Quindromeda family

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The quindromeda family tempers out [56 -28 -5⟩, equating the Pythagorean comma with a stack of five schismas, making it a member of the schismic–Pythagorean equivalence continuum.

Quindromeda

The name quindromeda comes from "quintans" (Latin for "one fifth") and "Andromeda", because the generator is 1/5 of the andromeda fourth (~4/3, about 497.6 cents).

Subgroup: 2.3.5

Comma: [56 -28 -5⟩ = 72057594037927936/71489976421753125

Mapping: [⟨1 2 0], ⟨0 -5 28]]

POTE generator: ~4428675/4194304 = 99.526

Optimal ET sequence: 12, 169, 181, 193, 205, 217, 422

Badness: 0.399849

Subgroup temperament

The quindromeda temperament works well for the 2.3.5.17.19 subgroup, tempering out the password comma (1216/1215), aureusma (1445/1444), and 6144/6137. An obvious 17-limit interpretation of the generator is ~18/17, equating three 18/17s with 19/16, five 18/17s with 4/3, and twenty-eight 18/17s with the fifth harmonic.

2.3.5.17

Subgroup: 2.3.5.17

Comma list: 24576/24565, 295936/295245

Gencom: [2 18/17; 24576/24565 295936/295245]

Gencom mapping: [⟨1 2 0 0 0 0 5], ⟨0 -5 28 0 0 0 -11]]

Sval mapping: [⟨1 2 0 5], ⟨0 -5 28 -11]]

POL2 generator: ~18/17 = 99.524

Optimal ET sequence: 12, 169, 181, 193, 205, 422

RMS error: 0.0757 cents

2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 1216/1215, 1445/1444, 6144/6137

Gencom: [2 18/17; 1216/1215 1445/1444 6144/6137]

Gencom mapping: [⟨1 2 0 0 0 0 5 4], ⟨0 -5 28 0 0 0 -11 3]]

Sval mapping: [⟨1 2 0 5 4], ⟨0 -5 28 -11 3]]

POL2 generator: ~18/17 = 99.524

Optimal ET sequence: 12, 169, 181, 193, 205, 422

RMS error: 0.0813 cents

Badness: 0.007140

Quintagar

The quintagar temperament (12&217) tempers out the hemimean comma (3136/3125) and the garischisma (33554432/33480783) in the 7-limit. In the 2.3.5.7.17.19 subgroup, 256/255 (the difference between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), 361/360 (between 19/18 and 20/19), and 400/399 (between 20/19 and 21/20) are equated together, and 476/475 (between 28/25 and 19/17) is tempered out. Immediate 2.3.5.7.11.17.19 extensions include quintoneum (12&217, tempering out 441/440), quinsandra (217&229, equating 385/384 with 400/399), and quinsandric (12&229, equating 400/399 with 441/440). Full 19-limit extensions include quintoneum (12f&217), quintoneoid (12&217), quinsandra (217&229), quinsandric (12f&229), and quinsandro (12&229). The name quintagar is so named because the generator is 1/5 of the garibaldi fourth (~4/3, about 497.8 cents).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 33554432/33480783

Mapping: [⟨1 2 0 -3], ⟨0 -5 28 70]]

Wedgie: ⟨⟨ 5 -28 -70 -56 -125 -84 ]]

POTE generator: ~200/189 = 99.555

Optimal ET sequence: 12, 217, 229, 446, 675c

Badness: 0.142897

Quintoneum

The name quintoneum is a play on the words "quintans" and "cotoneum".

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 7168000/7144929

Mapping: [⟨1 2 0 -3 -5], ⟨0 -5 28 70 102]]

POTE generator: ~35/33 = 99.539

Optimal ET sequence: 12, 205d, 217

Badness: 0.087157

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3136/3125, 13720/13689

Mapping: [⟨1 2 0 -3 -5 -7], ⟨0 -5 28 70 102 129]]

POTE generator: ~35/33 = 99.541

Optimal ET sequence: 12f, 205df, 217

Badness: 0.052361

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3136/3125, 3757/3750

Mapping: [⟨1 2 0 -3 -5 -7 5], ⟨0 -5 28 70 102 129 -11]]

POTE generator: ~18/17 = 99.540

Optimal ET sequence: 12f, 205df, 217

Badness: 0.035653

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 441/440, 476/475, 595/594, 1216/1215, 3757/3750

Mapping: [⟨1 2 0 -3 -5 -7 5 4], ⟨0 -5 28 70 102 129 -11 3]]

POTE generator: ~18/17 = 99.541

Optimal ET sequence: 12f, 205df, 217

Badness: 0.025782

Quintoneoid

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 59150/59049

Mapping: [⟨1 2 0 -3 -5 11], ⟨0 -5 28 70 102 -88]]

