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 #Godzilla
• Betic, {225/224, 1071875/1062882} → Sycamore family #Betic
• Pentorwell, {1728/1715, 179200/177147} → Orwellismic temperaments #Pentorwell
• Amicable, {2401/2400, 1600000/1594323} → Breedsmic temperaments #Amicable
• Parakleismic, {3136/3125, 4375/4374} → Ragismic microtemperaments #Parakleismic
• Septiquarter, {5120/5103, 420175/419904} → Hemifamity temperaments #Septiquarter
• Marthirds, {15625/15552, 2460375/2458624} → Kleismic family #Marthirds
• Kleischismic, {32805/32768, 1500625/1492992} → Schismatic family #Kleischismic
• Semiluna, {4802000/4782969, 95703125/95551488} → Luna family #Semiluna

Considered below are satin and superlimmal.

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.

5-limit

Subgroup: 2.3.5

Comma list: [104 -70 3⟩

Mapping: [⟨1 2 12], ⟨0 -3 -70]]

POTE generator: ~[-34 23 -1⟩ = 165.907

Badness: 2.853049

7-limit

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

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

Badness: 0.012158

Superlimmal

The superlimmal temperament uses an ever slightly sharpened large limma as the generator, nine exceed the octave by 126/125. It can be described as 80 & 311. It gets all the primes up to 29 reasonably covered, but still acceptible just as a 13-limit microtemperament, judging from its comma basis. The 80-tone MOS scale is presumably the place to start, and a 151-tone MOS is possible.

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

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

Badness: 0.013054