73 linear {2, 3, 7, 11} temperaments

I simply used every pair of standard vals up to 199&200, and took
those with an rms badness less than 500. I didn't prune out the very
complex or very inaccurate temperaments uncovered in this way. Here's
the list, in order of an increasing badness figure:

[18, 18, 9, -22, -48, -37] [[9, 14, 25, 31], [0, 2, 2, 1]]
rms: .225560 com: 18.973666 bad: 81.201736

[57, 12, -39, -141, -259, -151] [[3, -2, 7, 15], [0, 19, 4, -13]]
rms: .014655 com: 85.352211 bad: 106.762432

[11, -19, -17, -61, -65, 18] [[1, -2, 9, 9], [0, 11, -19, -17]]
rms: .200923 com: 26.130017 bad: 137.185477

[28, 13, -31, -58, -146, -132] [[1, 2, 3, 3], [0, -28, -13, 31]]
rms: .065296 com: 47.269911 bad: 145.900853

[1, -14, 23, -25, 33, 113] [[1, 2, -3, 13], [0, -1, 14, -23]]
rms: .374691 com: 19.734347 bad: 145.921462

[29, -1, -8, -83, -113, -19] [[1, -4, 3, 5], [0, 29, -1, -8]]
rms: .107705 com: 40.034707 bad: 172.627042

[7, 37, 26, 39, 17, -55] [[1, 2, 5, 5], [0, -7, -37, -26]]
rms: .389295 com: 21.972205 bad: 187.942992

[10, -5, -40, -36, -98, -95] [[5, 8, 14, 17], [0, -2, 1, 8]]
rms: .169192 com: 33.829639 bad: 193.631312

[2, 13, 5, 15, 1, -31] [[1, 1, -1, 2], [0, 2, 13, 5]]
rms: 3.645253 com: 7.378648 bad: 198.463761

[101, -76, 93, -404, -202, 524] [[1, 0, 4, 2], [0, 101, -76, 93]]
rms: .008459 com: 154.398115 bad: 201.659060

[4, 9, 10, 3, 2, -3] [[1, 3, 6, 7], [0, -4, -9, -10]]
rms: 5.583620 com: 6.036923 bad: 203.491919

[1, -2, -6, -6, -13, -10] [[1, 2, 2, 1], [0, -1, 2, 6]]
rms: 9.265724 com: 4.702245 bad: 204.875452

[6, -2, 15, -20, 3, 49] [[1, 1, 3, 2], [0, 6, -2, 15]]
rms: 1.882195 com: 10.790943 bad: 219.171183

[19, 4, 32, -47, -15, 76] [[1, -7, 1, -11], [0, 19, 4, 32]]
rms: .388051 com: 24.071652 bad: 224.853977

[2, 1, -2, -4, -10, -9] [[1, 2, 3, 3], [0, -2, -1, 2]]
rms: 21.273448 com: 3.265986 bad: 226.916781

[17, 32, -14, 3, -81, -150] [[1, 7, 13, -1], [0, -17, -32, 14]]
rms: .252731 com: 30.322342 bad: 232.371891

[1, -2, -1, -6, -5, 4] [[1, 2, 2, 3], [0, -1, 2, 1]]
rms: 43.753630 com: 2.333333 bad: 238.214210

[1, 3, 4, 2, 3, 1] [[1, 2, 4, 5], [0, -1, -3, -4]]
rms: 44.308075 com: 2.333333 bad: 241.232855

[0, 2, 0, 3, 0, -7] [[2, 3, 6, 7], [0, 0, -1, 0]]
rms: 140.888692 com: 1.333333 bad: 250.468786

[8, 23, 49, 14, 50, 58] [[1, 2, 4, 6], [0, -8, -23, -49]]
rms: .345838 com: 27.828842 bad: 267.832611

[6, 22, 15, 18, 3, -34] [[1, 3, 8, 7], [0, -6, -22, -15]]
rms: 1.783391 com: 12.400717 bad: 274.245975

[5, 7, 4, -3, -11, -13] [[1, 1, 2, 3], [0, 5, 7, 4]]
rms: 9.958945 com: 5.259911 bad: 275.530818

[39, -6, -48, -119, -211, -114] [[3, 1, 9, 15], [0, 13, -2, -16]]
rms: .056072 com: 70.107061 bad: 275.593363

