Topic: 25:30:35:45

1 scales

Thread (24 messages)

From: Ozan Yarman (2007-02-27)
Subject: 25:30:35:45

I find this to be a good half-diminished seventh, but could not see it
included in SCALA.

Oz.

From: Aaron Krister Johnson (2007-02-27)
Subject: Re: 25:30:35:45

--- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
>
> I find this to be a good half-diminished seventh, but could not see it
> included in SCALA.

All the terms are divisible by 5....

it's really a 5:6:7:9!

-A.

From: Danny Wier (2007-02-27)
Subject: Re: [tuning] 25:30:35:45

From: "Ozan Yarman" <[email protected]>
To: "Tuning List" <[email protected]>
Sent: Tuesday, February 27, 2007 3:25 AM
Subject: [tuning] 25:30:35:45


>I find this to be a good half-diminished seventh, but could not see it
> included in SCALA.

As Aaron said, it would be 5:6:7:9.

It would be some sort of "harmonic" half-dim seventh, since I think of 7/5 
as an augmented fourth more than a diminished fifth. It's also a harmonic 
ninth without the root.

~D.

From: Danny Wier (2007-02-27)
Subject: Re: [tuning] 25:30:35:45

From: "Danny Wier" <[email protected]>
To: <[email protected]>
Sent: Tuesday, February 27, 2007 10:56 AM
Subject: Re: [tuning] 25:30:35:45


> As Aaron said, it would be 5:6:7:9.
>
> It would be some sort of "harmonic" half-dim seventh, since I think of 7/5
> as an augmented fourth more than a diminished fifth. It's also a harmonic
> ninth without the root.

In fact, that's what Scala calls it: "harmonic half-diminshed seventh". And 
not because I think 7/5 is an augmented fourth.

From: Carl Lumma (2007-02-27)
Subject: Re: 25:30:35:45

--- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
>
> I find this to be a good half-diminished seventh, but could not
> see it included in SCALA.

Perhaps because it's not in lowest terms?

-Carl

From: Gene Ward Smith (2007-02-27)
Subject: Re: 25:30:35:45

--- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
>
> I find this to be a good half-diminished seventh, but could not see it
> included in SCALA.

Why are you writing it like that, and not as 5:6:7:9? Or you could use 
my favorite arrangement of this chord, 6:7:9:10, the point of which is 
that we have a fifth, and so 6 is more a root than 5 is. On the other 
hand, 5:6:7:9 wins in the sounding-harmonius department.

What is it you wanted to find in Scala?

From: Ozan Yarman (2007-02-27)
Subject: Re: [tuning] Re: 25:30:35:45

Ah, clumsy me. Thanks for the reminder!

Oz.

----- Original Message -----
From: "Aaron Krister Johnson" <[email protected]>
To: <[email protected]>
Sent: 27 \ufffdubat 2007 Sal\ufffd 17:49
Subject: [tuning] Re: 25:30:35:45


> --- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I find this to be a good half-diminished seventh, but could not see it
> > included in SCALA.
>
> All the terms are divisible by 5....
>
> it's really a 5:6:7:9!
>
> -A.
>
>

From: Ozan Yarman (2007-02-27)
Subject: Re: [tuning] Re: 25:30:35:45

SNIP

> 
> Why are you writing it like that, and not as 5:6:7:9? 


I'm not a math wiz like you, that's why.


Or you could use 
> my favorite arrangement of this chord, 6:7:9:10, the point of which is 
> that we have a fifth, and so 6 is more a root than 5 is. 



I find the inversion so zesty, that I like to employ it in 79 MOS 159-tET.



On the other 
> hand, 5:6:7:9 wins in the sounding-harmonius department.
> 


Better than 4:5:6:7?



> What is it you wanted to find in Scala?
> 
> 


Found it, thanks!
Oz.

From: Gene Ward Smith (2007-02-27)
Subject: Re: 25:30:35:45

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

> Or you could use 
> > my favorite arrangement of this chord, 6:7:9:10, the point of which 
is 
> > that we have a fifth, and so 6 is more a root than 5 is. 

> I find the inversion so zesty, that I like to employ it in 79 MOS 159-
tET.

Which one is the inversion?
 
> On the other 
> > hand, 5:6:7:9 wins in the sounding-harmonius department.

> Better than 4:5:6:7?

