User:Francium/Miscellaneous rank-2 temperaments

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

Berkenthin (19 & 239)

Subgroup: 2.3.5

Comma list: [-26 -29 31⟩

Mapping: [⟨1 13 13], ⟨0 -31 -29]]

POTE generator: ~39182082048/30517578125 = 441.8700

Optimal ET sequence: 19, 144bc, 163bc, 182b, 201, 220, 239, 258, 277, 535, 1882c

Badness: 19.828

7-limit

Subgroup: 2.3.5.7

Comma list: 10976/10935, 48828125/48771072

Mapping: [⟨1 13 13 33], ⟨0 -31 -29 -82]]

POTE generator: ~4032/3125 = 441.8483

Optimal ET sequence: 19, 144bcddd, 163bcdd, 182bdd, 201d, 220, 239, 497, 736

Badness: 3.566

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

19 & 349

Subgroup: 2.3.5

Comma list: [-32 -34 37⟩

Mapping: [⟨1 -14 -12], ⟨0 37 34]]

POTE generator: ~244140625/181398528 = 505.4605

Optimal ET sequence: 19, 178bc, 197bc, 216b, 235b, 254, 273, 292, 311, 641

Badness: 40.121

11-limit

Subgroup: 2.3.5.11

Comma list: 1953125/1948617, 4322682045/4294967296

Mapping: [⟨1 -14 -12 46], ⟨0 37 34 -101]]

POTE generator: ~1375/1024 = 505.4459

Optimal ET sequence: 19, 292ee, 311e, 330e, 349, 1028cee, 1377cceee, 1726bccceee

Badness: 12.504

19 & 563

Subgroup: 2.3.5

Comma list: [38 61 -58⟩

Mapping: [⟨1 26 28], ⟨0 -58 -61]]

POTE generator: ~19073486328125/14281868906496 = 505.1394

Optimal ET sequence: 19, 297bbcc, 316bbcc, 335bbcc, 354bcc, 373bc, 392bc, 411bc, 430c, 449, 468, 487, 506, 1537, 2043, 4592c

Badness: 49.390

31 & 227

Subgroup: 2.3.5.7

Comma list: 235298/234375, 65625/65536

Mapping: [⟨1 -2 3 3], ⟨0 37 -7 -2]]

POTE generator: ~2560/2401 = 116.2790

Optimal ET sequence: 31, 165, 196, 227, 258, 289

Badness: 2.917

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 151263/151250

Mapping: [⟨1 -2 3 3 -1], ⟨0 37 -7 -2 46]]

POTE generator: ~294/275 = 116.2717

Optimal ET sequence: 9, 22, 31, 53, 84e, 87, 118, 258e, 376de, 547dee

Badness: 2.101

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 352/351, 385/384, 16807/16731

Mapping: [⟨1 -2 3 3 -1 10], ⟨0 37 -7 -2 46 -65]]

POTE generator: ~294/275 = 116.2869

Badness: 2.393

31 & 317

Subgroup: 2.5.7

Comma list: 281484423828125/281474976710656

Mapping: [⟨1 0 6], ⟨0 8 -11]]

POTE generator: ~262144/214375 = 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: ~16807/13750 = 348.2809

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

Badness: 0.811

31 & 367

Subgroup: 2.3.5

Comma list: [-156 12 59⟩

Mapping: [⟨1 13 0], ⟨0 -59 12]]

POTE generator: ~2500/2187 = 232.1720

Optimal ET sequence: 31, 305b, 336, 367, 398, 429, 460, 889, 1349

Badness: 90.743

31 & 397

Subgroup: 2.3.5

Comma list: [55 -1 -23⟩

Mapping: [⟨1 9 2], ⟨0 -46 2]]

POTE generator: ~4096/3645 = 193.4315

Optimal ET sequence: 28b, 31, 90, 121, 152, 335, 822, 1157c, 1492c, 2649cc

Badness: 9.862

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

37 & 577

Subgroup: 2.5.7

Comma list: [20 -34 21⟩

Mapping: [⟨1 8 12], ⟨0 -42 -68]]

CTE tuning: ~1475789056/1220703125 = 324.444

Optimal ET sequence: 11cd, 26cdd, 37, 159, 196, 233, 270, 503, 773, 1816, 2589c

Badness: 7.966

11-limit

Subgroup: 2.5.7.11

Comma list: 102487/102400, 4302592/4296875

Mapping: [⟨1 8 12 4], ⟨0 -42 -68 -4]]

