Tricot family

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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]]

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

Septimal tricot aka trimot

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

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

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

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