POTE generator: ~35/33 = 99.537

Optimal ET sequence: 12, 205d, 217

Badness: 0.072826

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 1001/1000, 2601/2600, 3136/3125

Mapping: [⟨1 2 0 -3 -5 11 5], ⟨0 -5 28 70 102 -88 -11]]

POTE generator: ~18/17 = 99.537

Optimal ET sequence: 12, 205d, 217

Badness: 0.042339

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 441/440, 476/475, 595/594, 1001/1000, 1216/1215, 2601/2600

Mapping: [⟨1 2 0 -3 -5 11 5 4], ⟨0 -5 28 70 102 -88 -11 3]]

POTE generator: ~18/17 = 99.537

Optimal ET sequence: 12, 205d, 217

Badness: 0.028983

Quinsandra

The name quinsandra is a play on the words "quintans" and "cassandra". This temperament tempers out 19712/19683 and 41503/41472 in the 2.3.7.11 subgroup as the cassandra temperament, but with the hemimean comma rather than the schisma tempered out.

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 19712/19683, 41503/41472

Mapping: [⟨1 2 0 -3 13], ⟨0 -5 28 70 -115]]

POTE generator: ~200/189 = 99.551

Optimal ET sequence: 12e, 217, 446

Badness: 0.109908

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 3136/3125, 4096/4095, 19712/19683

Mapping: [⟨1 2 0 -3 13 11], ⟨0 -5 28 70 -115 -88]]

POTE generator: ~55/52 = 99.548

Optimal ET sequence: 12e, 217, 446, 663c

Badness: 0.067730

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1156/1155, 1377/1375, 3136/3125, 4096/4095

Mapping: [⟨1 2 0 -3 13 11 5], ⟨0 -5 28 70 -115 -88 -11]]

POTE generator: ~18/17 = 99.548

Optimal ET sequence: 12e, 217, 446, 663c

Badness: 0.038153

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 936/935, 1156/1155, 1216/1215, 1377/1375, 1729/1728

Mapping: [⟨1 2 0 -3 13 11 5 4], ⟨0 -5 28 70 -115 -88 -11 3]]

POTE generator: ~18/17 = 99.547

Optimal ET sequence: 12e, 217, 446, 663ch

Badness: 0.026654

Quinsandric

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 8019/8000, 15488/15435

Mapping: [⟨1 2 0 -3 -6], ⟨0 -5 28 70 114]]

POTE generator: ~200/189 = 99.570

Optimal ET sequence: 12, 217e, 229, 470cd, 699cd

Badness: 0.093971

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 3136/3125, 10648/10647

Mapping: [⟨1 2 0 -3 -6 -8], ⟨0 -5 28 70 114 141]]

POTE generator: ~55/52 = 99.578

Optimal ET sequence: 12f, 217ef, 229, 241, 470cd, 711ccd

Badness: 0.065701

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 3136/3125, 7744/7735

Mapping: [⟨1 2 0 -3 -6 -8 5], ⟨0 -5 28 70 114 141 -11]]

POTE generator: ~18/17 = 99.574

Optimal ET sequence: 12f, 217ef, 229, 241, 470cd

Badness: 0.046624

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 442/441, 476/475, 561/560, 627/625, 6144/6137

Mapping: [⟨1 2 0 -3 -6 -8 5 4], ⟨0 -5 28 70 114 141 -11 3]]

POTE generator: ~18/17 = 99.575

Optimal ET sequence: 12f, 217ef, 229, 241, 470cd

Badness: 0.033145

Quinsandro

Subgroup: 2.3.5.7.11.13

Comma list: 1573/1568, 3136/3125, 4096/4095, 4459/4455

Mapping: [⟨1 2 0 -3 -6 11], ⟨0 -5 28 70 114 -88]]

POTE generator: ~200/189 = 99.556

Optimal ET sequence: 12, 217e, 229, 446e, 675ceef

Badness: 0.100195

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 715/714, 1701/1700, 3136/3125, 4096/4095

Mapping: [⟨1 2 0 -3 -6 11 5], ⟨0 -5 28 70 114 -88 -11]]

POTE generator: ~18/17 = 99.556

Optimal ET sequence: 12, 217e, 229, 446e, 675ceef

Badness: 0.057851

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 476/475, 561/560, 627/625, 1216/1215, 1729/1728

Mapping: [⟨1 2 0 -3 -6 11 5 4], ⟨0 -5 28 70 114 -88 -11 3]]

