Full thread (36 messages)
From: monz (2001-05-15)
Subject:
I feel I've given enough attention to Blackjack; now it's
time for Canasta (a.k.a. MIRACLE-21).
Dave Keenan asked, somewhere in the mass of posts on this
thread today, about preserving the generator consistently
in a keyboard mapping.
I've come up with a mapping of Canasta to the Ztar keyboard
which does that *and* amazingly gives an approximation that
preserves something something very familiar to string players,
altho here it's the bowed string family and not the guitar:
If each key (or Ztar "fret") to the right is one 7/72-"octave"
higher than the one to its left, and we duplicate every 7th
"fret" on the next "string", each "string" will be a 12-EDO
"perfect 5th" higher than the previous one!
F^ 5&1/2
Bb^10&1/2 C< 11&2/3 C#- 5/6 D 2 Eb+ 3&1/6 E> 4&1/3 F^ 5&1/2
Eb^ 3&1/2 F< 4&2/3 F#- 5&5/6 G 7 G#+ 8&1/6 A> 9&1/3 Bb^10&1/2
G#^ 8&1/2 Bb< 9&2/3 B- 10&5/6 C 0 C#+ 1&1/6 D> 2&1/3 Eb^ 3&1/2
C#^ 1&1/2 Eb< 2&2/3 E- 3&5/6 F 5 F#+ 6&1/6 G> 7&1/3 G#^ 8&1/2
F#^ 6&1/2 G#< 7&2/3 A- 8&5/6 Bb 10 B+ 11&1/6 C> 1/3 C#^ 1&1/2
(sorry about squeezing the diagram so much... ASCII is so
unforgiving...)
Of course, the orchestral strings tune their strings to
*Pythagorean* 3:2 "perfect 5ths"... that's why I say this
is an approximation to that.
One problem here is that chords are only available in certain
inversions. I didn't say it was perfect... but it's another
mapping idea, and only my second attempt at Canasta.
It's interesting to me that this mapping is a perpendicular
cousin to the other Canasta mapping I made, where 6 x 5 = 30
keys form an array and there is one left over, with 5 keys
being "wasted".
-monz
http://www.monz.org
"All roads lead to n^0"
From: monz (2001-05-15)
Subject:
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/22793
>
> I feel I've given enough attention to Blackjack; now it's
> time for Canasta (a.k.a. MIRACLE-21).
Oops!... my bad again. Of course that's MIRACLE-31.
Blackjack is 21. Duh!
-monz
http://www.monz.org
"All roads lead to n^0"
From: monz (2001-05-15)
Subject:
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/22793
I was looking again at my most recent Canasta Ztar mapping:
F^ 5&1/2
Bb^10&1/2 C< 11&2/3 C#- 5/6 D 2 Eb+ 3&1/6 E> 4&1/3 F^ 5&1/2
Eb^ 3&1/2 F< 4&2/3 F#- 5&5/6 G 7 G#+ 8&1/6 A> 9&1/3 Bb^10&1/2
G#^ 8&1/2 Bb< 9&2/3 B- 10&5/6 C 0 C#+ 1&1/6 D> 2&1/3 Eb^ 3&1/2
C#^ 1&1/2 Eb< 2&2/3 E- 3&5/6 F 5 F#+ 6&1/6 G> 7&1/3 G#^ 8&1/2
F#^ 6&1/2 G#< 7&2/3 A- 8&5/6 Bb 10 B+ 11&1/6 C> 1/3 C#^ 1&1/2
(I should have stated when I posted it that this is notated
in ASCII 72-EDO with Semitones.)
Notice how each "fret" contains a series of notes with the
same type of accidental: the first one is all ^, the second
one is all <, etc.
So I started speculating. Seems to me that the reason *why*
the 7/72-"octave" generator is a MIRACLE (i.e., approximates
so many just harmonic structures so well *and* is melodically
even) is because it steps sequentially thru each of the
approximate-JI inflections ("bike gears") that 72-EDO provides:
^ representing factors of 11-otonal,
< giving 7-otonal,
- giving 5-otonal,
no accidental in the center giving the Pythagorean (3-limit) basis
+ giving 5-utonal,
> giving 7-utonal,
v giving 11-utonal.
Any thoughts on this?
-monz
http://www.monz.org
"All roads lead to n^0"
From: monz (2001-05-15)
Subject: second Monzo Canasta Ztar mapping
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/22793
In ASCII 72-EDO and Semitones:
F^ 5&1/2
Bb^10&1/2 C< 11&2/3 C#- 5/6 D 2 Eb+ 3&1/6 E> 4&1/3 F^ 5&1/2
Eb^ 3&1/2 F< 4&2/3 F#- 5&5/6 G 7 G#+ 8&1/6 A> 9&1/3 Bb^10&1/2
G#^ 8&1/2 Bb< 9&2/3 B- 10&5/6 C 0 C#+ 1&1/6 D> 2&1/3 Eb^ 3&1/2
C#^ 1&1/2 Eb< 2&2/3 E- 3&5/6 F 5 F#+ 6&1/6 G> 7&1/3 G#^ 8&1/2
F#^ 6&1/2 G#< 7&2/3 A- 8&5/6 Bb 10 B+ 11&1/6 C> 1/3 C#^ 1&1/2
For those following along in Graham's decimal notation, that's:
5v
9 0v 1v 2v 3v 4v 5v
3 4 5 6 7 8 9
7^ 8^ 9^ 0 1 2 3
1^ 2^ 3^ 4^ 5^ 6^ 7^
5^^ 6^^ 7^^ 8^^ 9^^ 0^ 1^
Right, Graham?
