Topic: the simplest pan-proportionally beating 12-tone temperament

17 scales

Thread (38 messages)

From: George D. Secor (2005-11-18)
Subject: the simplest pan-proportionally beating 12-tone temperament

This is in reply to Gene's message #62475 on the main list.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "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:

I saw some unexpected consequences when these got rationalized 
(details below).

> ! 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

Here are the brats and total absolute error for each triad of the 
above:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.5000 34.43
Bb... ------- ------- 1.5000 25.40 
F.... ------- ------- 1.5000 16.23
C....  1.6667  5.0000 3.0000 17.15
G....  1.6667  5.0000 3.0000 18.03
D....  2.5000  6.2500 2.5000 22.88
A....  5.0000 10.0000 2.0000 30.54
E....  6.2814 11.9221 1.8980 38.69
B.... 15.0000 25.0000 1.6667 43.01
F#... ------- ------- 1.5000 43.01
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 43.01
Bb... ------- ------- 1.0000 43.01
F.... ------- ------- 1.0000 43.01
C....  7.9678  9.9678 1.2510 43.01
G....  5.5975  7.5975 1.3573 34.43 (brat was 1.3353)
D....  3.5262  5.5262 1.5672 25.40 (brat was 1.5025)
A....  2.2314  4.2314 1.8963 16.23 (brat was 2.0050)
E....  2.2049  4.2049 1.9071 17.15 (brat was 2.0050)
B....  6.3224  8.3224 1.3163 18.03 (brat was 1.3367)
F#... ------- ------- 1.0000 22.88
C#... ------- ------- 1.0000 30.54
G#... ------- ------- 1.0000 38.69

All I can say is: Yikes, what happened?  The A, B, and F# major 
triads now have exact brats, but five of the minor triads having 
close-to-exact approximations were devastated!  I think that this 
would not have happened had the brat for the A major triad not been 
changed so drastically.

But the next one is a different story:

> ! 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

The temperament (extra)ordinaire fared much better:

Major -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 15.0000 20.0000 1.3333 33.86
Bb... ------- ------- 1.5000 21.44 
F....  5.0000 10.0000 2.0000 17.18
C....  1.6667  5.0000 3.0000 17.23
G....  1.6667  5.0000 3.0000 17.27
D....  1.6667  5.0000 3.0000 17.10
A....  3.3333  7.5000 2.2500 25.68
E....  5.0000 10.0000 2.0000 34.28
B....  7.6185 13.9278 1.8281 48.36
F#... ------- ------- 1.5000 52.14
C#... ------- ------- 1.5000 52.14
G#... 15.0000 20.0000 1.3333 47.84

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... 27.4021 25.4021 0.9270 52.14
Bb... ------- ------- 1.0000 52.14
F.... 22.0625 24.0625 1.0907 52.14
C....  7.8056  9.8056 1.2562 42.49
G....  4.9383  6.9383 1.4050 30.08
D....  3.0000  5.0000 1.6667 21.44
A....  2.0000  4.0000 2.0000 17.18
E....  2.0000  4.0000 2.0000 17.23
B....  1.9512  3.9512 2.0250 17.27
F#... ------- ------- 1.0000 17.10
C#... ------- ------- 1.0000 25.68
G#... 14.7479 12.7479 0.8644 39.63

Here the surprise is that the brats for the D, A, and E minor triads 
are now exact, without ruining anything else -- wonderful!!!

In the two original temperaments the pitches in a chain of 5ths from 
C to B were exactly the same, so I expect that the ratios for those 
tones in the well-temperament could be the same as the simpler ones 
in the (extra)ordinaire if the A major triad were not forced to have 
a brat of 2 (which would then also improve the five minor triad 
brats), yes?

--George

From: Gene Ward Smith (2005-11-19)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> In the two original temperaments the pitches in a chain of 5ths from 
> C to B were exactly the same, so I expect that the ratios for those 
> tones in the well-temperament could be the same as the simpler ones 
> in the (extra)ordinaire if the A major triad were not forced to have 
> a brat of 2 (which would then also improve the five minor triad 
> brats), yes?

There's no doubt lots which could be done along these lines; trying to
both solve for exact major brats and approximate minor ones opens up a
whole new level of complexity.

From: George D. Secor (2005-11-23)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> 
> > In the two original temperaments the pitches in a chain of 5ths 
from 
> > C to B were exactly the same, so I expect that the ratios for 
those 
> > tones in the well-temperament could be the same as the simpler 
ones 
> > in the (extra)ordinaire if the A major triad were not forced to 
have 
> > a brat of 2 (which would then also improve the five minor triad 
> > brats), yes?
> 
> There's no doubt lots which could be done along these lines; trying 
to
> both solve for exact major brats and approximate minor ones opens 
up a
> whole new level of complexity.

Starting with the two sets of ratios you supplied, I was able to 
arrive at the following:

! secorWT08.scl
!
George Secor's well-temperament, proportional beating (attempt #8), 
rational version
 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

With 8 exact and 2 approximate major, and 8 exact and 4 approximate 
minor proportional-beating triads, it's a reasonably satisfactory 
solution.

One thing I like about it is that none of the numbers gets above 4 
digits -- simple enough, perhaps, to trick someone into thinking it's 
JI.  ;-)

--George

From: George D. Secor (2005-11-27)
Subject: Re: the simplest pan-proportionally beating 12-tone temperament

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> There's no doubt lots which could be done along these lines; trying 
to
> both solve for exact major brats and approximate minor ones opens 
up a
> whole new level of complexity.

I'm hoping that your "rationalization" calculations aren't too 
difficult or time-consuming (a Maple program, I presume?), because 
I'm in the process of coming up with a few more proportional-beating 
temperaments that could use some rationalizing.

