More comma sequence thoughts

I'd proposed comma sequences as unique identifiers for linear
temperaments which displayed their family relationships, but there
were some problems with the definition. Another approach is to do what
I just did with equal temperaments; first use reduction to Hermite
normal form, giving a comma sequence whose first element is a 5-limit
comma. Then reduce the remaining commas with respect to this comma,
getting a list of, mostly, 5-limit bridge commas. Family similarity is
reflected in identity of the first n commas. Below is a list of comma
sequence identifiers for 11-limit linear temperaments.

[36, 54, 36, 18, 2, -44, -96, -68, -145, -74]
[7629394531250/7625597484987, 4375/4374, 1890625/1889568]
rms: .198798 comp: 36.000000

[6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
[34171875/33554432, 225/224, 4125/4096]
rms: 1.901466 comp: 12.351981

[12, 22, -4, -6, 7, -40, -51, -71, -90, -3]
[31381059609/31250000000, 4375/4374, 8019/8000]
rms: 1.249417 comp: 19.063803

[14, 6, 74, 52, -23, 78, 34, 155, 100, -110]
[6115295232/6103515625, 4375/4374, 5632/5625]
rms: .352551 comp: 42.006802

[12, -2, 20, -6, -31, -2, -51, 52, -7, -86]
[2197265625/2147483648, 225/224, 4125/4096]
rms: 1.584514 comp: 17.952318

[9, 5, -3, 7, -13, -30, -20, -21, -1, 30]
[1990656/1953125, 126/125, 3168/3125]
rms: 4.418576 comp: 10.525479

[18, 15, -6, 9, -18, -60, -48, -56, -31, 46]
[15625/15552, 1029/1024, 385/384]
rms: 1.357052 comp: 21.152507

[2, -4, -4, -12, -11, -12, -26, 2, -14, -20]
[2048/2025, 64/63, 100/99]
rms: 9.552922 comp: 8.349508

[24, 32, 40, 24, -5, -4, -45, 3, -55, -71]
[59604644775390625/59296646043258912, 4375/4374, 1953125/1948617]
rms: .768706 comp: 23.421602

[3, 0, -6, -6, -7, -18, -20, -14, -14, 4]
[128/125, 64/63, 100/99]
rms: 12.772525 comp: 6.989788

[7, -3, 8, 2, -21, -7, -21, 27, 15, -22]
[2109375/2097152, 225/224, 4125/4096]
rms: 5.548615 comp: 8.860023

[2, -4, -16, -24, -11, -31, -45, -26, -42, -12]
[2048/2025, 126/125, 3168/3125]
rms: 3.182070 comp: 16.388149

[1, 4, 10, 18, 4, 13, 25, 12, 28, 16]
[81/80, 126/125, 3168/3125]
rms: 6.584357 comp: 10.155927

[1, 4, 10, -13, 4, 13, -24, 12, -44, -71]
[81/80, 126/125, 4125/4096]
rms: 5.644271 comp: 11.093756

[5, 1, 12, -8, -10, 5, -30, 25, -22, -64]
[3125/3072, 225/224, 100/99]
rms: 4.730404 comp: 10.620062

[0, 12, 24, 36, 19, 38, 57, 22, 42, 18]
[531441/524288, 225/224, 8019/8000]
rms: 1.886540 comp: 22.219682

[38, -3, 8, 64, -93, -94, -30, 27, 159, 152]
[9903520314283042199192993792/9822542779147624969482421875,
29360128/29296875, 5632/5625]
rms: .383188 comp: 48.844505

[1, -1, 3, 4, -4, 2, 3, 10, 13, 1]
[16/15, 28/27, 55/54]
rms: 44.341252 comp: 2.725541

[2, 3, 1, -2, 0, -4, -10, -6, -15, -9]
[27/25, 21/20, 33/32]
rms: 40.160927 comp: 3.116775

[6, 5, 22, -21, -6, 18, -54, 37, -66, -135]
[15625/15552, 225/224, 4125/4096]
rms: 1.697137 comp: 20.989793

[10, 9, 7, 25, -9, -17, 5, -9, 27, 46]
[10077696/9765625, 126/125, 3168/3125]
rms: 3.316530 comp: 13.063034

[15, -2, -5, 22, -38, -50, -17, -6, 58, 79]
[274877906944/274658203125, 65625/65536, 5632/5625]
rms: 1.772097 comp: 20.707142