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/22799
> So I started speculating. Seems to me that the reason *why*
> the 7/72-"octave" generator is a MIRACLE (i.e., approximates
> so many just harmonic structures so well *and* is melodically
> even) is because it steps sequentially thru each of the
> approximate-JI inflections ("bike gears") that 72-EDO provides:
>
> ^ representing factors of 11-otonal,
> < giving 7-otonal,
> - giving 5-otonal,
> no accidental in the center giving the Pythagorean (3-limit) basis
> + giving 5-utonal,
> > giving 7-utonal,
> v giving 11-utonal.
Double-duh!!
This is pretty darn close to what Paul wrote in his original
post on the MIRACLE scale (WITH THE SCREAMING SUBJECT):
http://groups.yahoo.com/group/tuning/message/21894
Only difference is that Paul only specified the primary
otonal ratios, but the MIRACLE scale does include downward
generation which gives the primary utonal ratios too.
(Sorry about sending so many posts today without a subject
line... too many hours awake staring at the monitor...)
-monz
http://www.monz.org
"All roads lead to n^0"
From: Dave Keenan (2001-05-15)
Subject:
--- In tuning@y..., "monz" <joemonz@y...> wrote:
> So I started speculating. Seems to me that the reason *why*
> the 7/72-"octave" generator is a MIRACLE (i.e., approximates
> so many just harmonic structures so well *and* is melodically
> even) is because it steps sequentially thru each of the
> approximate-JI inflections ("bike gears") that 72-EDO provides:
>
> ^ representing factors of 11-otonal,
> < giving 7-otonal,
> - giving 5-otonal,
> no accidental in the center giving the Pythagorean (3-limit) basis
> + giving 5-utonal,
> > giving 7-utonal,
> v giving 11-utonal.
>
> Any thoughts on this?
No. Any generator that is not a multiple of 2 or 3 steps of 72-EDO
would do this. e.g. 5/72, 11/72. But these will not give as many
hexads per note nor will the necessarily be as even in as few notes.
-- Dave Keenan
From: [email protected] (2001-05-15)
Subject: Re: second Monzo Canasta Ztar mapping
--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Only difference is that Paul only specified the primary
> otonal ratios, but the MIRACLE scale does include downward
> generation which gives the primary utonal ratios too.
Huh? Everything I've posted on the MIRACLE scale has been exactly
symmetrical between otonal and utonal, just as the scale is. 5 otonal
shizbots, 5 utonal shizbots. What have I written that is any
different?
From: monz (2001-05-15)
Subject: Re: second Monzo Canasta Ztar mapping
--- In tuning@y..., paul@s... wrote:
http://groups.yahoo.com/group/tuning/message/22820
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > Only difference is that Paul only specified the primary
> > otonal ratios, but the MIRACLE scale does include downward
> > generation which gives the primary utonal ratios too.
>
> Huh? Everything I've posted on the MIRACLE scale has been
> exactly symmetrical between otonal and utonal, just as the
> scale is. 5 otonal shizbots, 5 utonal shizbots. What have
> I written that is any different?
Hi Paul,
In your original MIRACLE post, you wrote:
http://groups.yahoo.com/group/tuning/message/21894
> Stacking six of these upward gives you the 3/2. So you need
> a chain of 6 to yield a 3-limit dyad. 31 - 6 = 25 -- that's
> why there are 25 dyads.
>
> Stacking seven of these _downward_ gives you the 4/5. So you
> need a chain of 7+6=13 to yield a 5-limit triad. 31-13=18 --
> that's why there are 18 major triads and 18 minor triads.
>
> Stacking only two of these downward gives you the 7/8. That's
> why all the triads can be completed into 7-limit tetrads.
>
> Stacking twelve of these upward gives you the 9/4. 12+7=19,
> and 31-19=12 -- that's why there are 12 major pentads and
> 12 minor pentads.
>
> Stacking fifteen of these upward gives you the 11/4. 15+7=22,
> and 31-22=9 -- that why there are 9 major hexads and 9 minor
> hexads.
Of course I realize that you intend for the symmetry to be
recognized the half of the scale you don't mention, but my
point was that you *did* only describe one direction in each
of these.
By explicity pointing out that it works the same in both
directions, the reader can relate the concept directly to
what can be seen on my Ztar mapping. That's all I was
saying.
And I'm sure that in later MIRACLE posts you *have* pointed
out the symmetry, but I was referring only to this post.
At this point, I think I really need to quit posting to the
list for today. Good night.
-monz
From: [email protected] (2001-05-15)
Subject: Re: second Monzo Canasta Ztar mapping
--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Hi Paul,
>
> In your original MIRACLE post, you wrote:
>
> http://groups.yahoo.com/group/tuning/message/21894
>
>
> > Stacking six of these upward gives you the 3/2. So you need
> > a chain of 6 to yield a 3-limit dyad. 31 - 6 = 25 -- that's
> > why there are 25 dyads.