POTE generator: ~18/17 = 99.557

Optimal ET sequence: 12, 217e, 229, 446e, 675ceefh

Badness: 0.040410

Quintakwai

The quintakwai temperament (12&193) tempers out the hemifamity comma (5120/5103) and the compass comma (9765625/9680832, quinruyoyo) in the 7-limit; 1375/1372 and 4375/4356 in the 11-limit. In the 2.3.5.7.17.19 subgroup, 225/224 (the difference between 15/14 and 16/15), 256/255 (between 16/15 and 17/16), 289/288 (between 17/16 and 18/17), 324/323 (between 18/17 and 19/18), and 361/360 (between 19/18 and 20/19) are equated together, and 400/399 (between 20/19 and 21/20) is tempered out. The name quintakwai is so named because the generator is 1/5 of the kwai fourth (~4/3, about 497.4 cents).

Subgroup: 2.3.5.7

Comma list: 5120/5103, 9765625/9680832

Mapping: [⟨1 2 0 -2], ⟨0 -5 28 58]]

Wedgie: ⟨⟨ 5 -28 -58 -56 -106 -56 ]]

POTE generator: ~625/588 = 99.483

Optimal ET sequence: 12, 169, 181, 193

Badness: 0.155536

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 5120/5103

Mapping: [⟨1 2 0 -2 -4], ⟨0 -5 28 58 90]]

POTE generator: ~35/33 = 99.472

Optimal ET sequence: 12, 181, 193, 374, 567ce

Badness: 0.073158

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1375/1372, 1575/1573, 4096/4095

Mapping: [⟨1 2 0 -2 -4 10], ⟨0 -5 28 58 90 -76]]

POTE generator: ~35/33 = 99.468

Optimal ET sequence: 12, 181, 193, 374, 567ce, 941bce

Badness: 0.062737

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 375/374, 595/594, 1275/1274, 4096/4095

Mapping: [⟨1 2 0 -2 -4 10 5], ⟨0 -5 28 58 90 -76 -11]]

POTE generator: ~18/17 = 99.469

Optimal ET sequence: 12, 181, 193, 374, 567ce, 941bceg

Badness: 0.037855

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 375/374, 400/399, 595/594, 1216/1215, 1275/1274

Mapping: [⟨1 2 0 -2 -4 10 5 4], ⟨0 -5 28 58 90 -76 -11 3]]

POTE generator: ~18/17 = 99.469

Optimal ET sequence: 12, 181, 193, 374, 567ce, 941bcegh, 1508bccdeegghh

Badness: 0.025861

Quinkwai

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1375/1372, 4375/4356

Mapping: [⟨1 2 0 -2 -4 -5], ⟨0 -5 28 58 90 105]]

POTE generator: ~35/33 = 99.456

Optimal ET sequence: 12f, 169e, 181

Badness: 0.061873

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 375/374, 595/594, 833/832, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5], ⟨0 -5 28 58 90 105 -11]]

POTE generator: ~18/17 = 99.458

Optimal ET sequence: 12f, 169e, 181, 374ff, 555cff

Badness: 0.043506

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 375/374, 400/399, 495/494, 595/594, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5 4], ⟨0 -5 28 58 90 105 -11 3]]

POTE generator: ~18/17 = 99.459

Optimal ET sequence: 12f, 169e, 181, 374ff, 555cff

Badness: 0.031313

Quintakwoid

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 625/624, 1375/1372, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6], ⟨0 -5 28 58 90 117]]

POTE generator: ~35/33 = 99.484

Optimal ET sequence: 12f, 181f, 193

Badness: 0.057357

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 442/441, 595/594, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6 5], ⟨0 -5 28 58 90 117 -11]]

POTE generator: ~18/17 = 99.484

Optimal ET sequence: 12f, 181f, 193

Badness: 0.039289

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215

Mapping: [⟨1 2 0 -2 -4 -6 5 4], ⟨0 -5 28 58 90 117 -11 3]]

POTE generator: ~18/17 = 99.485

Optimal ET sequence: 12f, 181f, 193

Badness: 0.028101

Quindro

The quindro temperament (205&217) is an extension of the quindromeda temperament which tempers out the ragisma (4375/4374), the vishdel comma (5632/5625), and the ibnsinma (2080/2079).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 72057594037927936/71489976421753125

Mapping: [⟨1 2 0 15], ⟨0 -5 28 -147]]

Wedgie: ⟨⟨ 5 -28 147 -56 219 420 ]]

POTE generator: ~4428675/4194304 = 99.529

Optimal ET sequence: 205, 217, 422, 639, 1061

Badness: 0.316850

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 25165824/25109315

Mapping: [⟨1 2 0 15 -5], ⟨0 -5 28 -147 102]]