For starters, here's a new modern (low-contrast) well-temperament I 
devised this past week.  It contains a chain of seven ~1/7-(Didymus-)
comma fifths and four just fifths (not all consecutive, which results 
in each and every triad having less total absolute error than 
Pythagorean):

! secor_WT10.scl
!
George Secor's 12-tone well-temperament, proportional beating 
(attempt #10)
 12
!
 95.22303
 197.75601
 297.21199
 395.51202
 501.12199
 593.26803
 698.87801
 797.17803
 896.63402
 999.16699
 1094.39003
 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 30.70
Bb... ------- ------- 1.5000 24.55 
F....  5.0000 10.0000 2.0000 24.55
C....  5.0000 10.0000 2.0000 24.55
G....  5.0000 10.0000 2.0000 24.55
D....  5.0000 10.0000 2.0000 24.55
A....  6.6785 12.5178 1.8743 30.70
E....  8.3600 15.0401 1.7990 36.86
B....  8.9925 15.9888 1.7780 39.17
F#... ------- ------- 1.5000 39.17
C#... ------- ------- 1.5000 39.17
G#... 14.3988 24.0982 1.6736 36.86

Minor -----Beat Ratios------ Total abs.
Triad M3/5th. m3/5th. m3/M3  error (cents)
----- ------- ------- ------ -------------
Eb... ------- ------- 1.0000 39.17
Bb... ------- ------- 1.0000 39.17
F.... 10.6696 12.6696 1.1874 39.17
C....  9.9258 11.9258 1.2015 36.86
G....  7.9435  9.9435 1.2518 30.70
D....  5.9576  7.9576 1.3357 24.55
A....  5.9576  7.9576 1.3357 24.55
E....  5.9576  7.9576 1.3357 24.55
B....  5.9576  7.9576 1.3357 24.55
F#... ------- ------- 1.0000 24.55
C#... ------- ------- 1.0000 30.70
G#... 17.0956 19.0956 1.1170 36.86

I'm hoping that this can be "rationalized" in such a way that the 
major triads in a chain of fifths from F to D and their relative 
minors have exact brats of 2 and 4/3, respectively, and that the four 
just fifths be kept.  After that, I would hope for exact brats for 
one of more of the following: G# major (5/3), G minor (5/4), C minor 
(6/5), E major (9/5), B major (16/9), and A major (15/8).

--George

From: George D. Secor (2005-11-29)
Subject: Request for Gene

Gene,

Since you haven't yet responded to the message below, I just wanted to 
make sure you noticed that I was asking whether it would be too much 
trouble for you to calculate rational versions of a few more 
temperaments:

http://groups.yahoo.com/group/tuning-math/message/13507

I was thinking that the particular temperament that I gave in the above 
message might be of interest to Aaron Johnson, inasmuch as it consists 
of a 1/7-comma temperament, modified in order to minimize the total 
absolute error of the worst triads (while keeping the brats reasonably 
simple for most of the major & minor triads).

--George

From: Gene Ward Smith (2005-11-29)
Subject: Re: Request for Gene

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> 
> Gene,
> 
> Since you haven't yet responded to the message below, I just wanted to 
> make sure you noticed that I was asking whether it would be too much 
> trouble for you to calculate rational versions of a few more 
> temperaments:

Sorry, I came down with a cold and haven't been feeling too ambitious.

From: George D. Secor (2005-11-29)
Subject: Re: Request for Gene

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> >
> > 
> > Gene,
> > 
> > Since you haven't yet responded to the message below, I just wanted 
to 
> > make sure you noticed that I was asking whether it would be too 
much 
> > trouble for you to calculate rational versions of a few more 
> > temperaments:
> 
> Sorry, I came down with a cold and haven't been feeling too ambitious.

Okay, no problem (and no hurry).

Best,

--George

From: Gene Ward Smith (2005-12-04)
Subject: Re: Request for Gene

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> 
> Gene,
> 
> Since you haven't yet responded to the message below, I just wanted to 
> make sure you noticed that I was asking whether it would be too much 
> trouble for you to calculate rational versions of a few more 
> temperaments:

I don't know if you will find this useful or not, but here is a
circulating temperament with exact values and some exact major and
minor brats. However, the exact values are mostly algebraic numbers,
not rational numbers.

! george.scl
George Secor inspired circulating temperament
12
!
98.314208
203.910002
295.986682
401.659957
499.896684
599.424914
701.955001
797.199419
902.787037
997.941683
1100.528751
1200.000000

From: Gene Ward Smith (2005-12-04)
Subject: Re: Request for Gene

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

Here's a rational version of george which is so close to the same
thing Scala calls them the same.

! geo.scl
George Secor style circulating temperament
12
!
55881/52796
9/8
15660/13199
332913/263980
35235/26398
18660/13199
3/2
83673/52796
88935/52796
23490/13199
372/197
2

From: George D. Secor (2005-12-05)
Subject: Re: Request for Gene

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
> 
> Here's a rational version of george which is so close to the same
> thing Scala calls them the same.
> 
> ! geo.scl
> George Secor style circulating temperament
> 12
> !
> 55881/52796
> 9/8
> 15660/13199
> 332913/263980
> 35235/26398
> 18660/13199
> 3/2
> 83673/52796
> 88935/52796
> 23490/13199
> 372/197
> 2

Gene, this is super!

The one thing I would want is to transpose it upward a minor 3rd so 
that the best major triads are on Bb, F, C, G, and D, thus:

! secor_WT2-11R.scl
Secor 2/11-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

Before I saw your solution, I thought I would try my hand at 
rationalizing this temperament and came up with the following ratios:

4469/4230
843/752
20089/16920
945/752
20089/15040
8475/6016
563/376
4469/2820
5049/3008
20089/11280
1415/752
2/1

The E and B minor brats are exact, while D and A minor are very 
close, but I ended up with 4 inexact major brats (on A, E, B, and 
F#).  I was disappointed that I had to make the fifth of F# inexact 
in order to get an exact brat on D major, so I like yours better.

--George

From: George D. Secor (2005-12-05)
Subject: Re: Request for Gene

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...> 
wrote:
> 
> ! secor_WT2-11R.scl
> Secor 2/11-comma well-temperament, Gene Ward Smith rational version

Oops, sorry!  I should have called it:

! secor_WT1-7R.scl
> Secor 1/7-comma well-temperament, Gene Ward Smith rational version

--George

From: George D. Secor (2005-12-07)
Subject: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> Here's a rational version of george which is so close to the same
> thing Scala calls them the same. ...

Gene, in the process of enthusing over how much I liked this latest 
one (even if I had to transpose it), I neglected to say "thanks!"

