Yahoo Tuning Groups Ultimate Backup tuning-math Some 11-limit bases and associated temperament bases

Here are 11-limit bases for 22,31,41,46,58, and 72. Associated to these are four linear temperaments, which can also be regarded as forming a basis for the et; these are obtained by leaving out one of the commas. These are by no means necessarily the four best linear temperaments for that et, but they are interesting and do define it.

In the case of the 22-et, we get two versions of pajara; these are the same for 22, but the [2,-4,-4,-12,-11,-12,-26,2,-14,-20] version is consistent with h10-v11, h12, h34+v7, while the
[2,-4,-4,10,-11,-12,9,2,37,42] version is consistent with h10,
h12-v11, and h34+v7-v11. The temperament consisting of two chains of 9/7 a half-ocatave apart still looks good for the 22-et, and I wonder if Paul has ever had occasion to try it. A scale of 8 or 14 tones suggests itself for this.

The temperament with a compromise 12/11 or 11/10 generator in the 31-et seems to go with the discussion of Arabic, though it is quite different, and the 11/7 generator system looks worth exploring.

The 333333333333334 MOS of the 46-et suggested by the temperament with a 77.963 cent generator looks interesting, if a little melodically bland.

The temperament consisting of two chains of major thirds a half-octave apart seems like an interesting 72-et alternative.

22: [50/49, 55/54, 64/63, 99/98]

[99/98, 64/63, 55/54]

wedgie [1, 9, -2, -6, 12, -6, -13, -30, -45, -10]

map [[0, -1, -9, 2, 6], [1, 2, 6, 2, 1]]

generators [490.8051577, 1200.]

bad 344.0525564 rms 12.62392639 g 7.265377779

[99/98, 64/63, 50/49]

wedgie [2, -4, -4, -12, -11, -12, -26, 2, -14, -20]

map [[0, -1, 2, 2, 6], [2, 4, 3, 4, 2]]

generators [492.8941324, 600.]

bad 328.2749546 rms 9.552922731 g 8.349508111

[99/98, 55/54, 50/49]

wedgie [6, 10, 10, 8, 2, -1, -8, -5, -16, -12]

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

generators [434.9412914, 600.]

bad 238.7261365 rms 11.89273381 g 6.047431569

[64/63, 55/54, 50/49]

wedgie [2, -4, -4, 10, -11, -12, 9, 2, 37, 42]

map [[0, -1, 2, 2, -5], [2, 4, 3, 4, 11]]

generators [490.1172150, 600.]

bad 373.4699058 rms 12.26714783 g 7.764387569

31: [81/80, 99/98, 121/120, 126/125]

[126/125, 121/120, 99/98]

wedgie [11, 13, 17, 12, -5, -4, -19, 3, -17, -25]

map [[0, -11, -13, -17, -12], [1, 3, 4, 5, 5]]

generators [154.5139994, 1200.]

bad 299.8803381 rms 6.113337322 g 10.33717287

[126/125, 121/120, 81/80]

wedgie [2, 8, 20, 5, 8, 26, 1, 24, -16, -55]

map [[0, 2, 8, 20, 5], [1, 1, 0, -3, 2]]

generators [348.1847713, 1200.]

bad 331.9892755 rms 6.645961894 g 10.45056389

[126/125, 99/98, 81/80]

wedgie [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]

map [[0, -1, -4, -10, -18], [1, 2, 4, 7, 11]]

generators [502.9994276, 1200.]

bad 313.6023849 rms 6.584357338 g 10.15592719

[121/120, 99/98, 81/80]

wedgie [4, 16, 9, 10, 16, 3, 2, -24, -32, -3]

map [[0, -4, -16, -9, -10], [1, 3, 8, 6, 7]]

generators [425.8501579, 1200.]

bad 218.9540099 rms 6.965622568 g 7.914724072

41: [100/99, 225/224, 243/242, 385/384]

[385/384, 243/242, 225/224]

wedgie [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]

map [[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]]

generators [116.6722644, 1200.]

bad 125.5016755 rms 1.901465778 g 12.35198075

[385/384, 243/242, 100/99]

wedgie [4, 9, -15, 10, 5, -35, 2, -60, -8, 80]

map [[0, 4, 9, -15, 10], [1, 1, 1, 5, 2]]

generators [175.4823391, 1200.]