> >
> > Stacking seven of these _downward_ gives you the 4/5. So you
> > need a chain of 7+6=13 to yield a 5-limit triad. 31-13=18 --
> > that's why there are 18 major triads and 18 minor triads.
> >
> > Stacking only two of these downward gives you the 7/8. That's
> > why all the triads can be completed into 7-limit tetrads.
> >
> > Stacking twelve of these upward gives you the 9/4. 12+7=19,
> > and 31-19=12 -- that's why there are 12 major pentads and
> > 12 minor pentads.
> >
> > Stacking fifteen of these upward gives you the 11/4. 15+7=22,
> > and 31-22=9 -- that why there are 9 major hexads and 9 minor
> > hexads.
>
See that? 5 major shizbots, 5 minor shizbots . . .
>
> Of course I realize that you intend for the symmetry to be
> recognized the half of the scale you don't mention,
Huh? What half is that? There is no fixed "starting point" -- this is
an MOS and not a Tonality Diamond or anything like that.
> but my
> point was that you *did* only describe one direction in each
> of these.
Yes, because one direction is sufficient. But this direction is
neither otonal nor utonal. For example, the interval 11/4 can be
thought of as otonal, as the 11th and 4th harmonics, with the 4th
harmonic octave-equivalent to (and closer to) the root. Or it can be
thought of as utonal, as the 11th and 4th subharmonics, with the 4th
subharmonic octave-equivalent (and closer to) the guide tone.
Only triads or larger chords can have a more otonal or more utonal
nature. As Partch pointed out, any dyad has a dual nature as otonal
and utonal, and an equal potential to act in either capacity.
(Now you know I don't believe in full dualism, but I'm acting as if I
do for all the MIRACLE stuff so far).
>
> By explicity pointing out that it works the same in both
> directions, the reader can relate the concept directly to
> what can be seen on my Ztar mapping. That's all I was
> saying.
OK, that's absolutely true and it's valuable if it can help people
understand the mapping for practical musicmaking.
From: [email protected] (2001-05-15)
Subject: Re: second Monzo Canasta Ztar mapping
monz wrote:
> In ASCII 72-EDO and Semitones:
>
> F^ 5&1/2
> Bb^10&1/2 C< 11&2/3 C#- 5/6 D 2 Eb+ 3&1/6 E> 4&1/3 F^ 5&1/2
> Eb^ 3&1/2 F< 4&2/3 F#- 5&5/6 G 7 G#+ 8&1/6 A> 9&1/3 Bb^10&1/2
> G#^ 8&1/2 Bb< 9&2/3 B- 10&5/6 C 0 C#+ 1&1/6 D> 2&1/3 Eb^ 3&1/2
> C#^ 1&1/2 Eb< 2&2/3 E- 3&5/6 F 5 F#+ 6&1/6 G> 7&1/3 G#^ 8&1/2
> F#^ 6&1/2 G#< 7&2/3 A- 8&5/6 Bb 10 B+ 11&1/6 C> 1/3 C#^ 1&1/2
>
>
> For those following along in Graham's decimal notation, that's:
>
> 5v
> 9 0v 1v 2v 3v 4v 5v
> 3 4 5 6 7 8 9
> 7^ 8^ 9^ 0 1 2 3
> 1^ 2^ 3^ 4^ 5^ 6^ 7^
> 5^^ 6^^ 7^^ 8^^ 9^^ 0^ 1^
>
>
> Right, Graham?
The decimal bit looks right. I'm not perfect with 72-EDO, but they look
like the same notes. It seems you're on to something. Have you borrowed
an instrument to try it out on?
Graham
From: [email protected] (2001-05-18)
Subject: Re: second Monzo Canasta Ztar mapping
--- In tuning@y..., paul@s... wrote:
http://groups.yahoo.com/group/tuning/message/22828
> OK, that's absolutely true and it's valuable if it can help people
> understand the mapping for practical musicmaking.
I remain unconvinced that all this "mapping" is of practical value
for "practical" musicmaking...
Gee... maybe this should be on the "practical" list!
HOWEVER, it would seem a generalized keyboard for the ENTIRE 72-tET
would be most appropriate, no... especially if one could figure out a
way to have more than only one octave?? Yes?
Then people could learn the various MIRACLES as SUBSETS and practice
them much the way people practice diatonic scales in different keys
today (??)
The thought is to keep the large general item invariant... (??)
_____________ _______ ____
Joseph Pehrson
From: monz (2001-05-18)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
--- In tuning@y..., jpehrson@r... wrote:
http://groups.yahoo.com/group/tuning/message/23119
> I remain unconvinced that all this "mapping" is of practical
> value for "practical" musicmaking...
>
> Gee... maybe this should be on the "practical" list!
I've joked about that a couple of times in my posts.
Well... maybe it wasn't a joke...
>
> HOWEVER, it would seem a generalized keyboard for the ENTIRE
> 72-tET would be most appropriate, no... especially if one
> could figure out a way to have more than only one octave??
> Yes?
