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: trimot (53 & 70), trident (53 & 229) and trillium (53 & 441). Tempering out hemifamity comma (5120/5103) leads to trimot, 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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~59049/40960 = 634.012
Optimal ET sequence: 53, 229, 282, 335, 388, 441, 1376, 1817, 2258
Badness: 0.046093
2.3.5.13 subgroup
See also No-fives subgroup temperaments#Threedic.
Subgroup: 2.3.5.13
Comma list: 2197/2187, 41067/40960
Gencom: [2 13/9; 2197/2187, 41067/40960]
Gencom mapping: [⟨1 0 -13 0 0 0], ⟨0 3 29 0 0 7]]
Sval mapping: [⟨1 0 -13 0], ⟨0 3 29 7]]
POL2 generator: ~13/9 = 633.997
Optimal ET sequence: 17c, 36c, 53
RMS error: 0.2342 cents
Trimot
Trimot, named by Petr Pařízek in 2011[1], can be described as the 53 & 70 temperament.
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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 634.0259
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~63/44 = 634.0273
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0115
Optimal ET sequence: 17c, 36ce, 53, 70, 123de
Badness: 0.032102
Trident
Trident, named by Xenllium in 2021, can be described as the 53 & 229 temperament.
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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~4096/2835 = 634.0480
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0669
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0652
Optimal ET sequence: 53, 176, 229
Badness: 0.046593
Trillium
Trillium, also named by Xenllium in 2021, can be described as the 53 & 441 temperament.
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 ]]
Optimal tuning (POTE): ~2 = 1\1, ~23625/16384 = 634.0118
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~3888/2695 = 634.0094
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0095
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0190
Optimal ET 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]]
Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0181
Optimal ET 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 ]]
Optimal tuning (POTE): ~63/50 = 1\3, ~100352/91125 = 165.9837
Optimal ET 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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9835
Optimal ET 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]]
Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9842
Optimal ET 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]]
Optimal tuning (POTE): ~34/27 = 1\3, ~11/10 = 165.9805
Optimal ET sequence: 159, 282, 441
Badness: 0.025972
Noletaland
Noletaland is described as 282 & 1323, and it combines the smallest consistent edo in the 29-odd-limit with the smallest uniquely consistent. It reaches 4/3 in nine generators (noleta-…) and tempers out the landscape comma (…-land). Noletaland reaches 13/11 in 2 generators, and 29/19 in 5. Then there is 44/25 in 4, and 152/115 in also 4.
Subgroup: 2.3.5.7.11
Comma list: 250047/250000, 56723625/56689952, 78675968/78594219
Mapping: [⟨3 6 19 30 35], ⟨0 -9 -87 -156 -178]]
- mappin generators: ~63/50, ~1936/1875
Optimal tuning (CTE): ~63/50 = 1\3, ~1936/1875 = 55.3290
Optimal ET sequence: 282, 759de, 1041, 1323, 4251e
Badness: 0.158
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 10648/10647, 43904/43875, 85750/85683, 250047/250000
Mapping: [⟨3 6 19 30 35 36], ⟨0 -9 -87 -156 -178 -180]]
Optimal tuning (CTE): ~63/50 = 1\3, ~1936/1875 = 55.3294
Optimal ET sequence: 282, 759def, 1041, 1323
Badness: 0.0725
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 2058/2057, 4914/4913, 8624/8619, 12376/12375, 250047/250000
Mapping: [⟨3 6 19 30 35 36 29], ⟨0 -9 -87 -156 -178 -180 -121]]
Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3295
Optimal ET sequence: 282, 759def, 1041, 1323
Badness: 0.0380
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 2058/2057, 2926/2925, 3136/3135, 4200/4199, 4914/4913, 250047/250000
Mapping: [⟨3 6 19 30 35 36 29 18], ⟨0 -9 -87 -156 -178 -180 -121 -38]]
Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3295
Optimal ET sequence: 282, 759def, 1041, 1323
Badness: 0.0269
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 2058/2057, 2926/2925, 3136/3135, 3381/3380, 3520/3519, 4914/4913, 18515/18513
Mapping: [⟨3 6 19 30 35 36 29 18 31], ⟨0 -9 -87 -156 -178 -180 -121 -38 -126]]
Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3296
Optimal ET sequence: 282, 759def, 1041, 1323
Badness: 0.0194
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 2058/2057, 2755/2754, 2926/2925, 3136/3135, 3381/3380, 3451/3450, 3520/3519, 4914/4913
Mapping: [⟨3 6 19 30 35 36 29 18 31 19], ⟨0 -9 -87 -156 -178 -180 -121 -38 -126 -32]]
Optimal tuning (CTE): ~63/50 = 1\3, ~351/340 = 55.3296
Optimal ET sequence: 282, 759def, 1041, 1323
Badness: 0.0168