[0, 0, 5, 0, 8, 14] [[5, 8, 14, 17], [0, 0, 0, 1]]
rms: 25.029469 com: 3.333333 bad: 278.105209

[2, -4, 5, -12, 1, 28] [[1, 1, 4, 2], [0, 2, -4, 5]]
rms: 10.740871 com: 5.291503 bad: 300.744374

[46, 31, -22, -80, -194, -169] [[1, -10, -5, 9], [0, 46, 31, -22]]
rms: .075288 com: 63.595947 bad: 304.497354

[1, 15, 11, 21, 14, -21] [[1, 2, 9, 8], [0, -1, -15, -11]]
rms: 3.010648 com: 10.071963 bad: 305.413560

[2, 1, 3, -4, -2, 5] [[1, 2, 3, 4], [0, -2, -1, -3]]
rms: 60.512514 com: 2.260777 bad: 309.286184

[5, 12, -8, 5, -30, -64] [[1, 0, -1, 6], [0, 5, 12, -8]]
rms: 2.221125 com: 11.921036 bad: 315.646596

[12, -33, 6, -86, -32, 131] [[3, 6, 5, 11], [0, -4, 11, -2]]
rms: .362928 com: 31.622777 bad: 362.928149

[3, -6, -1, -18, -12, 18] [[1, 3, 0, 3], [0, -3, 6, 1]]
rms: 8.431348 com: 6.641620 bad: 371.916139

[3, -1, -13, -10, -31, -33] [[1, 1, 3, 6], [0, 3, -1, -13]]
rms: 3.264106 com: 10.754844 bad: 377.548255

[7, 8, 2, -7, -21, -22] [[1, 0, 1, 3], [0, 7, 8, 2]]
rms: 6.153090 com: 7.852813 bad: 379.440557

[1, 5, -1, 5, -5, -20] [[1, 2, 5, 3], [0, -1, -5, 1]]
rms: 32.910659 com: 3.415650 bad: 383.957688

[47, 17, 1, -105, -161, -56] [[1, -20, -5, 3], [0, 47, 17, 1]]
rms: .119432 com: 56.945393 bad: 387.292474

[25, 55, 35, 17, -31, -92] [[5, 7, 12, 16], [0, 5, 11, 7]]
rms: .393660 com: 31.666667 bad: 394.753883

[6, 51, 3, 64, -16, -168] [[3, 4, 2, 10], [0, 2, 17, 1]]
rms: .409998 com: 31.144823 bad: 397.697590

[4, 2, 10, -8, 2, 21] [[2, 4, 6, 9], [0, -2, -1, -5]]
rms: 11.028354 com: 6.036923 bad: 401.922222

[13, 6, 17, -27, -18, 27] [[1, 2, 3, 4], [0, -13, -6, -17]]
rms: 1.977863 com: 14.286746 bad: 403.703765

[3, -1, -8, -10, -23, -19] [[1, 1, 3, 5], [0, 3, -1, -8]]
rms: 6.779348 com: 7.753136 bad: 407.514132

[4, 2, 5, -8, -6, 7] [[1, 2, 3, 4], [0, -4, -2, -5]]
rms: 21.953179 com: 4.320494 bad: 409.792675

[1, -4, -1, -9, -5, 11] [[1, 2, 1, 3], [0, -1, 4, 1]]
rms: 35.178471 com: 3.415650 bad: 410.415498

[12, 20, -6, -2, -51, -86] [[2, 1, 2, 8], [0, 6, 10, -3]]
rms: 1.192658 com: 18.666667 bad: 415.575097

[4, -15, 10, -35, 2, 80] [[1, 1, 5, 2], [0, 4, -15, 10]]
rms: 1.918009 com: 14.757296 bad: 417.699820

[8, 11, 20, -5, 4, 18] [[1, 1, 2, 2], [0, 8, 11, 20]]
rms: 3.444016 com: 11.135529 bad: 427.058020

[2, -11, 5, -23, 1, 52] [[1, 1, 6, 2], [0, 2, -11, 5]]
rms: 4.672463 com: 9.580072 bad: 428.828255

[2, 1, -7, -4, -18, -23] [[1, 2, 3, 2], [0, -2, -1, 7]]
rms: 11.244156 com: 6.200358 bad: 432.275338

[3, 6, 6, 1, -1, -4] [[3, 5, 9, 11], [0, -1, -2, -2]]
rms: 29.220844 com: 3.872983 bad: 438.312654

