10)
Subject: Re: 1/7-comma proportional-beating well-temperament

--- In [email protected], "George D. Secor" <gdsecor@...> wrote:
> http://launch.groups.yahoo.com/group/tuning/message/66183

George,

I haven't been checking out the main 'tuning list' lately, but I was
happy to come across you above mentioned post when I did a casual
glance the other day....

We both share an attraction for the beat-ratios in 1/7-comma...and I'm
intrigued by your rational variation that was inspired by my post a
while back on 1/7-comma a while back...thanks for post this, and I'll
check it out when I have time to be away from cleaning my daughter's
diapers! ;)

Best,
Aaron

From: George D. Secor (2006-05-10)
Subject: Re: 1/7-comma proportional-beating well-temperament

--- In [email protected], "Aaron Krister Johnson" <aaron@...> 
wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@> wrote:
> > http://launch.groups.yahoo.com/group/tuning/message/66183
> 
> George,
> 
> I haven't been checking out the main 'tuning list' lately, but I was
> happy to come across you above mentioned post when I did a casual
> glance the other day....
> 
> We both share an attraction for the beat-ratios in 1/7-comma...and 
I'm
> intrigued by your rational variation that was inspired by my post a
> while back on 1/7-comma a while back...thanks for post this, and 
I'll
> check it out when I have time to be away from cleaning my daughter's
> diapers! ;)
> 
> Best,
> Aaron

Aaron, thanks for taking time out from some highly important 
unschedulable tasks to reply.  This brings back memories of times 
past, when I was in the very same situation.  Once my daughter was 
born, I found not much time over the next 20 years to spend on 
microtonality (with no regrets :-).

Should you find a few spare moments ;-), I'd be interested in your 
opinion regarding the following.  I wrote (excerpt from msg. #62222):

<< Paul Bailey wrote me (off-list) that although he was initially 
impressed by it, he wouldn't be forwarding my 1/7-comma temperament 
to "a few other piano tuners" because he remembered that, "I've 
learned from other tuners [that] 18 cents is about the biggest 
[major] third that's acceptable to 'academic' pianists." >>

Since my 1/7-comma WT has a max M3 error of +19.66c, it was back to 
the drawing board.  After an insufficiently successful attempt to 
modify it, I came up with a rational temperament based on Ellis #2, 
a/k/a Broadwood's "usual" WT, with max M3 error of +18.20c:
http://groups.yahoo.com/group/tuning/message/66329

At the moment, I think I prefer this to the 1/7-comma RWT, but I'd 
like to get a few more opinions.

BTW, do you remember your challenge
http://groups.yahoo.com/group/tuning/message/61731
to find a temperament with the simplest proportional-beating major 
*and* minor triads?  If you revisit that thread, you'll find a bunch 
of replies with a few nice tries, none of which met all of the 
requirements.  I also inferred that the solution "should, at the very 
least, be reasonably useful from a musical standpoint [and] ... 
should sound good" (msg. #62031).

At the beginning of January, when I was messing around with the 
numbers in some spreadsheets containing equations that Gene had given 
me (as a "Christmas present for George"), I unexpectedly stumbled 
upon an answer (in the form of a very-high-contrast well-temperament):

http://groups.yahoo.com/group/tuning/message/63293

Best,

--George

From: Aaron Krister Johnson (2006-05-11)
Subject: the simplest pan-proportionally beating 12-tone temperament (remix)

--- In [email protected], "George D. Secor" <gdsecor@...> wrote:

> << Paul Bailey wrote me (off-list) that although he was initially 
> impressed by it, he wouldn't be forwarding my 1/7-comma temperament 
> to "a few other piano tuners" because he remembered that, "I've 
> learned from other tuners [that] 18 cents is about the biggest 
> [major] third that's acceptable to 'academic' pianists." >>

I can definately see how when M3rds get too wide, it might sound a bit
too subversive for most classical music. BTW, I assume you mean 18
cents wide of a 5/4?
 
> Since my 1/7-comma WT has a max M3 error of +19.66c, it was back to 
> the drawing board.  After an insufficiently successful attempt to 
> modify it, I came up with a rational temperament based on Ellis #2, 
> a/k/a Broadwood's "usual" WT, with max M3 error of +18.20c:
> http://groups.yahoo.com/group/tuning/message/66329
> 
> At the moment, I think I prefer this to the 1/7-comma RWT, but I'd 
> like to get a few more opinions.