No, but that's a different chord.

From: Ozan Yarman (2007-02-27)
Subject: Re: [tuning] Re: 25:30:35:45

----- Original Message -----
From: "Gene Ward Smith" <[email protected]>
To: <[email protected]>
Sent: 28 \ufffdubat 2007 \ufffdar\ufffdamba 0:41
Subject: [tuning] Re: 25:30:35:45


> --- In [email protected], "Ozan Yarman" <ozanyarman@...> wrote:
>
> > Or you could use
> > > my favorite arrangement of this chord, 6:7:9:10, the point of which
> is
> > > that we have a fifth, and so 6 is more a root than 5 is.
>
> > I find the inversion so zesty, that I like to employ it in 79 MOS 159-
> tET.
>
> Which one is the inversion?
>


Yours.


> > On the other
> > > hand, 5:6:7:9 wins in the sounding-harmonius department.
>
> > Better than 4:5:6:7?
>
> No, but that's a different chord.
>
>

Then I am pleased that 79 MOS 159-tET does so well a job with it.

Oz.

From: Billy Gard (2007-02-28)
Subject: Re: 25:30:35:45

<<< It would be some sort of "harmonic" half-dim seventh, since I think of
7/5
as an augmented fourth more than a diminished fifth. It's also a harmonic
ninth without the root. >>>

I like to call this (5:6:7:9) the just half-diminished 7th chord. There is
no more consonant tuning in the harmonic series for it. Everything else has
something of a buzz. And being in reality the upper four notes of the just
dominant 9th chord, this chord is in good company. Even the just minor triad
(10:12:15) is harmonically a rootless major 7th chord.

With the first inversion (6:7:9:10), we're really looking at a sub-minor (or
septimal-minor) triad with a major sixth. Attempting to subsitute the just
minor will produce a mess (30:36:45:50).

Speaking of the half-diminished, I found an enharmonic of this chord in the
Hungarian-minor scale, in the form of an augmented-sixth chord. But I don't
know of any names given for it. I kind of coined it as a Neptune 6th,
because Holst uses it in his Planets. It is the mirror image of the German
6th chord.

Billy

From: Gene Ward Smith (2007-02-28)
Subject: Re: 25:30:35:45

--- In [email protected], "Billy Gard" <billygard@...> wrote:

> Speaking of the half-diminished, I found an enharmonic of this chord 
in the
> Hungarian-minor scale, in the form of an augmented-sixth chord. But I 
don't
> know of any names given for it. I kind of coined it as a Neptune 6th,
> because Holst uses it in his Planets. It is the mirror image of the 
German
> 6th chord.

In a meantone tuning, that emerges as the utonal tetrad. One voicing 
for it was made popular by Wagner--the Tristan chord: F-B-D#-G#. How 
does the voicing on the Neptune chord go?

Here's another question: what do you call 1/5:1/6:1/7:1/9 or
1/6:1/7:1/9:1/10. Multiplying through the latter by 9 gives
9/10-1-9/7-3/2, and reducing that to an octave gives 1-9/7-3/2-9/5.

From: Billy Gard (2007-03-01)
Subject: Re: 25:30:35:45

<<< In a meantone tuning, that emerges as the utonal tetrad. One voicing for
it was made popular by Wagner--the Tristan chord: F-B-D#-G#. How does the
voicing on the Neptune chord go? >>>

The Tristan chord you gave above would be a properly spelled Neptune 6th
based on the Hungarian A-minor scale. As such It would want to resolve to
E-C-E-A. The question is whether the chord is defined as a Tristan on the
basis of spelling or of voicing. Revoicing it as a series of thirds would
give you G#BD#F, although it sounds like a half-diminished in the 1st
inversion. Revoicing it to sound like thirds would give you FG#BD#, which
would reveal the actual augmented sixth interval. But would these still be
the Tristan chord?

It was while examining that same Hungarian minor scale (ABCD#EFG#A) that I
found what I now know to be the French sixth and German sixth chords. But at
least those have been named in the music world.

