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 or 81/80) → Meantone family
- Betic (+225/224) → Sycamore family
- Pentorwell (+1728/1715) → Orwellismic temperaments
- Amicable (+2401/2400) → Breedsmic temperaments
- Parakleismic (+3136/3125 or 4375/4374) → Ragismic microtemperaments
- Septiquarter (+5120/5103) → Hemifamity temperaments
- Marthirds (+15625/15552) → Kleismic family
- Kleischismic (+32805/32768) → Schismatic family
- Turkey (+5250987/5242880) → Vulture family
- Hemiquindromeda (+67108864/66976875) → Quindromeda family
- Semiluna (+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]]
Optimal tuning (POTE): ~2 = 1\1, ~8575/7776 = 165.913
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.915
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.913
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.914
Optimal ET sequence: 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 acceptable 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0464
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0455
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0446
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0462
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0464
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0458
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 135.0460
Optimal ET sequence: 80, 231, 311
Badness: 0.010901