Tetracot family
The parent of the tetracot family is tetracot, the 5-limit temperament tempering out 20000/19683 = [5 -9 4⟩, the minimal diesis or tetracot comma. The dual of this comma is the wedgie ⟨⟨ 4 9 5 ]], which tells us ~10/9 is a generator, and that four of them give ~3/2. In fact, (10/9)4 = 20000/19683 × 3/2. We also have (10/9)9 = (20000/19683)2 × 5/2. From this it is evident we should flatten the generator a bit, and 34edo does this and makes for a recommendable tuning. Another possibility is to use (5/2)1/9 for a generator. The 13-note mos gives enough space for eight triads, with the 20-note mos supplying many more.
The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).
Tetracot
Subgroup: 2.3.5
Comma list: 20000/19683
Mapping: [⟨1 1 1], ⟨0 4 9]]
- 5-odd-limit: ~10/9 = [-1/9 0 1/9⟩
Optimal ET sequence: 7, 20c, 27, 34, 75, 109
Badness: 0.048518
Overview to extensions
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- 875/864, the keema, gives monkey;
- 179200/177147 (or equivalently 225/224) gives bunya;
- 245/243 gives octacot, which splits the generator in half.
Monkey and bunya
Monkey tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the ~7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34 & 41 temperament. 41edo is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.
Bunya adds 225/224 to the list of commas and may be described as the 34d & 41 temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 141/26 as a generator, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.
Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives 11-limit monkey, ⟨⟨ 4 9 -15 10 … ]] and 11-limit bunya, ⟨⟨ 4 9 26 10 … ]]. Again, 41edo can be used as a tuning, making the two identical, which is also the case if we turn to the 2.3.5.11 subgroup temperament, dispensing with 7. However, 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 141/26 generator supplies, or even sharper yet, as for instance by the val ⟨355 563 823 997 1230], with a 52/355 generator.
Since 16/13 is shy of (10/9)2 by just 325/324, it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us ⟨⟨ 4 9 -15 10 -2 … ]] for 13-limit monkey and ⟨⟨ 4 9 26 10 -2 … ]] for 13-limit bunya. Once again, 41edo is recommended as a tuning for monkey, while bunya can with advantage tune the fifth sharper: 17\116 as a generator with a fifth a cent and a half sharp or 11\75 with a fifth two cents sharp.
2.3.5.11 subgroup
As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which it tempers out 100/99, 144/143 and 243/242.
The S-expression-based comma list of this temperament is {S9/S11, S10}.
Subgroup: 2.3.5.11
Comma list: 100/99, 243/242
Sval mapping: [⟨1 1 1 2], ⟨0 4 9 10]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.7765
- POTE ~2 = 1\1, ~10/9 = 175.985
Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e
Badness: 0.014706
2.3.5.11.13 subgroup
Subgroup: 2.3.5.11.13
Comma list: 100/99, 144/143, 243/242
Sval mapping: [⟨1 1 1 2 4], ⟨0 4 9 10 -2]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.8150
- POTE ~2 = 1\1, ~10/9 = 176.196
Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e, 109ef
Badness: 0.012311
2.3.5.13 subgroup
Subgroup: 2.3.5.13
Comma list: 325/324, 512/507
Mapping: [⟨1 1 1 4], ⟨0 4 9 -2]]
Optimal ET sequence: 7, 20c, 27, 34, 245bff, 279bfff
Badness: 0.0165
Monkey
Subgroup: 2.3.5.7
Comma list: 875/864, 5120/5103
Mapping: [⟨1 1 1 5], ⟨0 4 9 -15]]
Wedgie: ⟨⟨ 4 9 -15 5 -35 -60 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.676
- POTE:~2 = 1\1, ~10/9 = 175.659
Optimal ET sequence: 7, 34, 41
Badness: 0.073437
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 243/242, 385/384
Mapping: [⟨1 1 1 5 2], ⟨0 4 9 -15 10]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.598
- POTE ~2 = 1\1, ~10/9 = 175.570
Optimal ET sequence: 7, 34, 41