CTE tuning: ~3773/3125 = 324.451

Optimal ET sequence: 11cd, 37, 159, 196, 233, 270, 503

Badness: 0.511

13-limit

Subgroup: 2.5.7.11.13

Comma list: 1001/1000, 10985/10976, 75712/75625

Mapping: [⟨1 8 12 4 11], ⟨0 -42 -68 -4 -54]]

CTE tuning: ~169/140 = 324.446

Optimal ET sequence: 11cdf, 37, 159, 196, 233, 270, 503, 773, 1276f

Badness: 0.290

41 & 167

Subgroup: 2.3.5.7

Comma list: 2401/2400, 179200/177147

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

CTE tuning: ~80/63 = 409.639

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]]

CTE tuning: ~80/63 = 409.630

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]]

CTE tuning: ~195/154 = 409.617

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 & 233

Subgroup: 2.3.5.7

Comma list: 2401/2400, 124556484375/123363917824

Mapping: [⟨1 9 -19 -6], ⟨0 -16 46 19]]

POTE generator: ~30375/21952 = 556.1960

Optimal ET sequence: 41, 151, 192, 233, 274, 315, 589bcd

Badness: 4.209

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 540/539, 5053640625/5035261952

Mapping: [⟨1 9 -19 -6 22], ⟨0 -16 46 19 -40]]

POTE generator: ~2475/1792 = 556.2083

41, 151, 192, 233, 274

Badness: 2.494

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 540/539, 21168/21125, 81675/81536

Mapping: [⟨1 9 -19 -6 22 38], ⟨0 -16 46 19 -40 -74]]

POTE generator: ~91/66 = 556.2131

Optimal ET sequence: 41, 192, 233, 274, 507b

Badness: 2.034

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: ~784/675 = 263.5183

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

Badness: 3.414

41 & 359

Subgroup: 2.3.5

Comma list: [155 -89 -6⟩

Mapping: [⟨1 1 11], ⟨0 6 -89]]

POTE generator: ~2187/2048 = 117.0069

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

Badness: 46.849

7-limit

Subgroup: 2.3.5.7

Comma list: 2401/2400, 11785390260224/11767897353375

Mapping: [⟨1 1 11 7], ⟨0 6 -89 -43]]

POTE generator: ~15/14 = 117.0067

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, 200704/200475, 46656/46585

Mapping: [⟨1 1 11 7 -2], ⟨0 6 -89 -43 56]]

POTE generator: ~77/72 = 117.0029

Optimal ET sequence: 41, 318c, 359, 400, 841e, 1241e

Badness: 1.716

41 & 421

Subgroup: 2.3.7

Comma list: [-125 47 18⟩

Mapping: [⟨1 -5 20], ⟨0 18 -47]]

POTE generator: ~9/7 = 438.9747

Optimal ET sequence: 41, 298, 339, 380, 421, 462, 503, 544, 585, 2966bddd

Badness: 78.897

11-limit

Subgroup: 2.3.7.11

Comma list: 2893401/2883584, 208971104256/208422380089

Mapping: [⟨1 -5 20 -28], ⟨0 18 -47 86]]

POTE generator: ~9/7 = 438.9681

Optimal ET sequence: 41, 380e, 421, 462, 503, 1047, 1550b

Badness: 6.355

13-limit

Subgroup: 2.3.7.11.13

Comma list: 85293/85184, 53361/53248, 216513/215306

Mapping: [⟨1 -5 20 -28 -38], ⟨0 18 -47 86 114]]

POTE generator: ~156/121 = 438.9595

Optimal ET sequence: 41, 380e, 421, 462

Badness: 4.703

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

43 & 233

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 21168/21125, 81675/81536

Mapping: [⟨1 8 -8 -17 -13 9], ⟨0 -23 37 71 59 -19]]

POTE generator: ~40/33 = 334.7481

Optimal ET sequence: 43, 147de, 190, 423d, 613bde

Badness: 1.813

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 561/560, 1275/1274, 1156/1155, 21168/21125

Mapping: [⟨1 8 -8 -17 -13 9 -16], ⟨0 -23 37 71 59 -19 72]]

POTE generator: ~17/14 = 334.7511

Optimal ET sequence: 43, 147deg, 190g, 233, 423dg

Badness: 1.592

Mehlem (53 & 179)