[1, 4, -2, -6, 4, -6, -13, -16, -28, -10]
[81/80, 36/35, 288/275]
rms: 18.933027 comp: 5.099020

[1, -3, -4, -1, -7, -9, -5, -1, 8, 11]
[135/128, 21/20, 33/32]
rms: 36.627981 comp: 3.380617

[1, -8, -14, -18, -15, -25, -32, -10, -14, -2]
[32805/32768, 225/224, 100/99]
rms: 5.290179 comp: 12.432676

[0, 0, 7, 0, 0, 11, 0, 16, 0, -24]
[2187/2048, 81/80, 33/32]
rms: 22.636347 comp: 4.183300

[24, 20, 16, -12, -24, -42, -102, -19, -97, -89]
[15625/15552, 2401/2400, 385/384]
rms: 1.086899 comp: 29.626243

[12, 34, 20, 30, 26, -2, 6, -49, -48, 15]
[129140163/128000000, 19683/19600, 8019/8000]
rms: 1.462302 comp: 17.952318

[23, -1, 13, 42, -55, -44, -13, 33, 101, 73]
[36028797018963968/35762786865234375, 29360128/29296875, 5632/5625]
rms: 1.013640 comp: 29.051678

[3, 17, -1, -13, 20, -10, -31, -50, -89, -33]
[131072000/129140163, 5120/5103, 2200/2187]
rms: 3.005389 comp: 14.320315

[3, -5, -6, -1, -15, -18, -12, 0, 15, 18]
[32768/30375, 64/63, 512/495]
rms: 13.781284 comp: 6.676184

[1, -8, -14, 23, -15, -25, 33, -10, 81, 113]
[32805/32768, 225/224, 2200/2187]
rms: 2.447559 comp: 18.413892

[6, -19, -26, -21, -44, -58, -54, -7, 17, 31]
[18160335421875/17592186044416, 225/224, 91125/90112]
rms: 1.883084 comp: 23.311554

[9, 10, -3, 2, -5, -30, -28, -35, -30, 16]
[1953125/1889568, 875/864, 100/99]
rms: 5.302952 comp: 11.119481

[2, -2, -2, 0, -8, -9, -7, 1, 7, 7]
[16/15, 50/49, 77/75]
rms: 46.396405 comp: 3.380617

[2, -16, -40, -60, -30, -69, -102, -48, -84, -30]
[32805/32768, 3136/3125, 5632/5625]
rms: .979643 comp: 38.500464

[3, 5, -6, 4, 1, -18, -4, -28, -8, 32]
[250/243, 64/63, 55/54]
rms: 11.793935 comp: 6.430952

[6, 0, 3, 3, -14, -12, -16, 7, 7, -2]
[128/125, 49/48, 77/75]
rms: 14.472091 comp: 6.414270

[0, 5, 0, -5, 8, 0, -8, -14, -29, -14]
[256/243, 28/27, 55/54]
rms: 22.089446 comp: 4.629100

[2, 8, 8, 12, 8, 7, 12, -4, 0, 6]
[81/80, 50/49, 45/44]
rms: 13.344995 comp: 6.502747

[4, 2, 2, -4, -6, -8, -20, -1, -16, -18]
[25/24, 49/48, 33/32]
rms: 22.632565 comp: 5.756983

[6, -12, 10, -14, -33, -1, -43, 57, 9, -74]
[2048/2025, 1728/1715, 540/539]
rms: 2.986631 comp: 16.212870

[42, 47, 34, 33, -23, -64, -93, -53, -86, -25]
[227373675443232059478759765625/223043140842065783739956330496,
68359375/68024448, 30517578125/30304891584]
rms: .579520 comp: 38.060103

[5, -11, -12, -3, -29, -33, -22, 3, 31, 33]
[553584375/536870912, 225/224, 66825/65536]
rms: 5.154394 comp: 12.671678

[8, 6, 6, -4, -9, -13, -34, -3, -30, -32]
[390625/373248, 875/864, 100/99]
rms: 8.163015 comp: 9.856108

[17, 6, 15, 27, -30, -24, -16, 18, 42, 24]
[782757789696/762939453125, 110592/109375, 6912/6875]
rms: 2.412281 comp: 17.968226