Joe, *please* read my posts on this! I've been advocating
mapping these scales to the Starr Labs Ztar and Zboard.
For the entire 72-EDO superset, these instruments would
give a range of 2 and 4 "octaves" (audible 2:1s, that is),
respectively, not just one! They have 144 and 288 keys,
respectively.
>
> Then people could learn the various MIRACLES as SUBSETS and
> practice them much the way people practice diatonic scales
> in different keys today (??)
>
> The thought is to keep the large general item invariant... (??)
This is pretty much how I feel about it. But the Canasta
scale (MIRACLE-31) has so much going for it that it really
can be considered "the large general item" from which smaller
scales can be derived. And mapping it onto the smaller Ztar,
even with 5 wasted keys [*] per "octave" (2:1), as here:
<http://groups.yahoo.com/group/tuning/message/22712>
still gives a range of nearly 4 "octaves" (2:1s) on an instrument
the size of a guitar neck! And there are 20 keys left over
to use for other stuff.
The nicest thing about a Ztar (or Zboard, or even a MicroZone)
is that you can map either the full 72-EDO or Canasta to it
at will, since it's all done in software.
And for those really strapped by lack of hardware: there are
59 keys on a computer keyboard that can be programmed to
play individual pitches with my JustMusic software. That
gives nearly 2 "octaves" of Canasta and nearly 3 "octaves"
of Blackjack. Only problem is that JustMusic so far can
only program rational scales 13-limit and under.
So here's a challenge: can anyone come up with 13-limit
rational pitch sets which will approximate Blackjack and
Canasta to good advantage? Then you can use my software
to put these scales on your computer keyboard, and play away!
(Hmmm... that's pretty weird... looking for a rational
approximation for subsets of an equal-temperament which
itself is supposed to be good at approximating ratios...
I see Escher pictures in my mind...)
-monz
http://www.monz.org
"All roads lead to n^0"
From: Kraig Grady (2001-05-18)
Subject: Re: [tuning] Re: second Monzo Canasta Ztar mapping
Joseph!
Here is another way of mapping the Hanson 72 pattern. Notice that the 7/72 intervals run in a
row. Sorry i have only this version as a work sheet as opposed to the Bosanquet. Quality is low
too but it is there.
[email protected] wrote:
>
>
> HOWEVER, it would seem a generalized keyboard for the ENTIRE 72-tET
> would be most appropriate, no... especially if one could figure out a
> way to have more than only one octave?? Yes?
>
>
-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com
The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm
From: Kraig Grady (2001-05-18)
Subject: Re: [tuning] Re: second Monzo Canasta Ztar mapping
http://www.anaphoria.com/images/hebdo15rank.GIF
here is the link! duh!
Kraig Grady wrote:
> Joseph!
> Here is another way of mapping the Hanson 72 pattern. Notice that
> the 7/72 intervals run in a row. Sorry i have only this version as a
> work sheet as opposed to the Bosanquet. Quality is low too but it is
> there.
>
> [email protected] wrote:
>
>>
>>
>> HOWEVER, it would seem a generalized keyboard for the ENTIRE 72-tET
>> would be most appropriate, no... especially if one could figure out
>> a
>> way to have more than only one octave?? Yes?
>>
>>
>
>
> -- Kraig Grady
> North American Embassy of Anaphoria island
> http://www.anaphoria.com
>
> The Wandering Medicine Show
> Wed. 8-9 KXLU 88.9 fm
>
>
>
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com
The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm
From: Dave Keenan (2001-05-19)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
--- In tuning@y..., "monz" <joemonz@y...> wrote:
> So here's a challenge: can anyone come up with 13-limit
> rational pitch sets which will approximate Blackjack and
> Canasta to good advantage? Then you can use my software
> to put these scales on your computer keyboard, and play away!
Err. How about you just fix your software. ;-)
From: monz (2001-05-19)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
http://groups.yahoo.com/group/tuning/message/23160
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > So here's a challenge: can anyone come up with 13-limit
> > rational pitch sets which will approximate Blackjack and
> > Canasta to good advantage? Then you can use my software
> > to put these scales on your computer keyboard, and play away!
>
> Err. How about you just fix your software. ;-)
Much easier said than done. I've been begging *you* to join
the group for 8 months, Dave... how about it now? We'd love
to have you aboard.
Anyway, in the meantime if I want to explore Canasta on a
keyboard it will have to be rational version that JustMusic
can map to my computer keyboard. Here's one that's very
close to the "standard" 72-EDO-based Canasta subset.
(I believe within 1 cent... check if you'd like; I'd
love to see a comparison.)
It's 13-limit, and all exponent limits are quite low:
no higher than 7 on the 3-axis, 2 on the 5-axis, and
1 on the 7-, 11- and 13-axes. And even this is giving
us problems in the current versions of JustMusic...
we're working on it.
I *could* substitute smaller-integer ratios that would
work fine in JustMusic (and I might, just to create some
music with it tonight), but I don't want to get any further
deviation from the 72-EDO-based Canasta.
Copy and paste everything between the lines below, not
including the lines, and save as "rational_canasta.scl".
-monz
http://www.monz.org
"All roads lead to n^0"
-----------------------Scala file begins below this line----
!\rational_canasta.scl
!
