38 linear 7-limit temperaments compatible with 12-et

Here is raw material for exotic 12-tone tunings and multiple keyboard
experiments, not to mention a lot of old friends.

I think I've seen this one before:
[1, 4, 10, 12, -13, 4] [[1, 0, -4, -13], [0, 1, 4, 10]]

comp 5.322447240 rms 3.665035228 bad 103.8247475

Here's a microtemperament for multi-keyboard enthusiasts:
[3, -24, -54, -58, 94, -45] [[3, 0, 45, 94], [0, 1, -8, -18]]

comp 31.51783075 rms .1469057415 bad 145.9322934

[2, -4, -4, 2, 12, -11] [[2, 0, 11, 12], [0, 1, -2, -2]]

comp 3.938677761 rms 10.90317755 bad 169.1429833

[3, 0, -6, -14, 18, -7] [[3, 0, 7, 18], [0, 1, 0, -2]]

comp 4.631825456 rms 8.100678834 bad 173.7904007

[4, 4, 4, -2, 5, -3] [[4, 0, 3, 5], [0, 1, 1, 1]]

comp 3.144366918 rms 19.13699259 bad 189.2082747

[1, 4, -2, -16, 6, 4] [[1, 0, -4, 6], [0, 1, 4, -2]]

comp 3.128478105 rms 20.16328150 bad 197.3456024

It worked for Helmholtz
[1, -8, -14, -10, 25, -15] [[1, 0, 15, 25], [0, 1, -8, -14]]

comp 8.612526914 rms 2.859336356 bad 212.0930465

[3, 0, 6, 14, -1, -7] [[3, 0, 7, -1], [0, 1, 0, 2]]

comp 3.675273386 rms 16.59867843 bad 224.2088808

This has possibilities
[0, 12, 24, 22, -38, 19] [[12, 19, 0, -22], [0, 0, 1, 2]]

comp 13.76571634 rms 1.496892545 bad 283.6535726

Another interesting one
[3, -12, -30, -36, 56, -26] [[3, 0, 26, 56], [0, 1, -4, -10]]

comp 17.83027719 rms .8942129314 bad 284.2870884

[2, 8, 8, -4, -7, 8] [[2, 0, -8, -7], [0, 1, 4, 4]]

comp 5.343650829 rms 11.21894132 bad 320.3524287

[2, -4, -16, -26, 31, -11] [[2, 0, 11, 31], [0, 1, -2, -8]]

comp 9.469818377 rms 3.821630536 bad 342.7141199

[4, -8, -20, -24, 43, -22] [[4, 0, 22, 43], [0, 1, -2, -5]]

comp 12.75555760 rms 2.220377240 bad 361.2648129

[5, -4, -10, -12, 30, -18] [[1, 2, 2, 2], [0, 5, -4, -10]]

comp 8.009157500 rms 6.041345016 bad 387.5317655

[6, 0, 0, 0, 17, -14] [[6, 0, 14, 17], [0, 1, 0, 0]]

comp 4.716550378 rms 18.04292374 bad 401.3801294

[0, 0, 12, 28, -19, 0] [[12, 19, 28, 0], [0, 0, 0, 1]]

comp 6.904855942 rms 9.840803062 bad 469.1803177

[4, -20, -44, -46, 81, -41] [[4, 0, 41, 81], [0, 1, -5, -11]]

comp 26.35358322 rms .6908190406 bad 479.7816635

[2, -16, -40, -48, 69, -30] [[2, 0, 30, 69], [0, 1, -8, -20]]

comp 23.01717204 rms .9641797248 bad 510.8129776

[5, 8, 2, -18, 11, 1] [[1, 2, 3, 3], [0, 5, 8, 2]]

comp 5.083424305 rms 21.64417648 bad 559.3115508

[1, -8, -2, 18, 6, -15] [[1, 0, 15, 6], [0, 1, -8, -2]]

comp 5.398824730 rms 19.66911204 bad 573.3016760

[3, 12, 18, 8, -20, 12] [[3, 0, -12, -20], [0, 1, 4, 6]]