And I hope you're feeling better now, because I have a proposal for a 
joint project.  I was in the process of organizing a selection of my 
best 8 circulating proportional-beating temperaments into a suite 
when you came up with a rational version of my original temperament 
(extra)ordinaire that improved it to the point that it completely 
blew me away!  With this latest "rational version of george" you've 
now rationalized 4 of the temperaments in my suite -- so your part of 
the proposed project is already half done.

My idea is to organize the circulating temperaments into 3 
categories, with Scala files named as follows (noting that both our 
initials are "GS"):

1) Modern (low-contrast) well-temperaments:

GS1_7WT.scl - 1/7-comma modern well-temperament (done!)
GS2_11WT.scl - 2/11-comma modern well-temperament

2) Baroque (high-contrast) well-temperaments:

GS1_5WT.scl - 1/5-comma well-temperament (done!)
GS5_23WT.scl - 5/23-comma well-temperament (not finalized)
GS1_4WT.scl - 1/4-comma well-temperament

3) Temperaments _extraordinaire_

GS1_5TE.scl - 1/5-comma temperament extraordinaire (done!)
GS5_23TE.scl - 5/23-comma temperament extraordinaire (done!)
GS1_4TE.scl - 1/4-comma temperament extraordinaire

Within these are various trade-offs, e.g., the 5/23-comma 
temperaments have only a few proportional-beating minor triads (but 
the best major brats), the 1/5-comma temperaments have the best brats 
for major and minor taken together, while the 1/4-comma temperaments 
will have the greatest key contrast.  The 1/7-comma temperament has 
the lowest key contrast and also the least amount of dissonance in 
the worst keys.

The description for each could be as in the following example:

"Gene Ward Smith's rational version of George Secor's 1/5-comma well-
temperament"

Are you feeling ambitious enough to solve equations for rational 
values for the remaining 4 temperaments?

--George

From: Gene Ward Smith (2005-12-07)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Are you feeling ambitious enough to solve equations for rational 
> values for the remaining 4 temperaments?

I'll take a look. Can you repost them here?

From: George D. Secor (2005-12-07)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> 
> > Are you feeling ambitious enough to solve equations for rational 
> > values for the remaining 4 temperaments?
> 
> I'll take a look. Can you repost them here?

Thanks, Gene!  8-))  (big smile)

Only two of these are ready for you to tackle.

The first one (2/11-comma WT) I already posted here:
http://groups.yahoo.com/group/tuning/message/61800
but I've since reworked it in order to get a greater number of exact 
brats.

The problem for me is that this seems to require a lot of 
simultaneous equations in order to arrive at an exact solution.  
Since you've already an expert at this, I figured there was no point 
in my attempting to re-invent the wheel.  The following cents 
listing, arrived at by trial and error, will produce brats accurate 
to 3 decimal places:

92.002
196.025
252.912
392.910
499.822
590.047
698.026
793.957
894.112
997.687
1090.255
1200.000

The intended major-triad brats are:
C, G: 2.5
D: 2.25
A: 2
E: ~1.875 (this one need not be exact)
B: 5/3
F#, C#, Ab, Eb, Bb: 1.5
F: 11/6

If you get the major-triad brats right, then most of the minor-triad 
brats will come close to simple-number ratios.

The second one is my 1/4-comma temperament extraodinaire:

86.76888
195.84306
296.15421
388.99991
500.06421
587.51670
699.26463
791.74537
892.42148
998.10921
1088.79629
1200.00000

The intended major-triad brats are:
Eb, Bb: 1.5
F: 2
C, D: 3
G: 4
A: 2.5
E: ~1.7
B: ~1.75 (I doubt you'll be able to make this one exact.)
F#: 5/3
C#, Ab: 4/3

For this one I employed a strategy of deliberately tempering all of 
the fifths in the 4 worst major triads (B, F#, C#, Ab) in order to 
cover up the harshness of the rapidly beating thirds; hence there are 
only 2 just fifths.  The 1/4-comma fifths are 3 in number, from G to 
E.  The C major triad has a total absolute error equivalent to a 1/4-
comma meantone triad, but with a slower-beating fifth (2.7 cents 
false).

Feel free to tinker with the E and B major brats, in case you find a 
way to make one of them exact.

As with the 2/11 temperament, if you get the major-triad brats right, 
then a majority of the minor-triad brats will come reasonably close 
to simple-number ratios.

Good luck!

--George

From: Gene Ward Smith (2005-12-07)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> 92.002
> 196.025
> 252.912
> 392.910
> 499.822
> 590.047
> 698.026
> 793.957
> 894.112
> 997.687
> 1090.255
> 1200.000

See what you think of this; the brats you want are there, at least.

! secrat.scl
Rationalized Secor well-temperament
12
!
8040000/7623887
17075795/15247774
9045000/7623887
76498155/60991096
10175625/7623887
10720000/7623887
11409902/7623887
12060000/7623887
102226155/60991096
13567500/7623887
14311200/7623887
2

From: George D. Secor (2005-12-07)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
> 
> See what you think of this; the brats you want are there, at least.

Golly, that was fast!

> ! secrat.scl
> Rationalized Secor well-temperament
> 12
> !
> 8040000/7623887
> 17075795/15247774
> 9045000/7623887
> 76498155/60991096
> 10175625/7623887
> 10720000/7623887
> 11409902/7623887
> 12060000/7623887
> 102226155/60991096
> 13567500/7623887
> 14311200/7623887
> 2

Bravo!!!  It's perfect!!!  8-)))

--George

From: Gene Ward Smith (2005-12-08)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> The second one is my 1/4-comma temperament extraodinaire:
> 
> 86.76888
> 195.84306
> 296.15421
> 388.99991
> 500.06421
> 587.51670
> 699.26463
> 791.74537
> 892.42148
> 998.10921
> 1088.79629
> 1200.00000
> 
> The intended major-triad brats are:
> Eb, Bb: 1.5
> F: 2
> C, D: 3
> G: 4
> A: 2.5
> E: ~1.7
> B: ~1.75 (I doubt you'll be able to make this one exact.)
> F#: 5/3
> C#, Ab: 4/3

You gave me more target flexibility with this, so I've listed three
possibilities below, for you to cherry-pick from. The rule is that you
can specify 11 out of the 12 brats, though if the brats are not
well-chosen the result might be far from satisfactory as a tuning.
There's actually a general solution to this problem in rational functions.