Subgroup: 2.3.5

Comma list: [12 -31 16⟩

Mapping: [⟨1 4 7], ⟨0 -16 -31]]

POTE generator: ~10/9 = 181.0964

Optimal ET sequence: 20c, 33c, 53, 338, 391, 444, 497, 550, 603, 656, 1259b

Badness: 10.511

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, 65536/64827

Mapping: [⟨1 4 7 1], ⟨0 -16 -31 12]]

POTE generator: ~10/9 = 181.0314

Optimal ET sequence: 20c, 53, 126, 179, 232d

Badness: 3.333

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 & 467

Subgroup: 2.3.5

Comma list: [-36 11 8⟩

Mapping: [⟨1 4 -1], ⟨0 -8 11]]

POTE generator: ~10125/8192 = 362.3210

Optimal ET sequence: 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc

Badness: 3.055

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, 69198046875/68719476736

Mapping: [⟨1 4 -1 33], ⟨0 -8 11 -100]]

POTE generator: ~10125/8192 = 362.3181

Optimal ET sequence: 53, 255d, 308d, 361, 414, 2123bccdd

Badness: 4.674

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 & 167

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3025/3024, 1716/1715, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6], ⟨0 74 51 44 82 27]]

POTE generator: ~440/343 = 431.1062

Badness: 0.662

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1001/1000, 1225/1224, 715/714, 936/935, 4225/4224

Mapping: [⟨1 -25 -16 -13 -26 -6 -11], ⟨0 74 51 44 82 27 42]]

POTE generator: ~440/343 = 431.1087

Optimal ET sequence: 39bce, 64be, 103, 167, 270

Badness: 1.068

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

113 & 409

Subgroup: 2.3.7.11

Comma list: 5038848/5021863, 235782657/234881024

Mapping: [⟨1 4 16 -1], ⟨0 -13 -71 24]]

POTE generator: ~8192/7203 = 222.9728

Optimal ET sequence: 70, 113, 183, 296

Badness: 5.092

13-limit

Subgroup: 2.3.7.11.13

Comma list: 729/728, 19773/19712, 718848/717409

Mapping: [⟨1 4 16 -1 5], ⟨0 -13 -71 24 -7]]

POTE generator: ~91/80 = 222.9746

Optimal ET sequence: 70, 113, 183, 296

Badness: 1.829

193 & 547

Subgroup: 2.3.5.7

Comma list: 589824/588245, 23066015625/23018340352

Mapping: [⟨1 0 4 2], ⟨0 51 -54 26]]

POTE generator: ~49/48 = 37.2913

Optimal ET sequence: 32, 129, 161, 193, 354, 547, 901d, 1255d

Badness: 7.020

239 & 569

Subgroup: 2.3.5.7

Comma list: 823543/820125, 68359375/68024448

Mapping: [⟨1 8 9 13], ⟨0 -73 -76 -116]]

POTE generator: ~729/686 = 105.4479

Optimal ET sequence: 34bcdd, 57bcd, 91, 148d, 239, 330, 569

Badness: 8.274

11-limit

Subgroup: 2.3.5.7.11

Comma list: 42875/42768, 42592/42525, 151263/151250

Mapping: [⟨1 8 9 13 22], ⟨0 -73 -76 -116 -211]]

POTE generator: ~729/686 = 105.4456

Optimal ET sequence: 57bcdeee, 91e, 148dee, 239, 569, 808

Badness: 3.518

311 & 571

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 2401/2400, 294885080956928/294197433834375

Mapping: [⟨1 11 -22 -7 37], ⟨0 -48 124 50 -171]]

POTE generator: ~63/55 = 235.3736

Optimal ET sequence: 10, 18bcd, 21e, 23bc, 31, 41, 72, 167, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee

Badness: 2.441

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 2401/2400, 47432/47385, 114345/114244

Mapping: [⟨1 11 -22 -7 37 19], ⟨0 -48 124 50 -171 -78]]

POTE generator: ~63/55 = 235.3738

Badness: 1.528

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2431/2430, 12376/12375, 2601/2600, 8624/8619, 2401/2400

Mapping: [⟨1 11 -22 -7 37 19 -22], ⟨0 -48 124 50 -171 -78 133]]

POTE generator: ~63/55 = 235.3741

Optimal ET sequence: 260, 311, 571, 882

Badness: 1.212

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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