Tricot family
The generator for tricot temperament is the real cube root of third harmonic, 31/3, tuned between 63/44 and 13/9. Tricot temperament can be described as 53&70 temperament, tempering out the tricot comma, [39 -29 3⟩ in the 5-limit. There are some mappings for 7-limit extension of this temperament: septimal tricot (53&70, also called as "trimot"), trident (53&229) and trillium (53&441). Tempering out hemifamity comma (5120/5103) leads to septimal tricot, porwell comma (6144/6125) leads to trident, and ragisma (4375/4374) leads to trillium.
Tricot
Subgroup: 2.3.5
Comma list: [39 -29 3⟩ = 68719476736000/68630377364883
Mapping: [⟨1 0 -13], ⟨0 3 29]]
Mapping generators: ~2, ~59049/40960
Wedgie: ⟨⟨3 29 39]]
POTE generator: ~59049/40960 = 634.012
Optimal GPV sequence: 53, 229, 282, 335, 388, 441, 1376, 1817, 2258
Badness: 0.046093
Septimal tricot
- See also: Hemifamity temperaments #Tricot
Subgroup: 2.3.5.7
Comma list: 2430/2401, 5120/5103
Mapping: [⟨1 0 -13 -3], ⟨0 3 29 11]]
Wedgie: ⟨⟨3 29 11 39 9 -56]]
POTE generator: ~81/56 = 634.0259
Optimal GPV sequence: 17c, 36c, 53, 70, 229dd, 282dd
Badness: 0.100127
11-limit
Subgroup: 2.3.5.7.11
Comma list: 99/98, 121/120, 5120/5103
Mapping: [⟨1 0 -13 -3 -5], ⟨0 3 29 11 16]]
POTE generator: ~63/44 = 634.0273
Optimal GPV sequence: 17c, 36ce, 53, 70, 123de
Badness: 0.056134
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 121/120, 169/168, 352/351
Mapping: [⟨1 0 -13 -3 -5 0], ⟨0 3 29 11 16 7]]
POTE generator: ~13/9 = 634.0115
Optimal GPV sequence: 17c, 36ce, 53, 70, 123de
Badness: 0.032102
Trident
Subgroup: 2.3.5.7
Comma list: 6144/6125, 14348907/14336000
Mapping: [⟨1 0 -13 25], ⟨0 3 29 -42]]
Wedgie: ⟨⟨3 29 -42 39 -75 -179]]
POTE generator: ~4096/2835 = 634.0480
Optimal GPV sequence: 53, 176, 229, 282, 511
Badness: 0.101694
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3388/3375, 6144/6125, 8019/8000
Mapping: [⟨1 0 -13 25 -33], ⟨0 3 29 -42 69]]
POTE generator: ~231/160 = 634.0669
Optimal GPV sequence: 53, 176, 229
Badness: 0.074272
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125
Mapping: [⟨1 0 -13 25 -33 0], ⟨0 3 29 -42 69 7]]
POTE generator: ~13/9 = 634.0652
Optimal GPV sequence: 53, 176, 229
Badness: 0.046593
Trillium
Subgroup: 2.3.5.7
Comma list: 4375/4374, 1099511627776/1098337086315
Mapping: [⟨1 0 -13 53], ⟨0 3 29 -95]]
Wedgie: ⟨⟨3 29 -95 39 -159 -302]]
POTE generator: ~23625/16384 = 634.0118
Optimal GPV sequence: 53, 441, 494, 935, 1376, 3193, 4569
Badness: 0.030852
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 131072/130977, 759375/758912
Mapping: [⟨1 0 -13 53 -89], ⟨0 3 29 -95 175]]
POTE generator: ~3888/2695 = 634.0094
Optimal GPV sequence: 53, 441, 494, 935, 1429
Badness: 0.046758
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078
Mapping: [⟨1 0 -13 53 -89 -28], ⟨0 3 29 -95 175 60]]
POTE generator: ~75/52 = 634.0095
Optimal GPV sequence: 53, 441, 494, 935, 1429
Badness: 0.019393
Pseudotrillium
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 5632/5625, 4108797/4096000
Mapping: [⟨1 0 -13 53 -61], ⟨0 3 29 -95 122]]
POTE generator: ~231/160 = 634.0190
Optimal GPV sequence: 53, 335, 388
Badness: 0.111931
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374
Mapping: [⟨1 0 -13 53 -61 -28], ⟨0 3 29 -95 122 60]]
POTE generator: ~75/52 = 634.0181
Optimal GPV sequence: 53, 335, 388
Badness: 0.054837
Tritricot
Subgroup: 2.3.5.7
Comma list: 250047/250000, 11785390260224/11767897353375
Mapping: [⟨3 6 19 30], ⟨0 -3 -29 -52]]
Wedgie: ⟨⟨9 87 156 117 222 118]]
POTE generator: ~100352/91125 = 165.9837
Optimal GPV sequence: 159, 282, 441, 2487, 2928, 3369
Badness: 0.086081
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4000/3993, 166698/166375, 200704/200475
Mapping: [⟨3 6 19 30 22], ⟨0 -3 -29 -52 -28]]
POTE generator: ~11/10 = 165.9835
Optimal GPV sequence: 159, 282, 441
Badness: 0.074002
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875
Mapping: [⟨3 6 19 30 22 36], ⟨0 -3 -29 -52 -28 -60]]
POTE generator: ~11/10 = 165.9842
Optimal GPV sequence: 159, 282, 441
Badness: 0.035641
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 1575/1573, 1701/1700, 2025/2023, 8624/8619
Mapping: [⟨3 6 19 30 22 36 16], ⟨0 -3 -29 -52 -28 -60 -9]]
POTE generator: ~11/10 = 165.9805
Optimal GPV sequence: 159, 282, 441
Badness: 0.025972