! ex1.scl
Secor extraordinary one
12
!
1405281/1336600
56124137/50122500
118946/100245
16732127/13366000
178419/133660
4691627/3341500
50048619/33415000
10558/6683
44759/26732
59473/33415
4701086/2506125
2

! ex2.scl
Secor extraordinary two
12
!
2313789/2200700
46204229/41263125
391688/330105
3443711/2750875
146883/110035
7724743/5501750
20600859/13754375
173837/110035
36848/22007
195844/110035
3096037/1650525
2

! ex3.scl
Secor extraordinary three
12
!
3132087/2978900
62547547/55854375
1590632/1340505
4662288/3723625
198829/148945
31370047/22341750
27880647/18618125
705949/446835
49884/29789
795316/446835
20947267/11170875
2

From: Carl Lumma (2005-12-08)
Subject: Re: [tuning-math] Re: Project for Gene? (was: Request for Gene)

>There's actually a general solution to this problem in rational functions.

What's that like?

-Carl

From: Gene Ward Smith (2005-12-09)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >There's actually a general solution to this problem in rational
functions.
> 
> What's that like?

I haven't tried to compute it.

From: George D. Secor (2005-12-11)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> ...
> > The intended major-triad brats are:
> > Eb, Bb: 1.5
> > F: 2
> > C, D: 3
> > G: 4
> > A: 2.5
> > E: ~1.7
> > B: ~1.75 (I doubt you'll be able to make this one exact.)
> > F#: 5/3
> > C#, Ab: 4/3
> 
> You gave me more target flexibility with this, so I've listed three
> possibilities below, for you to cherry-pick from. The rule is that 
you
> can specify 11 out of the 12 brats, though if the brats are not
> well-chosen the result might be far from satisfactory as a tuning.
> ...
> [snipped out the first two]
> 
> ! ex3.scl
> Secor extraordinary three
> 12
> !
> 3132087/2978900
> 62547547/55854375
> 1590632/1340505
> 4662288/3723625
> 198829/148945
> 31370047/22341750
> 27880647/18618125
> 705949/446835
> 49884/29789
> 795316/446835
> 20947267/11170875
> 2

Gene, your results were better than I expected -- excellent!

After listening to these, I found that I like #3 best, because the 
chain of fifths from E thru D# has the least variation in size, which 
makes the worst triads sound less dissimilar from one another.  (This 
could also have something to do with the fact that in temperament #3 
the brats have a steady progression ranging from 4 down to 4/3 
without the zig-zag that occurs in the other two temperaments.)

However, I'm a little disappointed that the brats for the minor 
triads didn't come out nearly as well as in my original.  I'm also 
undecided about whether I really like the tempered fifths in the F# 
and C# triads better than if they were just -- I probably should seek 
some opinions from others on the main tuning list about this.  I 
therefore would like to look further for another rational temperament 
that #3 could be compared with in a listening test.

I looked through my previous attempts with 1/4-comma fifths and came 
up with the following two, which are slight variations of one another:

Temperament extraordinaire attempt 11c
 85.53845
 194.05179
 293.76344
 387.20865
 499.61501
 583.58345
 697.47337
 787.49346
 890.63022
 997.66001
 1084.86039
 2/1

Temperament extraordinaire attempt 11d
 84.97762
 194.05179
 293.76344
 387.20865
 499.61501
 583.58345
 697.47337
 786.36833
 890.63022
 997.66001
 1084.86039
 2/1

After I came up with these, I continued looking for other options, 
because I wasn't able to get as many exact brats as I would have 
liked in either one.  Applying your equations to these, however, 
should take care of that.

For a rational synthesis of these I expect the major-triad brats to 
be:
F: 2
C, G: 9
D: 4
A: 2.5
E: 2
B: ~1.7 to 1.75
F#, C#: 1.5
Ab: ~1.2 to 1.25
Eb: 4/3
Bb: 1.5

Yes, that's a brat of 9 for C and G major, which has M3:5th and 
5th:m3 beat ratios of 1:3 -- very desirable, I think!  (The 5/21-
comma meantone temperament has this same brat for the major triads.)  
Also, many of the minor triad brats approximate simple ratios, and 
the total absolute error for the Bb, F, C, G, and D major triads 
(collectively) is lower than anything else I've tried so far, so this 
looks very promising for a high-contrast circulating temperament.

Since either B or Ab could be the unspecified brat, you once again 
have some target flexibility.  I suggest trying 1.7, 12/7, and 1.75 
for B -- or 1.2, 11/9, and 1.25 for Ab -- to see what you get for the 
remaining one.

Another possibility is to fix D at 3 and Ab at 4/3 and experiment 
with brats for A (~2.25 to 2.33) with B unspecified, or B (~1.6 to 
1.67) with A unspecified.  Inasmuch as this would significantly 
increase the total absolute error for the D, A, and E major triads 
(and would also probably have a detrimental effect on the minor triad 
brats), I believe I would be less enthusiastic about this option.

And you may think of other things to try.

--George

From: Gene Ward Smith (2005-12-11)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> For a rational synthesis of these I expect the major-triad brats to 
> be:
> F: 2
> C, G: 9
> D: 4
> A: 2.5
> E: 2
> B: ~1.7 to 1.75
> F#, C#: 1.5
> Ab: ~1.2 to 1.25
> Eb: 4/3
> Bb: 1.5

Here are some possibilities to consider:

! se1.scl
Secor extraordinare 1
12
!
19599/18668
208713/186680
1105971/933400
233473/186680
311403/233350
6533/4667
139641/93340
58797/37336
78053/46670
207602/116675
43669/23335
2

! se2.scl
Secor extraordinare 2
12
!
60125/57249
256093/228996
814039/686988
859195/686988
1146014/858735
240500/171747
171287/114498
60125/38166
287279/171747
4584056/2576205
642745/343494
2

! se3.scl
Secor extraordinare 3
12
!
60125/57249
256093/228996
814039/686988
859195/686988
1146014/858735
240500/171747
171287/114498
60125/38166
287279/171747
4584056/2576205
642745/343494
2

! sc1.scl
Secor1
12
!
708225/673984
1508109/1347968
1597375/1347968
1685885/1347968
2248769/1684960
236075/168496
1008201/673984
2124675/1347968
563809/336992
2248769/1263720
630475/336992
2