[38, 114, 76, 74, -11, -181] [[38, 60, 106, 131], [0, 1, 3, 2]]
rms: .110277 com: 63.333333 bad: 442.334481

[6, 10, 8, -1, -8, -12] [[2, 4, 7, 8], [0, -3, -5, -4]]
rms: 10.430068 com: 6.565905 bad: 449.651809

[7, -16, 38, -45, 36, 162] [[1, 5, -5, 22], [0, -7, 16, -38]]
rms: .517949 com: 29.490111 bad: 450.443049

[0, 7, 0, 11, 0, -24] [[7, 11, 20, 24], [0, 0, -1, 0]]
rms: 21.068470 com: 4.666667 bad: 458.824450

[26, 41, 58, -8, 2, 21] [[1, -4, -6, -9], [0, 26, 41, 58]]
rms: .410114 com: 33.532737 bad: 461.150400

[1, 0, -1, -3, -5, -3] [[1, 2, 3, 3], [0, -1, 0, 1]]
rms: 166.476077 com: 1.666667 bad: 462.433547

[21, -24, -57, -97, -163, -77] [[3, 6, 7, 7], [0, -7, 8, 19]]
rms: .142148 com: 57.140179 bad: 464.114220

[35, 50, -5, -19, -129, -187] [[5, 10, 17, 17], [0, -7, -10, 1]]
rms: .214064 com: 46.577295 bad: 464.399473

[24, 16, 24, -42, -45, 12] [[8, 12, 22, 27], [0, 3, 2, 3]]
rms: .778189 com: 24.585452 bad: 470.372270

[12, 20, 30, -2, 6, 15] [[2, 4, 7, 9], [0, -6, -10, -15]]
rms: 1.657538 com: 16.865481 bad: 471.477545

[3, 11, -1, 9, -12, -41] [[1, 3, 8, 3], [0, -3, -11, 1]]
rms: 9.148291 com: 7.187953 bad: 472.661687

[5, 24, 21, 24, 16, -24] [[1, 1, 0, 1], [0, 5, 24, 21]]
rms: 2.052205 com: 15.220600 bad: 475.427447

[12, -4, -6, -40, -51, -3] [[2, 5, 5, 6], [0, -6, 2, 3]]
rms: 1.410534 com: 18.378732 bad: 476.447050

[31, 77, 50, 35, -28, -126] [[1, -8, -21, -12], [0, 31, 77, 50]]
rms: .256216 com: 43.216509 bad: 478.525647

[1, 1, -1, -1, -5, -6] [[1, 2, 3, 3], [0, -1, -1, 1]]
rms: 172.348355 com: 1.666667 bad: 478.745430

[37, 22, 41, -69, -63, 39] [[1, 12, 9, 15], [0, -37, -22, -41]]
rms: .324114 com: 38.593321 bad: 482.750351

[6, 3, 3, -12, -16, -2] [[3, 4, 8, 10], [0, 2, 1, 1]]
rms: 11.506907 com: 6.480741 bad: 483.290111

[10, 7, 25, -17, 5, 46] [[1, -1, 1, -3], [0, 10, 7, 25]]
rms: 2.260804 com: 14.628739 bad: 483.812140

[4, 14, -2, 11, -17, -54] [[2, 3, 5, 7], [0, 2, 7, -1]]
rms: 5.409385 com: 9.475114 bad: 485.642585

[24, 40, 24, -4, -45, -71] [[8, 13, 23, 28], [0, -3, -5, -3]]
rms: .704924 com: 26.263621 bad: 486.240651

[9, -3, 7, -30, -20, 30] [[1, 1, 3, 3], [0, 9, -3, 7]]
rms: 3.447762 com: 11.893042 bad: 487.666730

[25, 2, 47, -67, -12, 125] [[1, 4, 3, 8], [0, -25, -2, -47]]
rms: .411475 com: 34.462540 bad: 488.695542

[30, -15, 15, -108, -80, 94] [[15, 24, 42, 52], [0, -2, 1, -1]]
rms: .279807 com: 41.833001 bad: 489.661749

[13, 59, 41, 57, 20, -89] [[1, -3, -18, -11], [0, 13, 59, 41]]
rms: .418301 com: 34.229617 bad: 490.109290

[46, 84, 137, 4, 58, 94] [[1, -12, -22, -37], [0, 46, 84, 137]]
rms: .088004 com: 75.297927 bad: 498.963279

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

> [18, 18, 9, -22, -48, -37] [[9, 14, 25, 31], [0, 2, 2, 1]]
> rms: .225560 com: 18.973666 bad: 81.201736

Commas 41503/41472 and 43923/43904, with a {2,3,7} comma
2^(-11) 3^(-9) 7^9.