[1, 4, 3, -1, 4, 2, -5, -4, -16, -13]
[81/80, 21/20, 33/32]
rms: 47.548543 comp: 2.725541

[1, 4, 10, 6, 4, 13, 6, 12, 0, -18]
[81/80, 126/125, 45/44]
rms: 18.965800 comp: 5.424811

[1, 33, 27, -18, 50, 40, -32, -30, -156, -144]
[5629499534213120/5559060566555523, 5120/5103, 14417920/14348907]
rms: 1.115730 comp: 27.846518

[3, -24, -1, 28, -45, -10, 34, 65, 148, 82]
[32805/32768, 1029/1024, 385/384]
rms: 1.499792 comp: 24.323710

[2, -4, -4, 0, -11, -12, -7, 2, 14, 14]
[2048/2025, 64/63, 45/44]
rms: 27.006882 comp: 4.659859

[18, 39, 42, 9, 20, 16, -48, -12, -114, -120]
[4052555153018976267/4000000000000000000, 100442349/100000000, 8019/8000]
rms: 1.049818 comp: 25.150973

[3, 12, -1, -8, 12, -10, -23, -36, -60, -19]
[81/80, 1029/1024, 385/384]
rms: 6.456792 comp: 9.895887

[4, 4, 4, 0, -3, -5, -14, -2, -14, -14]
[648/625, 36/35, 100/99]
rms: 27.265894 comp: 4.276180

[13, -10, 6, 17, -46, -27, -18, 42, 74, 27]
[72081298828125/70368744177664, 225/224, 4125/4096]
rms: 2.438407 comp: 18.413892

[18, -9, 18, 9, -56, -22, -48, 67, 52, -37]
[75084686279296875/72057594037927936, 225/224, 4125/4096]
rms: 1.689101 comp: 22.564194

[30, 13, 14, 3, -49, -62, -99, -4, -38, -40]
[931322574615478515625/897524058588526411776, 390625/387072,
21484375/21233664]
rms: 1.179375 comp: 31.559468

[6, 5, 3, -2, -6, -12, -24, -7, -22, -16]
[15625/15552, 126/125, 100/99]
rms: 14.472383 comp: 7.050836

[5, 3, 7, 4, -7, -3, -11, 8, -1, -13]
[3456/3125, 36/35, 132/125]
rms: 19.644403 comp: 4.840307

[4, 2, 2, 10, -6, -8, 2, -1, 16, 21]
[25/24, 49/48, 45/44]
rms: 19.453599 comp: 5.580579

[2, 1, 6, 5, -3, 4, 1, 11, 8, -7]
[25/24, 28/27, 45/44]
rms: 37.484665 comp: 3.422614

[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
[16/15, 28/27, 100/99]
rms: 43.037876 comp: 3.625308

[16, 2, 5, 40, -34, -37, 8, 6, 86, 95]
[393216/390625, 3136/3125, 5632/5625]
rms: 1.764586 comp: 24.161658

[10, 26, -34, -28, 18, -82, -79, -152, -155, 39]
[1600000/1594323, 4375/4374, 16384/16335]
rms: .881985 comp: 36.410360

[1, 4, -2, -1, 4, -6, -5, -16, -16, 4]
[81/80, 36/35, 33/32]
rms: 39.863722 comp: 3.150964

[2, 1, -4, 5, -3, -12, 1, -12, 8, 28]
[25/24, 64/63, 45/44]
rms: 27.107413 comp: 4.659859

[2, -4, -4, 10, -11, -12, 9, 2, 37, 42]
[2048/2025, 64/63, 55/54]
rms: 12.267148 comp: 7.764388

[6, 29, -2, -21, 32, -20, -54, -86, -149, -52]
[68630377364883/67108864000000, 413343/409600, 177147/176000]
rms: 1.555238 comp: 24.177912

[2, -6, 1, -2, -14, -4, -10, 19, 16, -9]
[135/128, 49/48, 33/32]
rms: 19.984071 comp: 5.077964

[18, -14, 30, -20, -64, -3, -94, 109, 2, -160]
[4294967296/4271484375, 6144/6125, 5632/5625]
rms: 1.000617 comp: 33.670674

[3, 5, 9, 4, 1, 6, -4, 7, -8, -20]
[250/243, 28/27, 55/54]
rms: 22.129858 comp: 4.342481