! sc2.scl
Secor2
12
!
320275/304956
682081/609912
2168125/1829736
2288395/1829736
3052313/2287170
320275/228717
456209/304956
320275/203304
765143/457434
6104626/3430755
427975/228717
2

! sc3.scl
Secor3
12
!
525975/500744
1120119/1001488
1186725/1001488
1252535/1001488
1670679/1251860
175325/125186
749091/500744
1577925/1001488
418819/250372
556893/312965
468475/250372
2

! sc4.scl
Secor4
12
!
1213425/1155752
2584377/2311504
2738875/2311504
2890905/2311504
3855857/2889380
404475/288938
1729053/1155752
3640275/2311504
966477/577876
3855857/2167035
1081425/577876
2

From: Carl Lumma (2005-12-11)
Subject: Re: [tuning-math] Re: Project for Gene? (was: Request for Gene)

Hi George,

I don't know if you saw this the first time around, but here's
an attempt at finding a well-temperament with good brats.  I
thought you might like to see how I went about it.  I don't know
how Gene does it, or how you do it, but this was my approach.
I realize the kind of scales you're looking for are a good bit
more complicated than the ones I was ... this is a tempered-
octaves scale with "power chord" brats, rather than trying to
hit brats on major and minor triads in the same scale.  Anyway...

http://groups.yahoo.com/group/tuning-math/message/12496

Well, the thread quickly goes off topic, and I see I didn't
actually very clearly show my work here :(, and I see that
all of it only applies to one particular rational octave, but
maybe it's of interest.

-Carl

From: Gene Ward Smith (2005-12-11)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> And you may think of other things to try.

How does this strike you? 

! circu.scl
A circulating temperament
12
!
25250/23829
26684/23829
37875/31772
29840/23829
63769/47658
11190/7943
35636/23829
12625/7943
39940/23829
32737/18330
14920/7943
2

From: George D. Secor (2005-12-14)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" 
<gdsecor@y...> wrote:
> 
> > And you may think of other things to try.
> 
> How does this strike you? 
> 
> ! circu.scl
> A circulating temperament
> 12
> !
> 25250/23829
> 26684/23829
> 37875/31772
> 29840/23829
> 63769/47658
> 11190/7943
> 35636/23829
> 12625/7943
> 39940/23829
> 32737/18330
> 14920/7943
> 2

The major brats *look* very nice!  In addition, four of the minor-
triad brats are exact, and four others come close to approximating 
simple ratios.  For overall simplicity I think it's going to be very 
difficult to improve on this.

However, looks aren't everything.  When I listened to it in Scala, I 
found that the A major triad was a bit difficult to accept -- it not 
only has the worst fifth (-6.66c error), but it's also one of the 3 
major triads with the greatest total absolute error.  I'm beginning 
to doubt whether you can get away with a fifth tempered much more 
than about 3 or 4 cents in a triad with such dissonant thirds.

For the major triads the smallest total absolute error is on Bb and 
the largest is on A, E, and B, so I would advise transposing this up 
a whole step (which would move your problem triad to B).

Assuming that's done, I would then compare the brats with this one:
http://groups.yahoo.com/group/tuning-math/message/62013
in search of a clue as to how the sound might be improved.

I was wondering what would happen if your leftover ~1.2 brat 
(transposed to G# major) were set to 4/3 and one of the 2 brats 
(e.g., the one that would be transposed to B) were unspecified; might 
it approximate 5/3?  (I haven't taken the time to work anything out 
along these lines -- this is just a thought about how you could 
further experiment with the brats.)

--George

From: George D. Secor (2005-12-14)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> 
> > For a rational synthesis of these I expect the major-triad brats 
to 
> > be:
> > F: 2
> > C, G: 9
> > D: 4
> > A: 2.5
> > E: 2
> > B: ~1.7 to 1.75
> > F#, C#: 1.5
> > Ab: ~1.2 to 1.25
> > Eb: 4/3
> > Bb: 1.5
> 
> Here are some possibilities to consider:
> 
> ! se1.scl
> ...
> ! se2.scl
> ...
> ! se3.scl
> ...
> ! sc1.scl
> Secor1
> 12
> !
> 708225/673984
> 1508109/1347968
> 1597375/1347968
> 1685885/1347968
> 2248769/1684960
> 236075/168496
> 1008201/673984
> 2124675/1347968
> 563809/336992
> 2248769/1263720
> 630475/336992
> 2
> ...
> ! sc2.scl
> ...
> ! sc3.scl
> ...
> ! sc4.scl
> ...

They're all very similar, of course, but here are my observations:

I noticed that se3 is the same as se2.  I'm guessing that you meant 
this one to be similar to sc3 (but with a different unspecified 
brat).  The brats in sc3 aren't as simple as the others, and the only 
advantage I could see is that it comes close to minimizing the 
maximum error of the fifths.  In any event, it wasn't my favorite.

I prefer sc1 over all of the others because:
1) The worst triads have less error;
2) The major brats are simplest;
3) No fifth is tempered by more than 5.7 cents;

These tunings all have 3 exact and 4 or 5 reasonably approximate 
minor brats, but overall I'd say those in sc1 are the simplest.

Thanks, Gene, for a job well done!  I think I'm about ready to ask 
respondents on the main list to evaluate sc1 vs. ex3 (in tm #13642) 
vs. secorteo4.scl (in tuning #62013), to determine whether the worst 
triads (on F# and C# major in each one) are judged more acceptable 
with just or tempered fifths.

--George

From: George D. Secor (2005-12-14)
Subject: Re: Project for Gene? (was: Request for Gene)

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> Hi George,
> 
> I don't know if you saw this the first time around, but here's
> an attempt at finding a well-temperament with good brats.  I
> thought you might like to see how I went about it.  I don't know
> how Gene does it, or how you do it, but this was my approach.
> I realize the kind of scales you're looking for are a good bit
> more complicated than the ones I was ... this is a tempered-
> octaves scale with "power chord" brats, rather than trying to
> hit brats on major and minor triads in the same scale.  Anyway...
> 
> http://groups.yahoo.com/group/tuning-math/message/12496

I didn't see anything there regarding how you went about anything -- 
it looked to me as if you were posing a question or problem for 
someone else to solve.  (Oh, there it is in your next message!)