<<< Here's another question: what do you call 1/5:1/6:1/7:1/9 or
1/6:1/7:1/9:1/10. Multiplying through the latter by 9 gives 9/10-1-9/7-3/2,
and reducing that to an octave gives 1-9/7-3/2-9/5. >>>

Sounds like dominant 7th tuned to the undertone series. I've listened to it,
but like other utonal chords it doesn't lock in like otonal versions. I've
also listened to the "utonal tetrad" tuning of the half-diminished
(1/4:1/5:1/6:1/7). But whatever consonance these have I would attribute to
the fact that they are rational, and can all be made integers by
multiplication, although really large ones.

From: Dave Keenan (2007-03-01)
Subject: Re: 25:30:35:45

--- In [email protected], "Billy Gard" <billygard@...> wrote:
>
> <<< It would be some sort of "harmonic" half-dim seventh, since I
think of
> 7/5
> as an augmented fourth more than a diminished fifth. It's also a
harmonic
> ninth without the root. >>>

Yes. If treated as a kind of fifth 5:7 must be called subdiminished.
http://users.bigpond.net.au/d.keenan/Music/IntervalNaming.htm
http://users.bigpond.net.au/d.keenan/Music/Miracle/MiracleIntervalNaming.txt

So 5:6:7 is the otonal subdiminished triad and 1/(7:6:5) (shorthand
for 1/7 : 1/6 : 1/5) is the utonal subdiminished triad.
http://users.bigpond.net.au/d.keenan/Music/DiatonicLattice.gif
http://users.bigpond.net.au/d.keenan/Music/ErlichDecChords.gif
http://users.bigpond.net.au/d.keenan/Music/ChainOfMinor3rdsLattice.gif
http://en.wikipedia.org/wiki/Chord_(music)

> I like to call this (5:6:7:9) the just half-diminished 7th chord. 

I call 5:6:7:9 an otonal half-subdiminished seventh or otonal
subdiminished minor seventh (osdm7).

> There is
> no more consonant tuning in the harmonic series for it. Everything
else has
> something of a buzz.

I consider the just half-diminished 7th to be 5 : 6 : 36/5 : 9.

> And being in reality the upper four notes of the just
> dominant 9th chord, this chord is in good company. Even the just
minor triad
> (10:12:15) is harmonically a rootless major 7th chord.
> 
> With the first inversion (6:7:9:10), we're really looking at a
sub-minor (or
> septimal-minor) triad with a major sixth.

Yes. Subminor major sixth (smM6).

> Attempting to subsitute the just
> minor will produce a mess (30:36:45:50).

But naming that is straightforward. Successive intervals from the root
are 5:6, 2:3, 3:5.
3(2(5:6):3):5
So it's a minor major sixth (mM6)

> Speaking of the half-diminished, I found an enharmonic of this chord
in the
> Hungarian-minor scale, in the form of an augmented-sixth chord. But
I don't
> know of any names given for it. I kind of coined it as a Neptune 6th,
> because Holst uses it in his Planets. It is the mirror image of the
German
> 6th chord.

I assume you mean the German augmented sixth whose Just-ification
would seem to be 4:5:6:7 otherwise known as a major subminor seventh
(Msm7) or harmonic seventh chord.

The utonal 1/(7:6:5:4) I guess I should call a utonal subdiminished
subminor seventh (usdsm7) or utonal subdiminished augmented sixth, and
in this voicing 1/(6:5:4:7/2) it's a minor supermajor sixth (mSM6) or
minor diminished seventh.

-- Dave Keenan

From: Dave Keenan (2007-03-01)
Subject: Re: 25:30:35:45

> <<< Here's another question: what do you call 1/5:1/6:1/7:1/9

1/(9:7:6:5) is a supermajor minor seventh (SMm7).

> or 1/6:1/7:1/9:1/10. Multiplying through the latter by 9 gives 
> 9/10-1-9/7-3/2,

1/(10:9:7:6) would probably just be a bad voicing of the above,
although if pushed you could call it a diminished major sixth
suspended second (dM6sus2).

> and reducing that to an octave gives 1-9/7-3/2-9/5. >>>

That's the supermajor minor seventh (SMm7) again.

--- In [email protected], "Billy Gard" <billygard@...> wrote:
> Sounds like dominant 7th tuned to the undertone series. I've
listened to it,
> but like other utonal chords it doesn't lock in like otonal
versions. I've
> also listened to the "utonal tetrad" tuning of the half-diminished
> (1/4:1/5:1/6:1/7). But whatever consonance these have I would
attribute to
> the fact that they are rational, and can all be made integers by
> multiplication, although really large ones.