I'm excited to check it out, even more so about what you write below!:


> BTW, do you remember your challenge
> http://groups.yahoo.com/group/tuning/message/61731
> to find a temperament with the simplest proportional-beating major 
> *and* minor triads?  If you revisit that thread, you'll find a bunch 
> of replies with a few nice tries, none of which met all of the 
> requirements.  I also inferred that the solution "should, at the very 
> least, be reasonably useful from a musical standpoint [and] ... 
> should sound good" (msg. #62031).
> 
> At the beginning of January, when I was messing around with the 
> numbers in some spreadsheets containing equations that Gene had given 
> me (as a "Christmas present for George"), I unexpectedly stumbled 
> upon an answer (in the form of a very-high-contrast well-temperament):
> 
> http://groups.yahoo.com/group/tuning/message/63293

Bravo! this looks incredible....from a pure-math point of view, is it
possible to rigorously prove that this or anything else is the optimal
solution?

I'll check out the *sound* when I get a chance later (diapers come first)

Best,
Aaron.

From: George D. Secor (2006-05-11)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament (remix)

--- In [email protected], "Aaron Krister Johnson" <aaron@...> 
wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@> wrote:
> 
> > << Paul Bailey wrote me (off-list) that although he was initially 
> > impressed by it, he wouldn't be forwarding my 1/7-comma 
temperament 
> > to "a few other piano tuners" because he remembered that, "I've 
> > learned from other tuners [that] 18 cents is about the biggest 
> > [major] third that's acceptable to 'academic' pianists." >>
> 
> I can definately see how when M3rds get too wide, it might sound a 
bit
> too subversive for most classical music. BTW, I assume you mean 18
> cents wide of a 5/4?

Yep.

> > Since my 1/7-comma WT has a max M3 error of +19.66c, it was back 
to 
> > the drawing board.  ...
> 
> I'm excited to check it out, even more so about what you write 
below!:
> 
> > BTW, do you remember your challenge
> > http://groups.yahoo.com/group/tuning/message/61731
> > to find a temperament with the simplest proportional-beating 
major 
> > *and* minor triads?  ...
> > 
> > At the beginning of January ... I unexpectedly stumbled 
> > upon an answer (in the form of a very-high-contrast well-
temperament):
> > 
> > http://groups.yahoo.com/group/tuning/message/63293
> 
> Bravo! this looks incredible....from a pure-math point of view, is 
it
> possible to rigorously prove that this or anything else is the 
optimal
> solution?

That's something Gene would be much better equipped to answer.  There 
would need to be some sort of constraint on how the pitches are 
related to one another.  For example, if all 12 tones had the ratio 
1/1, then there would be 24 extremely simple brats -- but this isn't 
what we had in mind.  Possibly it would need to be specified that the 
tones should be in a circle of fifths, with limits on how much the 
fifths could be tempered.

> I'll check out the *sound* when I get a chance later (diapers come 
first)

Have fun! ;-) -- and congratulations! :-D

--George

From: Gene Ward Smith (2006-05-12)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament (remix)

--- In [email protected], "George D. Secor" <gdsecor@...> wrote:

> > Bravo! this looks incredible....from a pure-math point of view, is 
> it
> > possible to rigorously prove that this or anything else is the 
> optimal
> > solution?
> 
> That's something Gene would be much better equipped to answer.

The most important requirement is a precise definition of "optimal".

From: Aaron Krister Johnson (2006-05-12)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament (remix)

--- In [email protected], "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@> wrote:
> 
> > > Bravo! this looks incredible....from a pure-math point of view, is 
> > it
> > > possible to rigorously prove that this or anything else is the 
> > optimal
> > > solution?
> > 
> > That's something Gene would be much better equipped to answer.
> 
> The most important requirement is a precise definition of "optimal".
>

we migght say 'minimal' instead of 'optimal'--perhaps the least sum of
numerators and denominators of the brats would serve?

From: Aaron Krister Johnson (2006-05-12)
Subject: Question for George

George,

In examining the 'WTPB-24a.scl' and 'WTPB-24b.scl' files, I see they
both contain chains of 6 pure 3/2s--putting it in the Valotti-Young
category...

question--would it be possible to have this low level of brats without
so many consecutive 3/2s? Or does it depend on them?

I notice that Wendell's Natural Synchronous doesn't use more than 3
3/2s in a row, but then again, it doesn't have the low brats for minor
triads that yours do.

Best,
Aaron.

From: George D. Secor (2006-05-12)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament (remix)

--- In [email protected], "Gene Ward Smith" <genewardsmith@...> 
wrote:
>
> --- In [email protected], "George D. Secor" <gdsecor@> wrote:
> 
> > > Bravo! this looks incredible....from a pure-math point of view, 
is it
> > > possible to rigorously prove that this or anything else is the 
optimal
> > > solution?
> > 
> > That's something Gene would be much better equipped to answer.
> 
> The most important requirement is a precise definition of "optimal".