Looking for proportional brats in a 2:3:4 "triad" with a tempered 
octave seems to me a bit self-defeating.  If the octave is *exact*, 
then the brats are *guaranteed* to be simple (1:2 for the 
fifth:fourth), regardless of the amount the fifth is tempered.

> Well, the thread quickly goes off topic, and I see I didn't
> actually very clearly show my work here :(, and I see that
> all of it only applies to one particular rational octave, but
> maybe it's of interest.

I didn't follow it, because I'm not very interested in tempered 
octaves (and consequently haven't attempted to calculate brats for 
these).  And, unfortunately, I don't have as much spare time as I 
would want to pursue things I *am* interested in.  :-( sorry )-:

--George

From: Carl Lumma (2005-12-14)
Subject: Re: [tuning-math] Re: Project for Gene? (was: Request for Gene)

>> Hi George,
>> 
>> I don't know if you saw this the first time around, but here's
>> an attempt at finding a well-temperament with good brats.  I
>> thought you might like to see how I went about it.  I don't know
>> how Gene does it, or how you do it, but this was my approach.
>> I realize the kind of scales you're looking for are a good bit
>> more complicated than the ones I was ... this is a tempered-
>> octaves scale with "power chord" brats, rather than trying to
>> hit brats on major and minor triads in the same scale.  Anyway...
>> 
>> http://groups.yahoo.com/group/tuning-math/message/12496
>
>I didn't see anything there regarding how you went about anything -- 
>it looked to me as if you were posing a question or problem for 
>someone else to solve.  (Oh, there it is in your next message!)
>
>Looking for proportional brats in a 2:3:4 "triad" with a tempered 
>octave seems to me a bit self-defeating.  If the octave is *exact*, 
>then the brats are *guaranteed* to be simple (1:2 for the 
>fifth:fourth), regardless of the amount the fifth is tempered.

The point was, since those scales have tempered octaves, any
special 4:5:6 brats would apply only in a given octave span --
extensions like 2:5:6 would break.

>> Well, the thread quickly goes off topic, and I see I didn't
>> actually very clearly show my work here :(, and I see that
>> all of it only applies to one particular rational octave, but
>> maybe it's of interest.
>
>I didn't follow it, because I'm not very interested in tempered 
>octaves (and consequently haven't attempted to calculate brats for 
>these).

My approach would have been the same regardless of what chord
I was solving for, which is why I thought you might be interested.

-Carl

From: Carl Lumma (2005-12-14)
Subject: Re: [tuning-math] Re: Project for Gene?

By the way, George, are you familiar with
Paul Bailey's EBWT?

!
 Paul Bailey's Equal-Beating Well Temperament.
 12
!
 90.23
 192.92
 294.14
 389.76
 498.05
 588.27
 696.19
 792.18
 891.85
 996.09
 1086.32
 2/1
!
! Seven pure fifths, good brats.

-Carl

From: George D. Secor (2005-12-15)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> By the way, George, are you familiar with
> Paul Bailey's EBWT?
> 
> !
>  Paul Bailey's Equal-Beating Well Temperament.
>  12
> !
>  90.23
>  192.92
>  294.14
>  389.76
>  498.05
>  588.27
>  696.19
>  792.18
>  891.85
>  996.09
>  1086.32
>  2/1
> !
> ! Seven pure fifths, good brats.
> 
> -Carl

Thanks, Carl.  I wasn't familiar with it.  It looks quite good, on a 
number of counts.

The major-triad brats are:
C: 4
G: unassigned (~3.5)
D, A, E: 2
B, F#, C#, G#, Eb, Bb, F: 1.5

Since there are 7 just fifths, there are also 7 minor brats of 1, and 
the G minor brat is very close to 1.5.

Last, but not least, it's a well-constructed well-temperament.

Gene:

If you rationalize it, I'll pass the numbers on to Paul.

And as long as I'm on the subject (and while you're at it), there was 
only one general category remaining in the rational proportional-
beating project, which I was leaving till last (since it looked like 
it might be a bit tricky) -- a well-temperament with characteristics 
similar to Paul's, above.  Looking through some of my recent 
attempts, I've selected one that suggests the following set of major-
triad brats for rationalizing (two possible ways):

F: 2
C: 3
G: 4
D: 2.5 or unassigned
A: 2
E: 5/3 or unassigned
B, F#, C#, G#, Eb, Bb: 1.5

Gene, if you'll work your magic, I'll be very interested to see what 
happens!

--George

From: Gene Ward Smith (2005-12-15)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> The major-triad brats are:
> C: 4
> G: unassigned (~3.5)
> D, A, E: 2
> B, F#, C#, G#, Eb, Bb, F: 1.5

G turns out to be exactly 32/9, which is pretty neat.

> If you rationalize it, I'll pass the numbers on to Paul.

I think he must already have them, but here it is:

! bailey.scl
Rationalized Paul Bailey well temperament
12
!
256/243
12224/10935
32/27
13696/10935
4/3
1024/729
16348/10935
128/81
18304/10935
16/9
4096/2187
2

From: Gene Ward Smith (2005-12-15)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> F: 2
> C: 3
> G: 4
> D: 2.5 or unassigned
> A: 2
> E: 5/3 or unassigned
> B, F#, C#, G#, Eb, Bb: 1.5
> 
> Gene, if you'll work your magic, I'll be very interested to see what 
> happens!