24
!
1/1
729/715
150/143
5632/5265
567/520
432/385
5005/4374
7/6
832/693
891/728
19712/15795
32768/25515
55/42
10935/8192
378/275
416/297
297/208
275/189
16384/10935
84/55
25515/16384
15795/9856
1456/891
5/3
12/7
110/63
385/216
11/6
243/130
143/75
108/155
--------------
From: monz (2001-05-19)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/23188
> Anyway, in the meantime if I want to explore Canasta on a
> keyboard it will have to be rational version that JustMusic
> can map to my computer keyboard. Here's one that's very
> close to the "standard" 72-EDO-based Canasta subset.
Oops! My bad. I told you all I was a Fish Brain.
Forget that Scala file I posted. Here's the correct one.
-monz
http://www.monz.org
"All roads lead to n^0"
-----------------------Scala file begins below this line----
! rational_canasta.scl
!
Rational version of Canasta MIRACLE-31 scale by Joe Monzo
31
!
729/715
150/143
5632/5265
567/520
432/385
5005/4374
7/6
832/693
891/728
19712/15795
32768/25515
55/42
10935/8192
378/275
416/297
297/208
275/189
16384/10935
84/55
25515/16384
15795/9856
1456/891
5/3
12/7
110/63
385/216
11/6
243/130
143/75
108/55
2/1
--------------------------
From: monz (2001-05-19)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/23189
> Oops! My bad. I told you all I was a Fish Brain.
>
> Forget that Scala file I posted. Here's the correct one.
OK, how about "Mollusk Brain"?
I guess I was just anxious to post this. The last version
had 3 pitches which were between 1 and 2 deviation from
the 72-EDO-based Canasta.
This is a much better version, greatest deviation only
3/4 of a cent.
Consider this one to be the rational canasta scale.
-monz
http://www.monz.org
"All roads lead to n^0"
-----------------------Scala file begins below this line----
! rational_canasta.scl
!
Rational version of Canasta MIRACLE-31 scale by Joe Monzo
31
!
729/715
150/143
5632/5265
567/520
432/385
5005/4374
7/6
832/693
891/728
19712/15795
32768/25515
55/42
10935/8192
378/275
416/297
297/208
275/189
16384/10935
84/55
25515/16384
15795/9856
1456/891
693/416
12/7
8748/5005
385/216
11/6
243/130
143/75
1430/729
2/1
--------------------------
From: monz (2001-05-19)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/23190
> OK, how about "Mollusk Brain"?
>
> I guess I was just anxious to post this. The last version
> had 3 pitches which were between 1 and 2 deviation from
> the 72-EDO-based Canasta.
"Insect Brain"? That should have said "between 1 and 2 *cents*
deviation...".
But the Scala file was good that time.
-monz
http://www.monz.org
"All roads lead to n^0"
From: monz (2001-05-19)
Subject: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/23190
> Consider this one to be the rational canasta scale.
Comparison of my Rational Canasta scale to
the "standard" 72-EDO-based Canasta:
Canasta rational
72-EDO canasta prime-factor deviation
degree cents degree vector ratio cents (cents)
0 0.0000 0 | 0 0 0 0 0| 1/1 0.000 +0.0000
2 33.3333 1 | 6 -1 0 -1 -1| 729/715 33.571 +0.2374
5 83.3333 2 | 1 2 0 -1 -1| 150/143 82.737 -0.5965
7 116.6667 3 |-4 -1 0 1 -1| 5632/5265 116.657 -0.0101
9 150.0000 4 | 4 -1 1 0 -1| 567/520 149.805 -0.1955
12 200.0000 5 | 3 -1 -1 -1 0| 432/385 199.407 -0.5926
14 233.3333 6 |-7 1 1 1 1| 5005/4374 233.300 -0.0331
16 266.6667 7 |-1 0 1 0 0| 7/6 266.871 +0.2042
19 316.6667 8 |-2 0 -1 -1 1| 832/693 316.474 -0.1929
21 350.0000 9 | 4 0 -1 1 -1| 891/728 349.784 -0.2156
23 383.3333 10 |-5 -1 1 1 -1| 19712/15795 383.527 +0.1941
26 433.3333 11 |-6 -1 -1 0 0| 32768/25515 433.130 -0.2030
28 466.6667 12 |-1 1 -1 1 0| 55/42 466.851 +0.1841
30 500.0000 13 | 7 1 0 0 0| 10935/8192 499.999 -0.0013
33 550.0000 14 | 3 -2 1 -1 0| 378/275 550.746 +0.7455
35 583.3333 15 |-3 0 0 -1 1| 416/297 583.345 +0.0114
37 616.6667 16 | 3 0 0 1 -1| 297/208 616.655 -0.0114
39 650.0000 17 |-3 2 -1 1 0| 275/189 649.254 -0.7455
42 700.0000 18 |-7 -1 0 0 0| 16384/10935 700.001 +0.0013
44 733.3333 19 | 1 -1 1 -1 0| 84/55 733.149 -0.1841
46 766.6667 20 | 6 1 1 0 0| 25515/16384 766.870 +0.2030
49 816.6667 21 | 5 1 -1 -1 1| 15795/9856 816.473 -0.1941
51 850.0000 22 |-4 0 1 -1 1| 1456/891 850.216 +0.2156
53 883.3333 23 | 2 0 1 1 -1| 693/416 883.526 +0.1929
56 933.3333 24 | 1 0 -1 0 0| 12/7 933.129 -0.2042
58 966.6667 25 | 7 -1 -1 -1 -1| 8748/5005 966.700 +0.0331
60 1000.0000 26 |-3 1 1 1 0| 385/216 1000.593 +0.5926
63 1050.0000 27 |-1 0 0 1 0| 11/6 1049.363 -0.6371
65 1083.3333 28 | 5 -1 0 0 -1| 243/130 1082.934 -0.3997
67 1116.6667 29 |-1 -2 0 1 1| 143/75 1117.263 +0.5965
70 1166.6667 30 |-6 1 0 1 1| 1430/729 1166.429 -0.2374
Total absolute difference 8.0656 cents
Average absolute difference 0.2602 cents
Root mean square difference 0.0622 cents
Highest absolute difference 0.7455 cents
I don't consider the integer ratio terms or the prime-factors
to denote any special significance. I simply had to put the
scale in this format in order for my software to read it.