For starters, look at Aaron's wording:

<< what is the simplest possible 12-note temperament where all 24 
major and minor triads have rationally proportional beating? 
here 'simplest' means that the brats (beat ratios for the un-
initiated) are the lowest numbers in the numerator and denominator 
that they can be..... >>

This leaves it pretty much up to your discretion as to what the order 
of simplicity for brats should be, but I think you already have some 
idea what this ought to be, since you've already put major brats of 
3/2 and 4 at the top of the list, followed by 2 and 3.  Other simple 
numbers are 7/3 (giving 3 for M3/5th and 7 for m3/5th) and 11/4 
(giving 2 for M3/5th and 11/2 for m3/5th).  I think it's important 
that all 3 ratios -- M3/5th, m3/5th, and m3/M3 -- be relatively 
simple: the more integers, the better.

The simplest minor brats can be determined similarly (with 1, 2, and 
3/2 topping the list).

My additional requirements are, at face value, a bit more subjective:

<< the solution should, at the very least, be reasonably useful from 
a musical standpoint [and] ... should sound good >>

For example, we could specify that no fifth be tempered by more than 
1/2-comma.  Also, in a well-constructed temperament the brats 
shouldn't zig-zag wildly as one goes around the circle of fifths.

If this isn't specific enough for you, then perhaps you could send me 
the simultaneous equations containing 11 independent major-triad brat 
variables that you've been using to determine the ratios for the 11 
pitches (relative to 1/1) that will produce those brats.  If I put 
those into a spreadsheet, then by trial and error I might find some 
more 24-simple-brat solutions, or at least, in the process, gravitate 
toward some sort of algorithm that you could use in a program to 
generate possible solutions.

We may want to continue this on tuning-math.

--George

From: George D. Secor (2006-05-12)
Subject: Re: Question for George

--- In [email protected], "Aaron Krister Johnson" <aaron@...> 
wrote:
> 
> George,
> 
> In examining the 'WTPB-24a.scl' and 'WTPB-24b.scl' files, I see they
> both contain chains of 6 pure 3/2s--putting it in the Valotti-Young
> category...

In general, it's a good practice in a well-temperament not to have 
the sizes of the fifths jumping back and forth as you go around the 
circle.  Instead, the narrowest ones should be between two natural 
notes, and the widest ones will usually be pure.  This will result in 
a smooth progression of mood or color as one goes around the circle.

24a is slightly less desirable than 24b in this respect -- but it 
does have better brat numbers (and higher contrast).

> question--would it be possible to have this low level of brats 
without
> so many consecutive 3/2s? Or does it depend on them?

Any triad with a pure 5th, either major or minor, will automatically 
have an exact brat (of 3/2 or 1, respectively), so having a lot of 
pure 5ths in a temperament is an easy way to get a lot of exact 
brats.  It has nothing to do with whether or not the 3/2's are 
consecutive.

> I notice that Wendell's Natural Synchronous doesn't use more than 3
> 3/2s in a row, but then again, it doesn't have the low brats for 
minor
> triads that yours do.

It's only because Robert didn't attempt to get low minor brats.  I 
don't think it has anything to do with the zig-zag fifth sizes.

Having the fifth sizes jump back and forth also tends to lower the 
key contrast.  It's rather wasteful IMO to have fifths tempered by 
more than 4 or 5 cents and not have any highly consonant triads to 
show for it.  (Compare this with my 2/11 or 1/5-comma rational WT's, 
which have all of the fifths tempered by less than 4.0 and 4.32 
cents, respectively.)

Pssst!  If there's any one trick for getting 24 simple brats, it's 
this: make the numerators and denominators in your pitch ratios 
fairly small, keeping as many as you can under 4 digits.  (Egad!  My 
secret is out!  I hope Gene doesn't read this. ;-)

--George

From: Carl Lumma (2006-05-12)
Subject: Re: Question for George

> In general, it's a good practice in a well-temperament not to have 
> the sizes of the fifths jumping back and forth as you go around the 
> circle.  Instead, the narrowest ones should be between two natural 
> notes, and the widest ones will usually be pure.  This will result
> in a smooth progression of mood or color as one goes around the
> circle.

Which might not be important to those who like to modulate by
thirds.

-Carl

From: Gene Ward Smith (2006-05-12)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament (remix)

--- In [email protected], "George D. Secor" <gdsecor@...> wrote:

> This leaves it pretty much up to your discretion as to what the order 
> of simplicity for brats should be, but I think you already have some 
> idea what this ought to be, since you've already put major brats of 
> 3/2 and 4 at the top of the list, followed by 2 and 3.

That's for brats in the vicinity of the 12-et brat. Simpler brats are
-1, 0, and infinity, of course.

From: Ozan Yarman (2006-05-19)
Subject: Re: [tuning] Re: 1/7-comma proportional-beating WT - and all the others as well!

Dear George,

SNIP

> >
> > ! Secor1_4TX.scl
> > !
> > George Secor's rational 1/4-comma temperament extraordinaire
> >  ...
> 
> This one is extra rugged.