Rationality of ratios is not audible or measurable. Think about it.
For every rational there's an irrational infinitesimally close and
vice versa.

Don't you think whatever consonance it does have might have something
to do with the pairwise consonance. i.e. every interval in it can be
exressed in small whole numbers, even though the whole thing can't.

-- Dave Keenan

From: Tom Dent (2007-03-01)
Subject: half-diminished in septimal pseudo-common practice

--- In [email protected], "Billy Gard" <billygard@...> wrote:
>
> <<< It would be some sort of "harmonic" half-dim seventh, since I
think of
> 7/5
> as an augmented fourth more than a diminished fifth. It's also a
harmonic
> ninth without the root. >>>
> 
> I like to call this (5:6:7:9) the just half-diminished 7th chord.
There is
> no more consonant tuning in the harmonic series for it. Everything
else has
> something of a buzz. And being in reality the upper four notes of
the just
> dominant 9th chord, this chord is in good company. 

With my proposal to tune the dominant 7th in common practice as
4:5:6:50/7, i.e. 28:35:42:50, the half-diminished seventh would be either

35:42:50:63 with pure fifth

or

70:84:100:125 with pure third but a rather poor fifth;

or one could 'temper' just the top note as

140:168:200:251 

shy by only half a septimal semicomma, scilicet a septimal
quarter-comma. This might be a reasonable model of what singers could
do within a meantone context (if, that is, anyone wrote root position
half-diminished chords in the meantone eta!).

Incidentally 200:224:251 is practically meantone: the discrepancy
between the two tones is 6275/6272. I mention 224 because the implied
root of the dominant 9th is 112 ... then it might also be a nice
solution to have the 7th and 9th equidistant from that root as

