Canousmic temperaments

These are rank-2 temperaments that temper out the canousma, 4802000/4782969 = [4 -14 3 4⟩. For the rank-3 temperament, see Canou family.

Note that 4802000/4782969 = 2×(10/9)³/(9/7)⁴, these intervals tend to have lower complexity.

Temperaments not dicussed here include:

• Godzilla ({49/48, 81/80})
• Betic ({225/224, 1071875/1062882})
• Pentorwell ({1728/1715, 179200/177147})
• Amicable ({2401/2400, 1600000/1594323})
• Parakleismic ({3136/3125, 4375/4374})
• Septiquarter ({5120/5103, 420175/419904})
• Kleischismic ({32805/32768, 1562500/1492992})

Discussed below are satin and semiluna.

Satin

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: 2.8530

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: 0.1972

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: 0.0580

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

Badness: 0.0303

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

Badness: 0.0200

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

Badness: 0.0145

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

Badness: 0.0122

Semiluna

See also: Luna family § Semiluna

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 95703125/95551488

Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]

POTE generator: ~2187/1960 = 193.1725

Badness: 0.1922

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 14641/14580

Mapping: [⟨2 8 4 23 14], ⟨0 -15 2 -54 -22]]

POTE generator: ~121/108 = 193.1732

Badness: 0.0678