bad 394.7928774 rms 5.007159926 g 13.74253044

[385/384, 225/224, 100/99]

wedgie [5, 1, 12, -8, -10, 5, -30, 25, -22, -64]

map [[0, 5, 1, 12, -8], [1, 0, 2, -1, 6]]

generators [380.7138125, 1200.]

bad 242.7224832 rms 4.730404304 g 10.62006188

[243/242, 225/224, 100/99]

wedgie [4, 9, 26, 10, 5, 30, 2, 35, -8, -62]

map [[0, 4, 9, 26, 10], [1, 1, 1, -1, 2]]

generators [175.7323939, 1200.]

bad 362.5442426 rms 5.060204371 g 12.97525116

46: [121/120, 126/125, 176/175, 245/243]

[176/175, 126/125, 245/243]

wedgie [7, 9, 13, 31, -2, 1, 25, 5, 41, 42]

map [[0, 7, 9, 13, 31], [1, -1, -1, -2, -8]]

generators [443.6497958, 1200.]

bad 461.2544269 rms 4.975617526 g 15.14454169

[176/175, 126/125, 121/120]

wedgie [9, 5, -3, 7, -13, -30, -20, -21, -1, 30]

map [[0, 9, 5, -3, 7], [1, 1, 2, 3, 3]]

generators [77.93200627, 1200.]

bad 223.3668950 rms 4.418576095 g 10.52547929

[176/175, 245/243, 121/120]

wedgie [4, -8, 14, -2, -22, 11, -17, 55, 23, -54]

map [[0, -2, 4, -7, 1], [2, 5, 1, 12, 6]]

generators [547.4864711, 600.]

bad 300.7076870 rms 5.273952598 g 11.31370850

[126/125, 245/243, 121/120]

wedgie [14, 18, 26, 16, -4, 2, -23, 10, -25, -45]

map [[0, 7, 9, 13, 8], [2, -2, -2, -4, 1]]

generators [443.4105637, 600.]

bad 464.7137988 rms 5.592402087 g 14.18248417

58: [126/125, 176/175, 243/242, 896/891]

[243/242, 896/891, 176/175]

wedgie [4, -8, 26, 10, -22, 30, 2, 83, 51, -62]

map [[0, -2, 4, -13, -5], [2, 4, 3, 11, 9]]

generators [248.2547452, 600.]

bad 458.2298129 rms 4.200153836 g 16.69901622

[243/242, 896/891, 126/125]

wedgie [6, 17, 39, 15, 13, 45, 3, 43, -24, -93]

map [[0, 6, 17, 39, 15], [1, -1, -5, -14, -3]]

generators [517.1519478, 1200.]

bad 541.0798888 rms 3.824357079 g 19.51739151

[243/242, 176/175, 126/125]

wedgie [10, 9, 7, 25, -9, -17, 5, -9, 27, 46]

map [[0, 10, 9, 7, 25], [1, -1, 0, 1, -3]]

generators [310.1470775, 1200.]

bad 240.3020098 rms 3.316530343 g 13.06303399

[896/891, 176/175, 126/125]

wedgie [2, -4, -16, -24, -11, -31, -45, -26, -42, -12]

map [[0, -1, 2, 8, 12], [2, 4, 3, -1, -3]]

generators [496.2164639, 600.]

bad 336.4543492 rms 3.182069339 g 16.38814903

72: [225/224, 243/242, 385/384, 4000/3993]

[4000/3993, 385/384, 243/242]

wedgie [6, 17, -26, 15, 13, -58, 3, -108, -24, 132]

map [[0, 6, 17, -26, 15], [1, -1, -5, 14, -3]]

generators [516.6837803, 1200.]

bad 353.9145617 rms 1.860428748 g 23.31155446

[4000/3993, 385/384, 225/224]

wedgie [12, -2, 20, -6, -31, -2, -51, 52, -7, -86]

map [[0, -6, 1, -10, 3], [2, 7, 4, 12, 5]]

generators [383.2159771, 600.]

bad 195.0280472 rms 1.584514409 g 17.95231779

[4000/3993, 243/242, 225/224]

wedgie [6, 17, 46, 15, 13, 56, 3, 59, -24, -117]

map [[0, 6, 17, 46, 15], [1, -1, -5, -17, -3]]

generators [516.6897696, 1200.]

bad 347.1579268 rms 1.824911029 g 23.31155446