You betcha!  Its maximum contrast is not for the faint of heart.  Did 
you notice how consonant the A major triad is?


~~~~~ Indeed, it is very similar to my 79MOS159-tET tempered A major triad:

1. Copy the scala file at www.ozanyarman.com\anonymous to the Scala folder.
2. Edit file and point to the actual scala folder on your hardrive.
3. Download and run the 79MOS159tET.cmd file in SCALA.

Also check out the incomplete Excel file at the same address and tell me what you think!


> ...
> > > Also, it might be a good idea to build a public scale archive 
for general
> > > use, much like in wikipedia?
> >
> > What's wrong with the Scala archive -- if Manuel will add these?
> 
> The issue is, this really should not be a job cast over the 
shoulders of one
> person, however hard-working. There are simply too many scales to 
track and
> catalogue. Minute changes and corrections may be forgotten or 
omitted
> altogether due to confusion. Scale archiving could be made a public 
effort
> within the tuning list. In this way, we could have a Wikipedia-like 
database
> and supply extra information for each scale. I for one would like to
> describe my 79 MOS 159-tET in detail in several pages. CVs are also
> important! It should be possible to view scales not only by the 
name of
> their creator, but date of creation, cardinality, equalness of 
intervals,
> etc...

Are you volunteering to chair a committee?  ;-)



~~~~~~~ Ack! no! I'm not the kind of guy competent for that stuff.


Best,

--George



~~~~~~

Cordially,
Oz.

From: Ozan Yarman (2006-05-19)
Subject: Re: [tuning] Re: 1/7-comma proportional-beating WT - and all the others as well!

George,

SNIP

>
> I'd be interested in your reaction to the "extra-rugged" 1/4-comma
> temperament, if Oz would be so kind as to make a recording of that
> one.
>

Unfortunately, I will be busy these few weeks, so the recording shall have
to wait a while.

Cordially,
Oz.

From: Ozan Yarman (2006-05-19)
Subject: Re: [tuning] Re: 1/7-comma proportional-beating WT - and all the others as well!

I'm pleased that you like it. Sorry for the late reply. I have an important
presentation to prepare in 9 days on my extended and improved 79-tone Qanun.
Returned from Izmir, will tell more later.

Oz.

----- Original Message -----
From: "Carl Lumma" <[email protected]>
To: <[email protected]>
Sent: 04 Mayıs 2006 Perşembe 6:09
Subject: [tuning] Re: 1/7-comma proportional-beating WT - and all the others
as well!


> > > ! Secor5_23TX.scl
> > > !
> > > George Secor's rational 5/23-comma temperament extraordinaire
> > >  12
> > > !
> > >  390/371
> > >  3321/2968
> > >  4397/3710
> > >  929/742
> > >  2476/1855
> > >  8325/5936
> > >  555/371
> > >  9365/5936
> > >  621/371
> > >  9904/5565
> > >  5559/2968
> > >  2/1
> >
> > This one is simply terrific, though lacking the brilliance of
> > the previous listed, which can be compensated by sheer volume
> > of sound I believe.
> >
> > I made a hasty recording using this temperament on FTS:
> >
> > http://www.ozanyarman.com/anonymous/Piyano-Secor.mp3
> >
> > My hands were too crude for the plastic keyboard of Yamaha P-200.
> > My apologies beforehand for the reviling pianism.
>
> Not at all.  This is quite nice.  Though for people like me,
> and I think we're in the minority around here, sounds like
> those at 1:02, though bareable and sometimes even flavorful,
> are in the end not worth the improved consonance of the 'near'
> keys relative to 12-equal.
>
> -Carl
>

From: George D. Secor (2006-05-19)
Subject: Re: 1/7-comma proportional-beating WT - and all the others as well!

--- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
>
> Dear George,
> 
> SNIP
> 
> > >
> > > ! Secor1_4TX.scl
> > > !
> > > George Secor's rational 1/4-comma temperament extraordinaire
> > >  ...
> > 
> > This one is extra rugged.
> 
> You betcha!  Its maximum contrast is not for the faint of heart.  
Did 
> you notice how consonant the A major triad is?
> 
> ~~~~~ Indeed, it is very similar to my 79MOS159-tET tempered A 
major triad:
> 
> 1. Copy the scala file at www.ozanyarman.com\anonymous to the Scala 
folder.
> 2. Edit file and point to the actual scala folder on your hardrive.
> 3. Download and run the 79MOS159tET.cmd file in SCALA.
> 
> Also check out the incomplete Excel file at the same address and 
tell me what you think!

Okay, I've downloaded these, but I'm taking a break (away from the 
Internet) for the next week or so and won't be able to respond until 
after next week.

Best,

--George