Badness: 0.038836
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 144/143, 243/242
Mapping: [⟨1 1 1 5 2 4], ⟨0 4 9 -15 10 -2]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.618
- POTE ~2 = 1\1, ~10/9 = 175.622
Optimal ET sequence: 7, 34, 41
Badness: 0.028410
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 144/143, 154/153, 170/169
Mapping: [⟨1 1 1 5 2 4 6], ⟨0 4 9 -15 10 -2 -13]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.754
Optimal ET sequence: 7, 34, 41
Badness: 0.025936
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 144/143, 154/153, 170/169, 171/169
Mapping: [⟨1 1 1 5 2 4 6 6], ⟨0 4 9 -15 10 -2 -13 -12]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.697
Optimal ET sequence: 7, 34, 41
Badness: 0.022158
Bunya
Subgroup: 2.3.5.7
Comma list: 225/224, 15625/15309
Mapping: [⟨1 1 1 -1], ⟨0 4 9 26]]
Wedgie: ⟨⟨ 4 9 26 5 30 35 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.785
- POTE:~2 = 1\1, ~10/9 = 175.741
Optimal ET sequence: 7d, …, 34d, 41, 116, 157c, 198c
Badness: 0.062897
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 243/242
Mapping: [⟨1 1 1 -1 2], ⟨0 4 9 26 10]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.738
- POTE ~2 = 1\1, ~10/9 = 175.777
Optimal ET sequence: 7d, …, 34d, 41, 116e
Badness: 0.031332
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 144/143, 225/224, 243/242
Mapping: [⟨1 1 1 -1 2 4], ⟨0 4 9 26 10 -2]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 175.748
- POTE ~2 = 1\1, ~10/9 = 175.886
Optimal ET sequence: 7d, 34d, 41, 116ef
Badness: 0.024886
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 120/119, 144/143, 170/169, 225/224
Mapping: [⟨1 1 1 -1 2 4 6], ⟨0 4 9 26 10 -2 -13]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.811
Optimal ET sequence: 34d, 41, 75e
Badness: 0.023404
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 120/119, 144/143, 170/169, 190/189, 225/224
Mapping: [⟨1 1 1 -1 2 4 6 0], ⟨0 4 9 26 10 -2 -13 29]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.802
Optimal ET sequence: 34dh, 41, 75e
Badness: 0.019460
Modus
Modus was named by Mike Battaglia in 2012 for its fantastic modmos structures[1].
Subgroup: 2.3.5.7
Comma list: 64/63, 4375/4374
Mapping: [⟨1 1 1 4], ⟨0 4 9 -8]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 176.818
- POTE:~2 = 1\1, ~10/9 = 177.203
Optimal ET sequence: 7, 20c, 27, 34d, 61d, 95dd
Badness: 0.068184
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 243/242
Mapping: [⟨1 1 1 4 2], ⟨0 4 9 -8 10]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 176.446
- POTE ~2 = 1\1, ~10/9 = 177.053
Optimal ET sequence: 7, 20ce, 27e, 34d
Badness: 0.035149
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 100/99, 144/143
Mapping: [⟨1 1 1 4 2 4], ⟨0 4 9 -8 10 -2]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 176.471
- POTE ~2 = 1\1, ~10/9 = 176.953
Optimal ET sequence: 7, 20ce, 27e, 34d
Badness: 0.023806
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 64/63, 78/77, 100/99, 120/119, 144/143
Mapping: [⟨1 1 1 4 2 4 1], ⟨0 4 9 -8 10 -2 21]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.453
Optimal ET sequence: 7g, …, 27eg, 34d
Badness: 0.021501
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 64/63, 78/77, 96/95, 100/99, 120/119, 144/143
Mapping: [⟨1 1 1 4 2 4 1 5], ⟨0 4 9 -8 10 -2 21 -5]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.538
Optimal ET sequence: 7g, …, 27eg, 34dh
Badness: 0.017941
- Music
- Tetracot Perc-Sitar by Dustin Schallert
- Tetracot Jam by Dustin Schallert
- Tetracot Pump by Dustin Schallert all in 27edo
Ponens
The error of 11 is about the same as that of modus, but flat instead of sharp, and much more abundant. Since the other primes are all sharp, however, this leads to a much larger error for other intervals involving 11.
