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 (a trivial case) and these:

• Betic
• Amicable
• Parakleismic
• Septiquarter
• Kleischismic

Discussed below are satin and semiluna.

Satin

5-limit

Comma list: [104 -70 3⟩

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

Mapping: [⟨1 2 12], ⟨0 -3 -70]]

Badness: 2.8530

7-limit

Comma list: 2100875/2097152, 4802000/4782969

POTE generator: ~8575/7776 = 165.913

Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]

Badness: 0.1972

11-limit

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

POTE generator: ~11/10 = 165.915

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

Badness: 0.0580

13-limit

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

POTE generator: ~11/10 = 165.914

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

Badness: 0.0303

17-limit

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

POTE generator: ~11/10 = 165.913

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

Badness: 0.0200

19-limit

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

POTE generator: ~11/10 = 165.913

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

Badness: 0.0145

23-limit

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

POTE generator: ~11/10 = 165.914

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

Badness: 0.0122

Semiluna

See also: Luna family § Semiluna

Comma list: 4802000/4782969, 95703125/95551488

POTE generator: ~2187/1960 = 193.1725

Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]

Badness: 0.1922

11-limit

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

POTE generator: ~121/108 = 193.1732

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

Badness: 0.0678