> [1, -14, 23, -25, 33, 113] [[1, 2, -3, 13], [0, -1, 14, -23]]
> rms: .374691 com: 19.734347 bad: 145.921462

Commas 19712/19683 and 41503/41472, with a {2,3,7} comma of
2^25 3^(-14) 7^(-1), fifth generator

> [2, 13, 5, 15, 1, -31] [[1, 1, -1, 2], [0, 2, 13, 5]]
> rms: 3.645253 com: 7.378648 bad: 198.463761

Commas 243/242 and 896/891, with a {2,3,7} comma of
2^15 3^(-13) 7^2, with half-fifth generator.

Well worth considering.

> [4, 9, 10, 3, 2, -3] [[1, 3, 6, 7], [0, -4, -9, -10]]
> rms: 5.583620 com: 6.036923 bad: 203.491919

Barton's system, with commas 99/98 and 243/242, and a {2,3,7} comma
of 19683/19208

> [1, -2, -6, -6, -13, -10] [[1, 2, 2, 1], [0, -1, 2, 6]]
> rms: 9.265724 com: 4.702245 bad: 204.875452

Commas of 64/63 and 99/98, with 64/63 the {2,3,7} comma. Sharp fifth
generator.

> [6, -2, 15, -20, 3, 49] [[1, 1, 3, 2], [0, 6, -2, 15]]
> rms: 1.882195 com: 10.790943 bad: 219.171183

Commas 243/242 and 1029/1024, the latter a {2,3,7} comma, secor
generator.

Thanks for the great work Gene; hope you don't mind some
questions/requests:

All of these had an octave period? Can you give more information on
the systems in your original list? And how about trying some of the
more recent damage and complexity measures too? You know George isn't
an rms kind of guy, and has also expressed support for the idea that
the simplest ratios should be weighted a little more heavily in the
optimization . . .

-Paul

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > [18, 18, 9, -22, -48, -37] [[9, 14, 25, 31], [0, 2, 2, 1]]
> > rms: .225560 com: 18.973666 bad: 81.201736
>
> Commas 41503/41472 and 43923/43904, with a {2,3,7} comma
> 2^(-11) 3^(-9) 7^9.
>
> > [1, -14, 23, -25, 33, 113] [[1, 2, -3, 13], [0, -1, 14, -23]]
> > rms: .374691 com: 19.734347 bad: 145.921462
>
> Commas 19712/19683 and 41503/41472, with a {2,3,7} comma of
> 2^25 3^(-14) 7^(-1), fifth generator
>
> > [2, 13, 5, 15, 1, -31] [[1, 1, -1, 2], [0, 2, 13, 5]]
> > rms: 3.645253 com: 7.378648 bad: 198.463761
>
> Commas 243/242 and 896/891, with a {2,3,7} comma of
> 2^15 3^(-13) 7^2, with half-fifth generator.
>
> Well worth considering.
>
> > [4, 9, 10, 3, 2, -3] [[1, 3, 6, 7], [0, -4, -9, -10]]
> > rms: 5.583620 com: 6.036923 bad: 203.491919
>
> Barton's system, with commas 99/98 and 243/242, and a {2,3,7} comma
> of 19683/19208
>
> > [1, -2, -6, -6, -13, -10] [[1, 2, 2, 1], [0, -1, 2, 6]]
> > rms: 9.265724 com: 4.702245 bad: 204.875452
>
> Commas of 64/63 and 99/98, with 64/63 the {2,3,7} comma. Sharp fifth
> generator.
>
>
> > [6, -2, 15, -20, 3, 49] [[1, 1, 3, 2], [0, 6, -2, 15]]
> > rms: 1.882195 com: 10.790943 bad: 219.171183
>
> Commas 243/242 and 1029/1024, the latter a {2,3,7} comma, secor
> generator.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> Thanks for the great work Gene; hope you don't mind some
> questions/requests:
>
> All of these had an octave period?

