Canousmic temperaments
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:
- Godzilla, {49/48, 81/80} → Meantone family
- Betic, {225/224, 1071875/1062882} → Sycamore family
- Pentorwell, {1728/1715, 179200/177147} → Orwellismic temperaments
- Amicable, {2401/2400, 1600000/1594323} → Breedsmic temperaments
- Parakleismic, {3136/3125, 4375/4374} → Ragismic microtemperaments
- Septiquarter, {5120/5103, 420175/419904} → Hemifamity temperaments
- Marthirds, {15625/15552, 2460375/2458624} → Kleismic family
- Kleischismic, {32805/32768, 1500625/1492992} → Schismatic family
- Turkey, {4802000/4782969, 5250987/5242880} → Vulture family
- Hemiquindromeda, {4802000/4782969, 67108864/66976875} → Quindromeda family
- Semiluna, {4802000/4782969, 95703125/95551488} → Luna family
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 sequence: 94, 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
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 sequence: 80, 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
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
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
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
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
Badness: 0.010901