comp 10.19699576 rms 5.782918708 bad 601.3004996

[7, 4, 10, 12, 4, -10] [[1, 1, 2, 2], [0, 7, 4, 10]]

comp 6.082925309 rms 16.44388527 bad 608.4563190

[2, -4, 8, 30, -7, -11] [[2, 0, 11, -7], [0, 1, -2, 4]]

comp 6.058298120 rms 18.06996660 bad 663.2215524

[1, -8, -26, -38, 44, -15] [[1, 0, 15, 44], [0, 1, -8, -26]]

comp 14.64779855 rms 3.106171476 bad 666.4539470

[0, 12, 12, -6, -19, 19] [[12, 19, 0, 6], [0, 0, 1, 1]]

comp 8.845819922 rms 10.15948550 bad 794.9648068

[5, -16, -34, -34, 68, -37] [[1, 2, 1, 0], [0, 5, -16, -34]]

comp 21.26748532 rms 1.787147240 bad 808.3372977

[6, 12, 12, -6, -2, 5] [[6, 0, -5, -2], [0, 1, 2, 2]]

comp 7.833907871 rms 13.63960404 bad 837.0640348

[9, 0, -6, -14, 35, -21] [[3, 1, 7, 11], [0, 3, 0, -2]]

comp 8.596026672 rms 11.94435731 bad 882.5885631

[8, -4, -4, 2, 29, -25] [[4, 1, 12, 14], [0, 2, -1, -1]]

comp 8.101297218 rms 14.29026872 bad 937.8848637

[6, -12, -24, -22, 55, -33] [[6, 0, 33, 55], [0, 1, -2, -4]]

comp 16.33270964 rms 3.600727660 bad 960.5207638

[7, -8, -14, -10, 42, -29] [[1, 1, 3, 4], [0, 7, -8, -14]]

comp 11.74162535 rms 7.012328960 bad 966.7601028

[3, 0, -18, -42, 37, -7] [[3, 0, 7, 37], [0, 1, 0, -6]]

comp 11.16933529 rms 8.439018022 bad 1052.801683

[9, 0, 6, 14, 16, -21] [[3, 1, 7, 6], [0, 3, 0, 2]]

comp 7.074825566 rms 21.62618964 bad 1082.459061

[6, -12, -36, -50, 74, -33] [[6, 0, 33, 74], [0, 1, -2, -6]]

comp 22.11900171 rms 2.367078438 bad 1158.093686

[9, 12, 6, -20, 16, -2] [[3, 1, 2, 6], [0, 3, 4, 2]]

comp 7.852968657 rms 21.26148578 bad 1311.177048

[10, 4, 16, 26, 3, -17] [[2, 4, 5, 7], [0, 5, 2, 8]]

comp 9.546317939 rms 16.25734866 bad 1481.567725

[1, -8, 10, 46, -13, -15] [[1, 0, 15, -13], [0, 1, -8, 10]]

comp 8.914766865 rms 18.78088561 bad 1492.574604

[2, -16, -16, 8, 31, -30] [[2, 0, 30, 31], [0, 1, -8, -8]]

comp 12.60828383 rms 10.10704662 bad 1606.705286

--- In tuning-math@y..., Gene W Smith <genewardsmith@j...> wrote:

> Here's a microtemperament for multi-keyboard enthusiasts:
> [3, -24, -54, -58, 94, -45] [[3, 0, 45, 94], [0, 1, -8, -18]]
>
> comp 31.51783075 rms .1469057415 bad 145.9322934

Let's see how this one might work. We can change the mapping given above to the equivalent

[12 3]
[19 5]
[28 5]
[34 4]

The generators for this are well approximated by 14/171 and 1/171, and we can use this as a keyboard system for the 171-et, with the 12-note keyboards tuned to semitones of size 14/171 ocatves, and the keyboards separated by 1/171. This non-ocatave tuning (12*(14/171) = 168/171, a comma less than the octave) can be modified to one which tunes each rank of 12-note keyboards slightly unevenly, to the 14/171 MOS, namely [14 14 14 15] repeated three times. Of course this gives four different patterns for chords, but you pays your money and you makes your choice.