Canousmic temperaments

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This is a collection of rank-2 temperaments that temper out the canousma, 4802000/4782969 = [4 -14 3 4. For the rank-3 temperament, see Canou family.

Temperaments discussed elsewhere are:

Considered below are satin and superlimmal.

Satin

For the 5-limit version of this temperament, see High badness temperaments #Satin.

The satin temperament (94 & 217) uses 11/10 as a generator, three of them gives 4/3, and tempers out both the rainy comma and the canousma.

Subgroup: 2.3.5.7

Comma list: 2100875/2097152, 4802000/4782969

Mapping: [1 2 12 -3], 0 -3 -70 42]]

POTE generator: ~8575/7776 = 165.913

Optimal GPV sequence94, 217, 311, 839, 1150

Badness: 0.197207

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 19712/19683, 41503/41472

Mapping: [1 2 12 -3 13], 0 -3 -70 42 -69]]

POTE generator: ~11/10 = 165.915

Vals: 94, 217, 311

Badness: 0.057972

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689

Mapping: [1 2 12 -3 13 -1], 0 -3 -70 42 -69 34]]

POTE generator: ~11/10 = 165.914

Vals: 94, 217, 311, 839e, 1150e

Badness: 0.030316

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095

Mapping: [1 2 12 -3 13 -1 11], 0 -3 -70 42 -69 34 -50]]

POTE generator: ~11/10 = 165.913

Vals: 94, 217, 311, 839e, 1150eg

Badness: 0.020007

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573

Mapping: [1 2 12 -3 13 -1 11 16], 0 -3 -70 42 -69 34 -50 -85]]

POTE generator: ~11/10 = 165.913

Vals: 94, 217, 311, 839e, 1150eg

Badness: 0.014479

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155

Mapping: [1 2 12 -3 13 -1 11 16 16], 0 -3 -70 42 -69 34 -50 -85 -83]]

POTE generator: ~11/10 = 165.914

Vals: 94, 217, 311, 839ei, 1150egi

Badness: 0.012158

Superlimmal

The superlimmal temperament (80 & 311) uses an ever slightly sharpened large limma as the generator, nine exceed the octave by 126/125. It gets all the primes up to 29 reasonably covered, but still acceptible just as a 13-limit microtemperament, judging from its comma basis. While the MOS scale may not be the most effective approach, the 80-tone MOS is presumably the place to start if it is used. It can also be extended to prime 37 by tempering out (27/25)/(40/37) = 1000/999, where 40/37 is notably the mediant of 27/25 and 13/12, which could be interpreted as an explanation of the sharpened limma.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 52734375/52706752

Mapping: [1 8 12 18], 0 -57 -86 -135]]

Wedgie⟨⟨57 86 135 3 53 72]]

POTE generator: ~27/25 = 135.0464

Optimal GPV sequence80, 231, 311, 1324b, 1635b

Badness: 0.252387

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 1479016/1476225

Mapping: [1 8 12 18 11], 0 -57 -86 -135 -67]]

POTE generator: ~27/25 = 135.0455

Vals: 80, 231, 311, 1013e, 1324be

Badness: 0.060667

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455

Mapping: [1 8 12 18 11 1], 0 -57 -86 -135 -67 24]]

POTE generator: ~27/25 = 135.0446

Vals: 80, 231, 311, 702, 1013e

Badness: 0.039017

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224

Mapping: [1 8 12 18 11 1 6], 0 -57 -86 -135 -67 24 -17]]

POTE generator: ~27/25 = 135.0462

Vals: 80, 231, 311

Badness: 0.030077

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499

Mapping: [1 8 12 18 11 1 6 11], 0 -57 -86 -135 -67 24 -17 -60]]

POTE generator: ~27/25 = 135.0464

Vals: 80, 231, 311

Badness: 0.020460

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495

Mapping: [1 8 12 18 11 1 6 11 7], 0 -57 -86 -135 -67 24 -17 -60 -22]]

POTE generator: ~27/25 = 135.0458

Vals: 80, 231, 311

Badness: 0.016146

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495

Mapping: [1 8 12 18 11 1 6 11 7 16], 0 -57 -86 -135 -67 24 -17 -60 -22 -99]]

POTE generator: ~27/25 = 135.0460

Vals: 80, 231, 311

Badness: 0.013054

No-31's 37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274

Mapping: [1 8 12 18 11 1 6 11 7 16 15], 0 -57 -86 -135 -67 24 -17 -60 -22 -99 -87]]

POTE generator: ~27/25 = 135.0460

Vals: 80, 231, 311

Badness: 0.010901