No, in fact the first one on the list (no fives hemiennealimmal) has a
1/9 octave period, and other nonoctave periods are present as well.

Can you give more information on
> the systems in your original list?

What information would be valuable?

And how about trying some of the
> more recent damage and complexity measures too? You know George isn't
> an rms kind of guy, and has also expressed support for the idea that
> the simplest ratios should be weighted a little more heavily in the
> optimization . . .

I'd have to write more code for that, so I need to know if it is
really worth it. I've got all these Maple functions which call other
Maple functions which call other Maple functions through layers of
recursiveness to sort through, since I didn't write code in the first
place which was generic.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> >
> > Thanks for the great work Gene; hope you don't mind some
> > questions/requests:
> >
> > All of these had an octave period?
>
> No, in fact the first one on the list (no fives hemiennealimmal)
has a
> 1/9 octave period, and other nonoctave periods are present as well.
>
> Can you give more information on
> > the systems in your original list?
>
> What information would be valuable?

Well, period and generator (for the octave-repeating case), for a
start.

> And how about trying some of the
> > more recent damage and complexity measures too? You know George
isn't
> > an rms kind of guy, and has also expressed support for the idea
that
> > the simplest ratios should be weighted a little more heavily in
the
> > optimization . . .
>
> I'd have to write more code for that, so I need to know if it is
> really worth it.

Really? I thought your program was already spitting out TOP
generators and L1-Tenney complexity measures . . . at least some of
your posts here made it seem that way . . . (?) I was trying to
suggest (though George should chime in) why TOP might be just right
for George above. And if anything, using L1-Tenney complexity would
favor systems with an octave period and fifth generator even more
than 'symmetrical' set-of-odd-numbers-based complexity measures, so
finding good systems under L1-Tenney complexity that don't follow
this octave&fifth pattern should serve as even stronger evidence
that George is missing out when he says "most of the useful systems
beyond the 7-limit are octave&fifth generated systems" or words to
that effect.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
.
>
> Really? I thought your program was already spitting out TOP
> generators and L1-Tenney complexity measures . . .

Yes, but not in the context of assuming 5 is not a number.

at least some of
> your posts here made it seem that way . . . (?) I was trying to
> suggest (though George should chime in) why TOP might be just right
> for George above. And if anything, using L1-Tenney complexity would
> favor systems with an octave period and fifth generator even more
> than 'symmetrical' set-of-odd-numbers-based complexity measures, so
> finding good systems under L1-Tenney complexity that don't follow
> this octave&fifth pattern should serve as even stronger evidence
> that George is missing out when he says "most of the useful systems
> beyond the 7-limit are octave&fifth generated systems" or words to
> that effect.

It seems to me this is backwards, actually--as complexity and the
number of primes being considered increase, the tendency for systems
to have either octaves as periods or fifths as generators goes down.
What you regard as useful is of course an interestign question--are
microtemperaments like hemiennealimmal or octoid useful? Well, they ar
e the sort of thing *I* use...but I don't use a keyboard other than in
a crude 1-finger way when playing with melody lines.

I'd say miracle is clearly useful, and unidec, minorsemi, wizard,
hemiwuerschmidt, etc etc through a whole raft of non-fifth generators
are reasonable systems. Most of the systems you find in 11-limit and
beyond are not fifth-generated.

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

> It seems to me this is backwards, actually--as complexity and the
> number of primes being considered increase, the tendency for systems
> to have either octaves as periods or fifths as generators goes down.

Exactly. It seems we need to convince George of this.

> Most of the systems you find in 11-limit and
> beyond are not fifth-generated.

And of this.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > Most of the systems you find in 11-limit and
> > beyond are not fifth-generated.
>
> And of this.

A while back I proved that if you set a bound on log-flat badness for
any particular generator, you end up with a finite list in a given
prime limit. This means that of the infinite number of, for example,
11-limit
linear temperaments below a certain log-flat badness cutoff, only
finitely many will be "bridgable", having an octave period and a fifth
generator. With increasing complexity and decreasing error, in other
words, fifths as generators become ever more rare, and finally at some
level of microtempering we don't find them at all among the best
systems. You get up to pontiac, in other words, and choke. This
theoretical fact is of course offset by the practical fact that we are
not normally concerned with extreme mictrotempering, and to start out
with temperaments with a consonance as a generator have an advantage.