User:Francium/Miscellaneous rank-2 temperaments

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On this page, I explore some unnamed rank-2 temperaments that catched my interest.

19 & 251

Subgroup: 2.3.5

Comma list: [22 33 -32⟩

Mapping: [⟨1 10 11], ⟨0 -32 -33]]

POTE generator: ~6/5 = 315.5643

Optimal ET sequence: 19, 156bc, 175bc, 194bc, 213, 232, 251, 270, 289, 559, 1407, 1966, 2525, 10659c

Badness: 2.830

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, 7710244864/7688671875

Mapping: [⟨1 10 11 27], ⟨0 -32 -33 -92]]

POTE generator: ~6/5 = 315.5565

Optimal ET sequence: 19, 175bcddd, 194bcdd, 213dd, 232d, 251, 270, 2449c, 2719c, 2989bc

Badness: 1.422

31 & 317

Subgroup: 2.5.7

Comma list: 281484423828125/281474976710656

Mapping: [⟨1 0 6], ⟨0 8 -11]]

POTE generator: ~49/40 = 348.2889

Optimal ET sequence: 7, 24, 31, 224, 255, 286, 317, 348, 379, 410, 789, 6691, 7480, 8269, 9058, 9847, 10636

Badness: 0.015

11-limit

Subgroup: 2.5.7.11

Comma list: 5767168/5764801, 537109375/536870912

Mapping: [⟨1 0 6 29], ⟨0 8 -11 -88]]

POTE generator: ~11/9 = 348.2809

Optimal ET sequence: 31, 255e, 286e, 317, 348, 379, 727, 3287, 4014, 4741, 5468

Badness: 0.811

31 & 449

Subgroup: 2.3.5.7

Comma list: 6144/6125, [7 -4 -16 13⟩

Mapping: [⟨1 -3 2 1], ⟨0 71 5 28]]

POTE generator: ~256/245 = 77.498

Optimal ET sequence: 31, 294bbd, 325bd, 356bd, 387bd, 418d, 449, 480, 991cd

Badness: 8.502

41 & 167

Subgroup: 2.3.5.7

Comma list: 2401/2400, 179200/177147

Mapping: [⟨1 5 17 11], ⟨0 -10 -43 -24]]

POTE generator: ~81/64 = 409.607

Optimal ET sequence: 38ccd, 41, 126, 167, 208, 375b

Badness: 2.943

11-limit

Subgroup: 2.3.5.7

Comma list: 896/891, 2200/2187, 2401/2400

Mapping: [⟨1 5 17 11 -2], ⟨0 -10 -43 -24 16]]

POTE generator: ~81/64 = 409.417

Optimal ET sequence: 41, 126, 167, 208, 375be

Badness: 1.816

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 896/891, 2401/2400

Mapping: [⟨1 5 17 11 -2 -12], ⟨0 -10 -43 -24 16 46]]

POTE generator: ~81/64 = 409.4451

Optimal ET sequence: 41, 126, 167, 208, 375be

Badness: 1.782

41 & 227

Subgroup: 2.3.5.7

Comma list: 5120/5103, 117649/116640

Mapping: [⟨1 0 -11 -1], ⟨0 5 42 12]]

POTE generator: ~56/45 = 380.6339

Optimal ET sequence: 19cc, 22c, 41, 104c, 145, 186, 413

Badness: 2.852

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 67228/66825

Mapping: [⟨1 0 -11 -1 19], ⟨0 5 42 12 -49]]

POTE generator: ~96/77 = 380.6090

Optimal ET sequence: 41, 145e, 186, 227

Badness: 2.774

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 325/324, 352/351, 67228/66825

Mapping: [⟨1 0 -11 -1 19 24], ⟨0 5 42 12 -49 -64]]

POTE generator: ~96/77 = 380.6135

Optimal ET sequence: 41, 145ef, 186, 227

Badness: 2.307

41 & 337

(better described as 41 & 255)