Perhaps those who are fond of using high-integer rational
tunings will disagree.
(Special thanks to Manuel and Robert for making this table
easy to do.)
-monz
http://www.monz.org
"All roads lead to n^0"
From: [email protected] (2001-05-19)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
Hi Monz.
On the idea of "rationalizing" the miracle scale so that it will work in your software.
Why use a prime limit of 13? You can get much more accurate approximations (for the size of
numbers you're using) if you drop that restriction.
From: [email protected] (2001-05-19)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Average absolute difference 0.2602 cents
That looks right.
> Root mean square difference 0.0622 cents
That doesn't. Perhaps you made an error in calculation?
From: monz (2001-05-19)
Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
--- In tuning@y..., paul@s... wrote:
http://groups.yahoo.com/group/tuning/message/23220
> Hi Monz.
>
> On the idea of "rationalizing" the miracle scale so that
> it will work in your software.
>
> Why use a prime limit of 13? You can get much more
> accurate approximations (for the size of numbers you're
> using) if you drop that restriction.
Thanks for that input, Paul. There's an overriding practical
(that word again...) reason for using 13-limit: that's all
JustMusic can handle at the moment.
My whole purpose was to get Canasta mapped to my computer
keyboard so that I could *play* it and *hear* it, and I've
achieved that, with less than 1 cent error.
But, just for the record, and for use in JustMusic when
we break the 13-barrier, how about a list of those ratios?
(Actually JustMusic will eventually be able to make full
use of all the scale-building power and options in Scala.
But right now there's no way to specify pitches except as
ratios or as a plain equal division of a ratio. So while
31-EDO would be possible, a subset of 72-EDO is not.)
-monz
http://www.monz.org
"All roads lead to n^0"
From: monz (2001-05-19)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., paul@s... wrote:
http://groups.yahoo.com/group/tuning/message/23221
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Average absolute difference 0.2602 cents
>
> That looks right.
>
> > Root mean square difference 0.0622 cents
>
> That doesn't. Perhaps you made an error in calculation?
I didn't do the calculation... Scala did. Manuel?
-monz
http://www.monz.org
"All roads lead to n^0"
From: [email protected] (2001-05-21)
Subject: Re: second Monzo Canasta Ztar mapping
--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
http://groups.yahoo.com/group/tuning/message/23157
> http://www.anaphoria.com/images/hebdo15rank.GIF
> here is the link! duh!
>
> Kraig Grady wrote:
>
> > Joseph!
> > Here is another way of mapping the Hanson 72 pattern. Notice
that the 7/72 intervals run in a row. Sorry i have only this version
as a work sheet as opposed to the Bosanquet. Quality is low too but
it is there.
> >
>
Thanks, Kraig... I was looking for this "missing link!"
_______ ______ ___ ____
Joseph Pehrson
From: [email protected] (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/23195
>
> Comparison of my Rational Canasta scale to
> the "standard" 72-EDO-based Canasta:
>
OK... this is probably a pretty "easy" question for somebody...
If the "Miracle" scales are constructed in order to find small
integer ratio intervals, why are the intervals in Monz' scale so
large??
Is it just the intervals measured from the "tonic" starting point
that come out larger like this, and many of the other chords and
intervals throughout the scale come out smaller??
Signed,
confused
________ _______ ______ _
Joseph Pehrson
From: Dave Keenan (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., jpehrson@r... wrote:
> OK... this is probably a pretty "easy" question for somebody...
>
> If the "Miracle" scales are constructed in order to find small
> integer ratio intervals, why are the intervals in Monz' scale so
> large??
I really wish Monz haddn't muddied the issue by publishing that silly
rationalisation of a tempering of a rational scale. However he tried
to make it clear that he was only doing it to work around a bug in his
software. Mind you, I see little point in bothering to work around it
since it still only plays one-note-at-a-time.
> Is it just the intervals measured from the "tonic" starting point
> that come out larger like this, and many of the other chords and
> intervals throughout the scale come out smaller??