! well1.scl
First well-temperament
12
!
28800/27307
30574/27307
32400/27307
4885/3901
36450/27307
38400/27307
5841/3901
43200/27307
45715/27307
48600/27307
51200/27307
2

! well2.scl
Second well-temperament
12
!
24000/22759
76606/68277
27000/22759
28480/22759
30375/22759
32000/22759
34101/22759
36000/22759
38080/22759
40500/22759
128000/68277
2

From: a_sparschuh (2005-12-16)
Subject: well1.scl rounded integral modification, was Re: Project for Gene? under razor

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> ! well1.scl
> First well-temperament
> 12
> !
> 28800/27307
> 30574/27307
> 32400/27307
> 4885/3901
> 36450/27307
> 38400/27307
> 5841/3901
> 43200/27307
> 45715/27307
> 48600/27307
> 51200/27307
> 2
that is for example concrete in absoulte pitch-frequenies: 
C  273.07 cps or Hz
C# 288
D  305.74  
Eb 324
E  341.85
F  364.5
F# 384
G  408.87
G# 432
A  457.15
Bb 486
B  512
C' 546.14

round all nonintegral ones to the next whole number except F=364.5cps
C  273 cps or Hz
C# 288,144,72,36,18,9
D  306,153
Eb 324,162,81
E  342,171
F  729/2
F# 384,192,96,48,24,12,6,3
G  409
G# 432,216,108,54,27
A  457
Bb 486,243
B  512,..,1
C' 546

or as chain of partial tempered 5ths F>C>G>D>A<E>B
C 273 cps or Hz
G (819)818,409(408,204,102,51)
D 153(152,76,38,19)
A (57,144,228,456)457
E 171
B (513)512,..,1
F# 3
C# 9 
G# 27
Eb 81
Bb 243
F  729(728,364,182,91)
C  273
yielding an tuning instruction similar like in
http://www.strukturbildung.de/Andreas.Sparschuh/

From: Gene Ward Smith (2005-12-16)
Subject: well1.scl rounded integral modification, was Re: Project for Gene? under razor

--- In tuning-math@yahoogroups.com, "a_sparschuh" <a_sparschuh@y...>
wrote:

> that is for example concrete in absoulte pitch-frequenies: 
> C  273.07 cps or Hz
> C# 288
> D  305.74  
> Eb 324
> E  341.85
> F  364.5
> F# 384
> G  408.87
> G# 432
> A  457.15
> Bb 486
> B  512
> C' 546.14

I was just remarking on MMM that 432 Hz might make a good choice for
A, so we could bring it up a step, and there we'd be. Of course, since
we want the keys with the flat fifths in the C vicinity and the keys
with the pure fifths in the F# vicinity, this problem is to be expected.

From: a_sparschuh (2005-12-16)
Subject: well1.scl rounded integral modification, was Re: Project for Gene? under razor

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> I was just remarking on MMM that 432 Hz might make a good choice for
> A, so we could bring it up a step, and there we'd be. Of course, since
> we want the keys with the flat fifths in the C vicinity and the keys
> with the pure fifths in the F# vicinity, this problem is to be
> expected.
oh, sorry, my misunderstanding:
i didn't knew about yours 432Hz preference,
taking yours common denominators in well1.scl: 
from 27307 to wrongly to C=273.07Hz,
yielding A=457.15Hz instead your correct 432Hz.

Presumably i had my similar Werckmeister
interpretation still to much in ears,
at baroque choirtone A'=456Hz

C  2173
G  (6561)6560,3280,1640,820,410,205(204,102,51)
D  153(152,76,38,19)
A  57
E  171
B  513(512,..,1) W:"...one quarter comma above the unity."
F# 3
C# 9
G# 27
Eb 81
Bb 243
F  729
C  2173

From: George D. Secor (2005-12-16)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> 
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> 
> > The major-triad brats are:
> > C: 4
> > G: unassigned (~3.5)
> > D, A, E: 2
> > B, F#, C#, G#, Eb, Bb, F: 1.5
> 
> G turns out to be exactly 32/9, which is pretty neat.

Yes, and the G minor brat is also exactly 1.5, and the one for D 
minor very close to 1.4.

> > If you rationalize it, I'll pass the numbers on to Paul.
> 
> I think he must already have them, …

Anyway, I wanted to have them, so thanks!

> but here it is:
> 
> ! bailey.scl
> Rationalized Paul Bailey well temperament
> 12
> !
> 256/243
> 12224/10935
> 32/27
> 13696/10935
> 4/3
> 1024/729
> 16348/10935
> 128/81
> 18304/10935
> 16/9
> 4096/2187
> 2

It's interesting to observe that the key contrast (as determined by 
the total absolute error of the triads) is very similar to this 2/11-
comma rational comma temperament:
http://groups.yahoo.com/group/tuning-math/message/13639
although many of the brats are quite different, as are the sizes of 
the fifths (3 tempered by more than 5.2 cents in Paul's, but none by 
more than 4 cents in the 2/11-comma temperament).

On the other hand, you might think, on the basis of the greater 
similarity of the brats and especially the sizes of the fifths, that 
Paul's WT would sound more like the well1 you posted yesterday:
http://groups.yahoo.com/group/tuning-math/message/13671
Instead, well1 (with 3 fifths tempered by more than 4.6 cents) has 
significantly higher key contrast than either of the other two.

BTW, well1 turned out to be pretty nice, while well2 (with D:A 
tempered over 10 cents) is junk (an errant guess on my part).  But I 
think well1 could be improved by making F:C just and B:F# tempered 
narrow by a small amount (or perhaps both fifths tempered by a small 
amount) -- back to the drawing board.

--George

From: a_sparschuh (2005-12-17)
Subject: improved roundning of well1.scl, was Re: Project for Gene?

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > ! well1.scl
> > First well-temperament
> > 12
> > !
> > 28800/27307
> > 30574/27307
> > 32400/27307
> > 4885/3901
> > 36450/27307
> > 38400/27307
> > 5841/3901
> > 43200/27307
> > 45715/27307
> > 48600/27307
> > 51200/27307
> > 2
> that is for example concrete in absoulte pitch-frequenies: 
> C  273.07 cps or Hz
> C# 288
> D  305.74  
> Eb 324
> E  341.85
> F  364.5
> F# 384
> G  408.87
> G# 432
> A  457.15
> Bb 486
> B  512
> C' 546.14
> 
> round all nonintegral ones to the next whole number except 
> F=364.5cps and
round 457.15 down to even lower 456 by replacing 457

> C  273 cps or Hz
> C# 288,144,72,36,18,9
> D  306,153
> Eb 324,162,81
> E  342,171
> F  729/2
> F# 384,192,96,48,24,12,6,3
> G  409
> G# 432,216,108,54,27

A 456,228,114,57 instead >457 as i had mistaken in my first attempt

> Bb 486,243
> B  512,..,1
> C' 546
> 
that avoids in the chain of partial tempered 5ths F>C>G>D>A,E>B
the obsolete over-wide sharp wolf 5th A<E
once amounting 457/456 to much large, but now just pure
> C 273 cps or Hz
> G (819)818,409(408,204,102,51)
> D 153(152,76,38,19)

A 57 please forget my formerly nonsene of> A (57,144,228,456)457

> E 171
> B (513)512,..,1
> F# 3
> C# 9 
> G# 27
> Eb 81
> Bb 243
> F  729(728,364,182,91)
> C  273
> yielding an other integral tuning instruction similar like in
> http://www.strukturbildung.de/Andreas.Sparschuh/
>
due to that further modification
the 5th A>E became now pure too, as already again before
F#>C#>G#>Eb>Bb>F especially inbetween the upper 5 
pythagorean-tuned black keys of the keyboard,
whrere the just 5ths should belong to and be placed,
according to common baroque tuning principles.