POTE generator: ~17325/16384 = 99.529

Optimal ET sequence: 205, 217, 422, 639, 1061e

Badness: 0.085219

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4375/4374, 5632/5625, 20480/20449

Mapping: [⟨1 2 0 15 -5 11], ⟨0 -5 28 -147 102 -88]]

POTE generator: ~4096/3861 = 99.529

Optimal ET sequence: 205, 217, 422, 639, 1061ef

Badness: 0.039203

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1156/1155, 2080/2079, 2431/2430, 4375/4374, 5632/5625

Mapping: [⟨1 2 0 15 -5 11 5], ⟨0 -5 28 -147 102 -88 -11]]

POTE generator: ~18/17 = 99.529

Optimal ET sequence: 205, 217, 422, 639, 1061ef

Badness: 0.024110

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1156/1155, 1216/1215, 1445/1444, 2080/2079, 2376/2375, 2431/2430

Mapping: [⟨1 2 0 15 -5 11 5 4], ⟨0 -5 28 -147 102 -88 -11 3]]

POTE generator: ~18/17 = 99.529

Optimal ET sequence: 205, 217, 422, 639h, 1061efh

Badness: 0.016764

Hemiquindromeda

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 67108864/66976875

Mapping: [⟨1 2 0 6], ⟨0 -10 56 -77]]

Wedgie: ⟨⟨ 10 -56 77 -112 94 336 ]]

POTE generator: ~36/35 = 49.756

Optimal ET sequence: 24, 169d, 193, 217, 410

Badness: 0.244589

11-limit

Subgroup: 2.3.5.7.11

Comma list: 14700/14641, 16384/16335, 19712/19683

Mapping: [⟨1 2 0 6 4], ⟨0 -10 56 -77 -13]]

POTE generator: ~36/35 = 49.757

Optimal ET sequence: 24, 169d, 193, 217, 410e

Badness: 0.093130

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 676/675, 4096/4095, 19712/19683

Mapping: [⟨1 2 0 6 4 2], ⟨0 -10 56 -77 -13 41]]

POTE generator: ~36/35 = 49.759

Optimal ET sequence: 24, 169d, 193, 217, 410e, 627e

Badness: 0.048715

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 595/594, 676/675, 1156/1155, 4096/4095

Mapping: [⟨1 2 0 6 4 2 5], ⟨0 -10 56 -77 -13 41 -22]]

POTE generator: ~34/33 = 49.759

Optimal ET sequence: 24, 169d, 193, 217, 410e, 627e

Badness: 0.028155

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 595/594, 676/675, 1156/1155, 1216/1215, 1729/1728

Mapping: [⟨1 2 0 6 4 2 5 4], ⟨0 -10 56 -77 -13 41 -22 6]]

POTE generator: ~34/33 = 49.759

Optimal ET sequence: 24, 169d, 193, 217, 410e

Badness: 0.019364

Semiquindromeda

The semiquindromeda temperament (410&422) is an extension of the quindromeda temperament with a half-octave period.

Subgroup: 2.3.5.7

Comma list: 102760448/102515625, 1220703125/1219784832

Mapping: [⟨2 4 0 -5], ⟨0 -5 28 64]]

POTE generator: ~1323/1250 = 99.521

Optimal ET sequence: 12, 398, 410, 422, 832, 1254d, 2086bd

Badness: 0.233140

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 85937500/85766121

Mapping: [⟨2 4 0 -5 -10], ⟨0 -5 28 64 102]]

POTE generator: ~1323/1250 = 99.525

Optimal ET sequence: 12, 410, 422

Badness: 0.093926

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 5632/5625, 831875/830466

Mapping: [⟨2 4 0 -5 -10 -13], ⟨0 -5 28 64 102 123]]

POTE generator: ~1323/1250 = 99.523

Optimal ET sequence: 12f, 410, 422, 1254df, 1676bdff, 2098bcddff

Badness: 0.053361

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1716/1715, 2080/2079, 2500/2499, 5632/5625, 15895/15876

Mapping: [⟨2 4 0 -5 -10 -13 10], ⟨0 -5 28 64 102 123 -11]]

POTE generator: ~18/17 = 99.522

Optimal ET sequence: 12f, 410, 422, 832, 1254df, 1676bdff

Badness: 0.034659

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1216/1215, 1445/1444, 1716/1715, 2080/2079, 2376/2375, 2500/2499

Mapping: [⟨2 4 0 -5 -10 -13 10 8], ⟨0 -5 28 64 102 123 -11 3]]

POTE generator: ~18/17 = 99.523

Optimal ET sequence: 12f, 410, 422, 1254dfhh, 1676bdffhh

Badness: 0.025439

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