A temperament is inherently irrational because it is trying to
approximate more just intervals in a certain number of notes than it
is possible to do with strict ratios. It relies on distributing the
commas, or bridges between different prime numbers. So the closer you
try to approximate an irrational scale with rationals, the bigger
the numbers must become.
But in general there's absolutely no point in trying to approximate
Miracle with ratios. In fact some of monz's rationals actually
corresponded to just intervals and would lead to undesirable
phase-locking if used with electronic instruments.
-- Dave Keenan
From: [email protected] (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> http://groups.yahoo.com/group/tuning/message/23195
>
> >
> > Comparison of my Rational Canasta scale to
> > the "standard" 72-EDO-based Canasta:
> >
>
> OK... this is probably a pretty "easy" question for somebody...
>
> If the "Miracle" scales are constructed in order to find small
> integer ratio intervals, why are the intervals in Monz' scale so
> large??
>
> Is it just the intervals measured from the "tonic" starting point
> that come out larger like this, and many of the other chords and
> intervals throughout the scale come out smaller??
That's partially true, but also, Monz is "breaking" a lot of the consonant intervals (probably most
of them) so that he can express the temperament in strict JI terms. To really express the
temperament in strict JI terms without breaking anything, you'd need a lot of "extra" notes.
For example, the diatonic scale in meantone temperament has six 5-limit consonant triads.
C E G
D F A
E G B
F A C
G B D
A C E
A JI diatonic scale can only have five 5-limit consonant triads, unless two ratios are used for D
(both a 9/8 above C and a 10/9 above C). So you'd need eight ratios, rather than seven, to
get across all the consonant triads in the diatonic scale. For the miracle scales, a lot of "extra"
ratios would be needed . . . and there would probably be several different ways to do it.
From: monz (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., jpehrson@r... wrote:
http://groups.yahoo.com/group/tuning/message/23388
> OK... this is probably a pretty "easy" question for somebody...
>
> If the "Miracle" scales are constructed in order to find small
> integer ratio intervals, why are the intervals in Monz' scale so
> large??
>
> Is it just the intervals measured from the "tonic" starting point
> that come out larger like this, and many of the other chords and
> intervals throughout the scale come out smaller??
>
> Signed,
>
> confused
>
> ________ _______ ______ _
> Joseph Pehrson
Yes, Joe, you are a little confused.
The MIRACLE temperaments *closely approximate* a number of
low-integer JI ratios. But they actually *are* irrational
tunings, since they are subsets of 72-EDO.
Any EDO is irrational.
My ratios have large numbers because I wanted to stay
within 1 cent of the actual EDO tuning of Canasta. I needed
to have them in rational form in order to input them into
JustMusic.
(That's a defect of the software that needs to be fixed...
but I wanted to hear this scale right away and so I did what
I had to do. Now I can play and record Canasta from my
computer, if less than 3/4 cent error still qualifies it
as Canasta, which I think it does.)
-monz
http://www.monz.org
"All roads lead to n^0"
From: [email protected] (2001-05-21)
Subject: Re: [tuning] Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
Joe Monzo wrote:
> > Root mean square difference 0.0622 cents
> I didn't do the calculation... Scala did. Manuel?
The formula used for root mean square difference is
the square root of the sum of squared logarithmic differences
and that divided by the number of tones (31 in Canasta).
Perhaps Paul was expecting something else?
Manuel
From: [email protected] (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> Joe Monzo wrote:
> > > Root mean square difference 0.0622 cents
> > I didn't do the calculation... Scala did. Manuel?
>
> The formula used for root mean square difference is
> the square root of the sum of squared logarithmic differences
> and that divided by the number of tones (31 in Canasta).
> Perhaps Paul was expecting something else?
>
> Manuel
Manuel, you should divide by the number of tones _before_ taking the
square root, not after. The RMS error should be directly comparable
to the MA error.
From: [email protected] (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., paul@s... wrote:
http://groups.yahoo.com/group/tuning/message/23400
> --- In tuning@y..., jpehrson@r... wrote:
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > http://groups.yahoo.com/group/tuning/message/23195
> >
> > >
> > > Comparison of my Rational Canasta scale to
> > > the "standard" 72-EDO-based Canasta:
> > >
> >
> > OK... this is probably a pretty "easy" question for somebody...
> >
> > If the "Miracle" scales are constructed in order to find small
> > integer ratio intervals, why are the intervals in Monz' scale so
> > large??
> >
> > Is it just the intervals measured from the "tonic" starting point
> > that come out larger like this, and many of the other chords and
> > intervals throughout the scale come out smaller??
>
> That's partially true, but also, Monz is "breaking" a lot of the
consonant intervals (probably most
> of them) so that he can express the temperament in strict JI terms.
To really express the
> temperament in strict JI terms without breaking anything, you'd
need a lot of "extra" notes.
>
> For example, the diatonic scale in meantone temperament has six 5-
limit consonant triads.
> C E G
> D F A
> E G B
> F A C
> G B D
> A C E
>
> A JI diatonic scale can only have five 5-limit consonant triads,
unless two ratios are used for D
> (both a 9/8 above C and a 10/9 above C). So you'd need eight
ratios, rather than seven, to
> get across all the consonant triads in the diatonic scale. For the
miracle scales, a lot of "extra"
> ratios would be needed . . . and there would probably be several
different ways to do it.