Subgroup: 2.3.5.7

Comma list: 16875/16807, 7381125/7340032

Mapping: [⟨1 4 -8 -4], ⟨0 -11 47 31]]

POTE generator: ~7/6 = 263.5183

Optimal ET sequence: 41, 173, 214, 255, 296, 551, 847c

Badness: 3.414

41 & 509

Subgroup: 2.3.5.7

Comma list: 2401/2400, [40 -27 0 1⟩

Mapping: [⟨1 1 -29 -13], ⟨0 2 107 54]]

POTE generator: ~49/40 = 351.278

Optimal ET sequence: 41, 263ccd, 304ccd, 345c, 386c, 427c, 468, 509, 1486bc

Badness: 6.884

41 & 523

(better known as 41 & 441)

Subgroup: 2.3.5.7

Comma list: 2401/2400, [35 -23 -3 3⟩

Mapping: [⟨1 1 11 7], ⟨0 6 -89 -43]]

POTE generator: ~15/14 = 117.006

Optimal ET sequence: 41, 236cc, 277c, 318c, 359, 400, 441

Badness: 1.383

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 19712/19683, 166375/165888

Mapping: [⟨1 1 11 7 -6], ⟨0 6 -89 -43 97]]

POTE generator: ~77/72 = 117.025

Optimal ET sequence: 41, 400e, 441e, 482, 1487cdee

Badness: 3.127

53 & 307

Subgroup: 2.9.5

Comma list: 32805/32768

Mapping: [⟨1 0 15], ⟨0 7 -28]]

POTE generator: ~512/375 = 543.3531

Optimal ET sequence: 6, 29, 35, 41, 47, 53, 59, 171, 230, 519, 749

Badness: 0.100

53 & 499

Subgroup: 2.3.5

Comma list: [63 -50 7⟩

Mapping: [⟨1 0 -9], ⟨0 7 50]

POTE generator: ~2560/2187 = 271.7276

Optimal ET sequence: 22c, 31c, 53, 393, 446, 499, 552, 605, 658, 711, 764, 6165b, 6929b, 7693b, 8457b, 9221bbc, 9985bbc, 10749bbc

Badness: 8.860

7-limit

Subgroup: 2.3.5.7

Comma list: 26873856/26796875, 33554432/33480783

Mapping: [⟨1 0 -9 25], ⟨0 7 50 -98]]

POTE generator: ~1024/875 = 271.7017

Optimal ET sequence: 53, 393, 446, 499, 552, 2155ccdd, 2707bcccdd

Badness: 6.432

103 & 503

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 46656/46585, 766656/765625

Mapping: [⟨1 -41 -34 -26 -60], ⟨0 102 87 69 152]]

POTE generator: ~4/3 = 500.9986

Optimal ET sequence: 91bcdee, 103, 297e, 400, 1303c, 1703ce

Badness: 2.344

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4225/4224, 75816/75625

Mapping: [⟨1 -41 -34 -26 -60 -13], ⟨0 102 87 69 152 40]]

POTE generator: ~4/3 = 500.9970

Optimal ET sequence: 91bcdee, 103, 297e, 400, 503, 903

Badness: 1.735

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1001/1000, 936/935, 1377/1375, 1716/1715, 4225/4224

Mapping: [⟨1 -41 -34 -26 -60 -13 2], ⟨0 102 87 69 152 40 5]]

POTE generator: ~4/3 = 500.9972

Badness: 1.402

311 & 1063

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3367254360064/3363025078125

Mapping: [⟨1 11 -22 -7], ⟨0 -24 62 25]]

POTE generator: ~21/16 = 470.7489

Optimal ET sequence: 51, 79c, 130, 311, 441, 2776, 3217, 3658, 4099, 4540, 4981

Badness: 1.127

11-limit

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 19712/19683, 9453125/9437184

Mapping: [⟨1 11 -22 -7 98], ⟨0 -24 62 25 -241]]

POTE generator: ~21/16 = 470.7429

Optimal ET sequence: 130e, 181e, 311, 752, 1063

Badness: 4.158

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