4:5:6:50/7:224/25

though not so consonant with respect to the upper tones!

~~~T~~~

From: Gene Ward Smith (2007-03-01)
Subject: Re: half-diminished in septimal pseudo-common practice

--- In [email protected], "Tom Dent" <stringph@...> wrote:

> With my proposal to tune the dominant 7th in common practice as
> 4:5:6:50/7, i.e. 28:35:42:50...

You are raising 16/9 by 225/224 to get 25/14, but that means you are 
sharp by a septimal kleisma of the subdominant. This means you are 
going to get into small but noticable comma shifts, as the septimal 
kleisma, at eight cents, is perceptible. An alternative approach 
would be to use marvel tempering--temper out 225/224 (for example by 
tuning in 72-et or 228-et or whatever floats your boat.)

the half-diminished seventh would be either
> 
> 35:42:50:63 with pure fifth
> 
> or
> 
> 70:84:100:125 with pure third but a rather poor fifth;
> 
> or one could 'temper' just the top note as
> 
> 140:168:200:251 

I don't see any reason to drag goofy primes like 251 into the picture.
If you want to raise 25/14 by about this amount, one choice would be 
to up it by 3136/3125 to 224/125: 140:168:200:250.88. Now at least 
you know that if you temper out 3136/3125, the difference will go 
away.

> 4:5:6:50/7:224/25

Now you're talkin'!

From: Gene Ward Smith (2007-03-01)
Subject: Re: half-diminished in septimal pseudo-common practice

--- In [email protected], "Tom Dent" <stringph@...> wrote:

> With my proposal to tune the dominant 7th in common practice as
> 4:5:6:50/7

I took the list of all 11-limit pairwise consonant tetrads in 72-et, 
and checked for the smallest "Tymoczko distance" to this chord. Here 
Tymoczko distance is a metric I put on Tymoczko's orbifold. Sadly, the 
only good candidate to emerge was the 72-et version of the otonal 
tetrad.

However, by lowering standards by decreasing the size of the et or 
increasing the prime limit, or both, we can change things. In 17-limit 
46-et, for example, the closest was the 46-et version of 1-5/4-3/2-
16/9, but nearly as close was 1-5/4-3/2-9/5.

From: Tom Dent (2007-03-02)
Subject: Re: half-diminished in septimal pseudo-common practice

--- In [email protected], "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> small but noticable comma shifts, as the septimal 
> kleisma, at eight cents, is perceptible.

Sure, but it's comparable with the 1/4-comma shifts of adaptive JI,
which I think are a reasonable model for flexible intonation in cases
where comma shifts are problematic, and I don't consider adaptive JI
shifts as obtrusive. In fact the shift away from 1/4 comma meantone
necessary to get this tuning of the 7th (given that the just 4:5 is
fixed) is even smaller. The difference 225/224 and half of 81/80 is
three cents. One might as well use adaptive JI and keep the sevenths
where they are.

> marvel tempering--temper out 225/224 (for example by 
> tuning in 72-et or 228-et or whatever floats your boat.)

Regular temperament floats, as you put it, my boat for instruments of
fixed tuning. (For which purposes septimal meantone is just fine.) But
it seems to me to be unsuitable as a model for flexible tuning where
vertical justness is actively sought. I think 225/224 is a small
enough horizontal shift in that context that it doesn't need tempering. 

If one wished to model a just intonation containing a 25/14 seventh
with an equal division of the octave, then I think 159-et would be
suitable. (Funny, really.) The various tones needed are 24, 26 and 27
steps. A very different musical purpose from the one (tempering out
all possible commas?) Gene seems to be pursuing with his et's, and to
me a more interesting. 

25/14 = 1003.80 cents
133 steps of 159/et = 1003.77 !

I wonder if this tuning could also be applied to deal with the comma
shifts of 5-limit JI. For example, with the infamous 'God save the
King' progression

G-B-D-G 
E-B-E-G
C-C-E-A
D-A-D-F#
(G-B-D-G)

requiring two different A's, one could keep each chord within a 53-et
subset, but try shifting a couple of subsequent chords up (with
respect to the common tones) by various numbers of 159-et steps. 

It might amount to adaptive 1/3 comma meantone, mightn't it!! That is:
take the cycle of roots G-E-A-D-G and specify that G-E be pure, then
the subsequent roots are derived by a series of 1/3-comma, scilicet
one-step-of-159-et-flat, tempered fifths. 

If the dominant chord were D-C-D-F# then a good septimal tritone
between C and F# could result from holding over the C and having a C-D
tone of 26 steps. In terms of deviations from Pythagorean, D lies at
-1/3 and its seventh C at 0. The previous chord puts A at -2/3 and C
at +1/3. But that only means A shifts up, and C down, by one step.

Conversely, in the bread-and-butter three-chord progression

C-E-G-C
F-F-A-C
G-F-G-B
(C-F-G-C)

one can tune all the roots Pythagorean, then F has to shift one step
*upwards* to fulful its septimal role.

So... with adaptive JI AND septimal intervals easily to hand, it seems
like 159-et has a lot going for it! The driving force is the
similarity between (225/224)^3 and 81/80 and 1/53 of an octave.

I guess I should thank Gene for nudging me towards this model, even
though he might not be too fond of it.


> > 4:5:6:50/7:224/25
> 
> Now you're talkin'!

... but I'm probably not singin', because the top note can't be tuned
to that ratio by ear, no more than 251/140 can be, they are mushy
compromises.

It struck me that one could also have a half-diminished chord, for example

42:50:60:75,

which would *not* have a good interpretation as the upper parts of a
JI dominant 9th. But it is the inversion of my 28:35:42:60 dominant 7th!

~~~T~~~

From: Gene Ward Smith (2007-03-02)
Subject: Re: half-diminished in septimal pseudo-common practice

--- In [email protected], "Tom Dent" <stringph@...> wrote:

> So... with adaptive JI AND septimal intervals easily to hand, it seems
> like 159-et has a lot going for it! The driving force is the
> similarity between (225/224)^3 and 81/80 and 1/53 of an octave.

(225/224)^3/(81/80) = 703125/702464, a comma which has been sitting 
around on my 7-limit comma list for years without me or anyone else 
paying it much mind. The dent comma? Dentisma? Dentistma??

I wouldn't associated it especially with 159. Notable equal 
temperaments which temper it out include 171, 311, 494, 665, and 1178; 
as you see you can get a whole lot more accurate than 159.

From: Tom Dent (2007-03-02)
Subject: septimal / 1/3 comma / adaptive JI

--- In [email protected], "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In [email protected], "Tom Dent" <stringph@> wrote:
> 
> > So... with adaptive JI AND septimal intervals easily to hand, it seems
> > like 159-et has a lot going for it! The driving force is the
> > similarity between (225/224)^3 and 81/80 and 1/53 of an octave.
> 
> (225/224)^3/(81/80) = 703125/702464, a comma which has been sitting 
> around on my 7-limit comma list for years without me or anyone else 
> paying it much mind. The dent comma? Dentisma?