That procedure divides the 
PC=3^12:2^19=531441:528244
exact into 5 superparticular(epimorphic) subfactors
(729:728)*(819:818)*(409:408)*(153:152)*(513:512)
Werckmeister called such parts in hollandaise tongue
"snipsel" as already J.A.Ban labeled them before him.

Control proof, by multiplying out the 5 "snipsels":
(3^6:(91*2^3))*
(91*3^2:(409*2))*
(409:(51*2^3)*
(51*3:(19*2^3)*
(19*3^3:2^9)
=3^12:2^19 q.e.d.

All prime factors herein, if not equal to powers of 2 or 3
do cancel each other by shortening 
the nominator versus the denominator,
so that only the bare PC remains alone,
leaving over the remaining dozen powers of 3 
over the 19 powers of 2.

That's presumably again only an other version of
the old "snipsel" trick, likely used already
by the gothic master Arnold Schlick 
in his circular tuning about 1511
now alomst half a millenium ago,
may be invented by
Hugo Spechtshart (ca. 1285-ca. 1359) 
http://vitrine.library.uu.nl/wwwroot/nl/teksten/Fqu348rar.htm
or even earlier?
Who knows here more about the history of that 
antediluvian method of subdividing the PC epimorphic?

From: George D. Secor (2005-12-19)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...> 
wrote:
> [To Gene:]
> 
> BTW, well1 turned out to be pretty nice, while well2 (with D:A 
> tempered over 10 cents) is junk (an errant guess on my part).  But 
I 
> think well1 could be improved by making F:C just and B:F# tempered 
> narrow by a small amount (or perhaps both fifths tempered by a 
small 
> amount) -- back to the drawing board.

Gene, here's another set of major-triad brats to try rationalizing:

F: 2
C: 4
G: unassigned (~16/9)
D: 2.75
A: 13/6
E, B: 5/3
F#, C#, G#, Eb, Bb, F: 1.5

I know that these numbers don't look very impressive, so if you're 
feeling ambitious, then you're welcome to tinker with some of the 
brats.

In arriving at the above numbers, I had a couple more objectives in 
mind, besides major brats:

1) To get some reasonable minor brats; in addition to the six exact 
1.0 brats, the above yields ones approximating 1.33, 1.5, 3.5, 1.75, 
1.375, and 1.2;

2) Carl's brat test seemed to indicate that the tempering of the 
fifths are a far more important factor than proportionality in the 
acceptability of the result, so I made an effort to get the maximum 
mileage from these by making the fifths with the greatest error 
consecutive.  The above brats should result in fifths between 5 and 
5.5 cents above G, D, and A, which would also make the C major triad 
the one with the lowest total absolute error.

--George

From: George D. Secor (2005-12-23)
Subject: Re: Project for Gene?

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...> 
wrote:
> 
> Gene, here's another set of major-triad brats to try rationalizing:
> 
> F: 2
> C: 4
> G: unassigned (~16/9)
> D: 2.75
> A: 13/6
> E, B: 5/3
> F#, C#, G#, Eb, Bb, F: 1.5

Oops!  I just noticed that F is there twice -- it's supposed to be 
1.5.

> I know that these numbers don't look very impressive, so if you're 
feeling 
> ambitious, then you're welcome to tinker with some of the brats.

I tried some more tinkering and found that the following looks like a 
promising alternative to well1 in message #13671:

C: 4
G: 3
D: 2.25
A: 2
E: 1.75
B: unassigned (~5/3)
F#, C#, G#, Eb, Bb, F: 1.5

You can also try E or D unassigned, which may make a difference in 
the minor brats.

Here's a different one, with brats of 4 for both C and G:

C, G: 4
D: 2.5
A: 2
E: unassigned (~23/12)
B, F#, C#, G#, Eb, Bb, F: 1.5

I'd also like to see if it would be more desirable to make the E brat 
23/12 and let B be unassigned.

IMO, these would be the ideal brats:

C, G: 4
D: unassigned (and hopefully ~7/3 or 2.5)
A, E: 2
B, F#, C#, G#, Eb, Bb, F: 1.5

This results, however, in the narrowest fifths being C:G and E:B, 
which lowers the key contrast too much.  I've attempted to remedy 
that by making the E brat ~1.9.

This is another one to try:

F: 5/3
C, G: 4
D: 9/3
A: 2
E: unassigned (I expect ~11/6)
B, F#, C#, G#, Eb, Bb: 1.5

I'm thinking that there could also be a couple of variations on this 
last one, setting E to 11/6 and making either D or F unassigned.  
Another possibility is to set E to 13/7 with F unassigned, which I 
find will give exact 2.6 and 7/3 brats on the A and E minor triads, 
respectively.

Still another possibility is this (hopefully the last one I'll be 
attempting):

F: unassigned (~1.6)
C, G: 4
D: 2.5
A: 2
E: 13/7
B, F#, C#, G#, Eb, Bb: 1.5

I'd also like to see a variation of this last one, setting F to 1.6 
and making E unassigned.

Gene, I hope I'm not "wearing out my welcome" with all of these 
variations on a theme, but I'm finding it a bit ironic that the 
constraint of wanting at least 6 consecutive just fifths in a well-
temperament (which should cut down the number of possible solutions) 
hasn't made it easier to home in on a couple of "preferred" 
solutions. Anyway, I expect to be away from the Internet for several 
days, so please take as much time as you need to respond.

Wishing you a Joyous Christmas,

--George