Oh I see... so this is why EVERYTHING is an approximation... but a
GOOD one... to eliminate all the "extra" notes...
This must mean that certain "unison vectors" are used in this
process.... (??) It's ESSENTIALLY a "tempering" process, like
meantone...
So THIS is where the "microtempering" comes in... (??)
_______ _____ ______
Joseph Pehrson
From: [email protected] (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., jpehrson@r... wrote:
>
> Oh I see... so this is why EVERYTHING is an approximation... but a
> GOOD one... to eliminate all the "extra" notes...
That's one valid way of looking at it, yes.
>
> This must mean that certain "unison vectors" are used in this
> process.... (??) It's ESSENTIALLY a "tempering" process, like
> meantone...
Exactly. You can think of the MIRACLES as having two unison vectors,
225:224 and 2400:2401, tempered out.
>
> So THIS is where the "microtempering" comes in... (??)
>
Yes. Since these unison vectors are small, and since they are
distributed over many intervals, the amount of tempering involved is
very small . . . microtempering.
From: [email protected] (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., "monz" <joemonz@y...> wrote:
http://groups.yahoo.com/group/tuning/message/23419
>
> Yes, Joe, you are a little confused.
>
> The MIRACLE temperaments *closely approximate* a number of
> low-integer JI ratios. But they actually *are* irrational
> tunings, since they are subsets of 72-EDO.
> Any EDO is irrational.
>
> My ratios have large numbers because I wanted to stay
> within 1 cent of the actual EDO tuning of Canasta. I needed
> to have them in rational form in order to input them into
> JustMusic.
>
This is a little humorous, isn't it? We're trying to find MIRACLE
approximations of low-integer ratios and come up with an EDO
temperament which is a close "compromise."
THEN, we take that temperament and convert that into ratios that,
mostly, turn out to be large...
Isn't there something a little "funny" (humorous) in that, or is my
levity misplaced...
_______ ______ ________
Joseph Pehrson
From: [email protected] (2001-05-21)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., jpehrson@r... wrote:
> This is a little humorous, isn't it? We're trying to find MIRACLE
> approximations of low-integer ratios and come up with an EDO
> temperament which is a close "compromise."
>
> THEN, we take that temperament and convert that into ratios that,
> mostly, turn out to be large...
>
> Isn't there something a little "funny" (humorous) in that, or is my
> levity misplaced...
>
Maybe . . . but it's not unprecedented.
We've heard about the Kirnberger II tuning that Lou Harrison likes.
In this well-tempered tuning, D-A and A-E are each flattened by 1/2
syntonic comma, while all other fifths are pure (actually, one of
them is a schisma off).
Or so we thought.
Last time I visited you, I went over to Johnny Reinhard's place
afterwards, and he showed be Kirnberger's writings.
The ratios for the Kirnberger tuning included ratios of 161. How odd,
Johnny and I thought.
Later, it became obvious to me what Kirnberger was doing. The
syntonic comma is 81/80. 1/2 of the syntonic comma would be the
square root of 81/80, not a rational number. But theorists in
Kirnberger's day, as in Zarlino's, still felt (irrationally) that
they had to provide rational numbers in theory even if they were
going to depart from them in practice. So Kirberger divided the
syntonic comma in half as follows:
81/80 = 162/161 * 161/160.
By multiplying the ratio for the note A by one of these "halves",
Kirnberger managed to get across his tuning using rational numbers.
What the numbers hide, though, is that Kirnberger was simply trying
to approximate as many simple-integer ratios as possible, while still
allowing playability in all keys, without any grossly unacceptable
compromises.
From: monz (2001-05-22)
Subject: Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
--- In tuning@y..., jpehrson@r... wrote:
http://groups.yahoo.com/group/tuning/message/23464
> This is a little humorous, isn't it? We're trying to find
> MIRACLE approximations of low-integer ratios and come up
> with an EDO temperament which is a close "compromise."
>
> THEN, we take that temperament and convert that into ratios
> that, mostly, turn out to be large...
>
> Isn't there something a little "funny" (humorous) in that,
> or is my levity misplaced...
>
> _______ ______ ________
> Joseph Pehrson
Hi Joe,
When I first suggested this rational version and mapping,
before I actually carried it out, I noted the irony in the idea:
http://groups.yahoo.com/group/tuning/message/23129
> (Hmmm... that's pretty weird... looking for a rational
> approximation for subsets of an equal-temperament which
> itself is supposed to be good at approximating ratios...
> I see Escher pictures in my mind...)
BTW, Paul, I found your follow-up on this fascinating!
http://groups.yahoo.com/group/tuning/message/23467
-monz
http://www.monz.org
"All roads lead to n^0"
From: [email protected] (2001-05-22)
Subject: Re: [tuning] Re: Ratios for MIRACLEs: rational and 72-EDO Canasta compared
>Manuel, you should divide by the number of tones _before_ taking the
>square root, not after. The RMS error should be directly comparable
>to the MA error.
Olala, that puts me to shame, and it really was unnoticed for a long time.
Manuel