...if you must. 'Tertiadec' is as good a name as any other.

The main reason why I would get excited about it is not any relation
with an et, rather the possibility of a very simple notation for
incorporating septimals into common practice quite accurately and
seamlessly within a flexible adaptive JI framework. The notation would
just say how many steps a note was above or below the Pythagorean
sequence, with one step playing the role of 1/3 comma or 225/224.

To the extent that 5-limit JI is fit to 53-equal, this leads to 159-et.

> I wouldn't associated it especially with 159. Notable equal 
> temperaments which temper it out include 171, (...)
> as you see you can get a whole lot more accurate than 159.

In cents SC/3 = 7.17, 225/224 = 7.71, 1200/159 = 7.55, 1200/171 = 7.02
... and it does turn out that 171-et is remarkably accurate for the
7-limit. (And easy to convert from 159-et, just add one step to each
semitone of the scale.)

Though as I said, for common practice harmony it's sufficient to take
a chain of pure fifths and then tweak by thirds of a comma.

Now how to test it out ...? Common practice pieces for flexible tuning
(i.e. voices, strings) which have a lot of seventh chords and
'problematic'-for-JI progressions.

~~~T~~~

From: Carl Lumma (2007-03-02)
Subject: Re: septimal / 1/3 comma / adaptive JI

Hiya Tom,

What's the mapping for this temperament?  Can you show
how a dominant 7th chord would be notated?

-Carl

From: Andreas Sparschuh (2007-03-02)
Subject: Bach-squiggle from Werckmeister's Septenarius, wasRe: half-diminished septima...

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

> You (it was: Tom Dent) are raising 16/9 by 225/224 to get 25/14,...
Just that subdivsion of the SC=81/80 into =(225/224)(126/125)
2 superparticular factors happens also too @ notes

!280! D. 28/25 =(10/9)(126/125) = (225/224)(9/8)
!420! A. 42/25 = (5/3)(126/125) = (225/224)(27/16) see down

in my first 1999 Bach-squiggle interpretation as an cycle of a dozen
partially temperated 5ths:

Subdivision of the PC:=3^12/2^19 into 8 superparticular factors: 
http://www.strukturbildung.de/Andreas.Sparschuh/

A. 105,210; !420! cps (or Hz) @ begin
E. 157; _314_ ;/(315:=105*3)
B. 235; _470_ ;/(471:=147*3)
F# 11,22,44,88,176; _352_ ;704/(705:=235*3)
C# 33:=11*3
G# 99:=33*3
Eb _297_ :=99*3
Bb ; _445_ ,890/(981:=297*3)
F. ; _333.5_ ;667,1334/(1335:=445*3)
C. 125; _250_ ;500,1000,2000/(2001:=667*3)
G. 187; _374_ ;/(375:=125*3)
D. 35,70,140; !280! ;560/(561:=187*3)
A. 105:=35*3 cycle terminates @ same pitch as already on start

short:
A 314/315 E 470/471 B 704/705 F# C# G# Eb 890/891 Bb 1334/1335 F
2000/2001 C 374/375 G 560/561 D A

Consiting in a decomposition of the PC into the 8-fold product:
PC = 3^12/2^19 = 531441/524288 =

(315/314)(471/470)(705/704)(891/890)(1335/1334)(2001/2001)(375/374)(560/561)

That's recombined in ascending pitch-order in
_abs_ frequency/pitch-name/relative ratio/deviation from 3&5 limits
..................................................................
_250_ C. __1/1__ =  middle 'C' has absolute  250 cps or Hz
_264_ C# 132/125 = (256/243)(8019/8000) = (99/100)(16/15) 
!280! D. _28/25_ = (10/9)(126/125) = (225/224)(9/8) see initial rem.
_297_ Eb 297/250 = (32/27)(8019/8000) = (99/100)(6/5)
_314
_333.5F. 667/500 = (4/3)(2001/2000) tiny sharp 4th
_352_ F# 176/125 = (10/7)(616/625) = (176/175)(7/5) ~sept. tritone 
_374_ G. 187/125 = (374/375)(3/2) flat 5th
_396_ G# 198/125 = (128/81)(3969/4000) = (49/50)(8/5 octaved down 3rd) 
!420! A. _42/25_ = (5/3)(126/125) = (225/224)(27/16) syntonic vs. pyth
_445_ Bb _89/50_ = (16/9)(801/800) = (89/90)(9/5)
_470_ B. _47/25_ = (15/8)(376/375) = (6016/6075)(243/128)
_500_ c' __2/1

attend also @ Bb: SC:=81/80=(801/800)(89/90)

!sparschuh1999.scl
!
Sparschuh's 1999 interpetation of J.S. Bach's 1722 WTC squiggles
12
!
! 1/1   ! 1.000
132/125 ! 1.056 
 28/25  ! 1.12 
297/250 ! 1.188
157/125 ! 1.256
667/500 ! 1.334
176/125 ! 1.408 
187/125 ! 1.496
198/125 ! 1.584  
 42/25  ! 1.68
 89/50  ! 1.78
 47/25  ! 1.88
2/1 ! 2.000


that was inspired by Andreas Werckmeisters squiggle on p.91 of
his "Musicalische Temperatur" Quedlinburg 1691
and his "Septenarius"-tuning Chap.  XXVII pp.71-74
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html


C. 393/392; _196_ ;98,49 := 7^2 initialization
G. 525/524,264; _131_ :=393/3
D. 351/350; _175_ :=525/3 original questionable 176 corrected to 175
A. _117_ :=351/3
E. _156_ ;78,39:=117/3
H. 417/416,208; _104_ ;52,26,13:=39/3
F# 279/278; _139_ :=417/3
C# _186_ ;93:=279/3
G# 495/496,248; _124_ ;62,31:=93/3
D# _165_ :=495/3
B. 441/440,220; _110_ ;55:=165/3
F _147_ :=441/3
C 49:=147/3 back home @ 7*7=49

PC decomposition:
C 392/393 G 524/525 D 350/351 A E H 416/417 F# 278/279 C# G# 496/495
D# B 441/440 F C

Yielding the recombined string-lengnts in descending counting-order:
 
196 C. __1/1__ 
186 C# _98/93_ = (256/243)(3969/3968) = (245/248)(16/15)
175 D. 196/175 = (10/9)(882/875) = (1568/1575)(9/8)
165 D# 196/165 = (32/27)(441/440) = (98/99)(6/5) superparticular
156 E. _49/39_ = (5/4)(196/195) = (3136/3159)(81/64)
147 F. __4/3__  
139 F# 196/139 = (7/5)(140/139) = (686/695)(10/7) tritone
131 G. 196/131 = (416/417)(3/2) tiny flat 5th 
124 G# _49/31_ = (128/81)(3969/3968) =(245/248)(8/5)
117 A. 196/117 = (5/3)(196/195) = (3136/3159)(27/16)
110 B. _98/55_ = (16/9)(441/440) = (98/99)(9/5) superparticular
104 H. _49/26_ = (15/8)(196/195) = (3136/3159)(243/128)
098 c' __2/1__

have a lot of fun with that

From: Tom Dent (2007-03-02)
Subject: Re: septimal / 1/3 comma / adaptive JI

--- In [email protected], "Carl Lumma" <clumma@...> wrote:
>
> Hiya Tom,
> 
> What's the mapping for this temperament?  Can you show
> how a dominant 7th chord would be notated?
> 
> -Carl
>

I'm not sure what 'mapping' means in this context. If we take the unit
to be 1/3 comma then a pure third is

C E_3

now the 5-limit 'just' tritone is

C F#_3

but the septimal tritone is 

C F#_4

i.e. one 'unit' lower.

The 7/4 is then

Ab^3 F#_4 or equivalently Ab F#_7 

The 7/6 is

F G#_7

the 9/7 is for example

F#_3 Bb^4

So a dom 7 to my way of thinking is

C E_3 G Bb^1

an aug 6 would be

C E_3 G A#_7

etc. etc.

If you are prepared to forget the schisma (a la 159- or 171- equal),
then Bb=A#_3, therefore the aug 6 might be rewritten as a 'harmonic 7th'

C E_3 G Bb_4

which corresponds to the approximation that 64/63 is 4/3 of 81/80. 

Hope that is clear enough...
~~~T~~~