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 363/350
Mapping: [⟨1 1 1 4 3], ⟨0 4 9 -8 3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 176.990
- POTE ~2 = 1\1, ~10/9 = 177.200
Optimal ET sequence: 7, 20c, 27, 34de, 61dee
Badness: 0.063077
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 64/63, 66/65, 143/140
Mapping: [⟨1 1 1 4 3 4], ⟨0 4 9 -8 3 -2]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 177.017
- POTE ~2 = 1\1, ~10/9 = 177.197
Optimal ET sequence: 7, 20c, 27, 34de, 61dee
Badness: 0.039043
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 52/51, 55/54, 64/63, 66/65, 143/140
Mapping: [⟨1 1 1 4 3 4 5], ⟨0 4 9 -8 3 -2 -6]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.378
Optimal ET sequence: 7, 20c, 27g
Badness: 0.035200
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 52/51, 55/54, 64/63, 66/65, 77/76, 143/140
Mapping: [⟨1 1 1 4 3 4 5 5], ⟨0 4 9 -8 3 -2 -6 -5]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.505
Optimal ET sequence: 7, 20c, 27g
Badness: 0.028026
Wollemia
Subgroup: 2.3.5.7
Comma list: 126/125, 2240/2187
Mapping: [⟨1 1 1 0], ⟨0 4 9 19]]
Wedgie: ⟨⟨ 4 9 19 5 19 19 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 176.900
- POTE:~2 = 1\1, ~10/9 = 177.357
Optimal ET sequence: 7d, 20cd, 27, 34, 61, 95d, 156bcd
Badness: 0.070522
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 100/99, 243/242
Mapping: [⟨1 1 1 0 2], ⟨0 4 9 19 10]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 176.704
- POTE ~2 = 1\1, ~10/9 = 177.413
Optimal ET sequence: 7d, 20cde, 27e, 34, 95dee
Badness: 0.037551
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 91/90, 100/99, 243/242
Mapping: [⟨1 1 1 0 2 4], ⟨0 4 9 19 10 -2]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/9 = 176.716
- POTE ~2 = 1\1, ~10/9 = 177.231
Optimal ET sequence: 7d, 20cde, 27e, 34, 95dee
Badness: 0.031219
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 56/55, 91/90, 100/99, 136/135, 154/153
Mapping: [⟨1 1 1 0 2 4 1], ⟨0 4 9 19 10 -2 21]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.641
Badness: 0.024471
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 56/55, 76/75, 91/90, 100/99, 136/135, 154/153
Mapping: [⟨1 1 1 0 2 4 1 1], ⟨0 4 9 19 10 -2 21 22]]
Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.749
Optimal ET sequence: 27eg, 34, 95deegh
Badness: 0.021108
Octacot
Octacot cuts the Gordian knot of deciding between the monkey and bunya mappings for 7 by cutting the generator in half and splitting the difference. It adds 245/243 to the normal comma list, and also tempers out 2401/2400. It may also be described as 41 & 68. 68edo or 109edo can be used as tunings, as can (5/2)1/18, which gives just major thirds. Another tuning is 150edo, which has a generator, 11\150, of exactly 88 cents. This relates octacot to the 88cET non-octave temperament, which like Carlos Alpha arguably makes more sense viewed as part of a rank-2 temperament with octaves rather than rank-1 without them.
Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, giving ⟨⟨ 8 18 11 20 -4 … ]] as the octave part of the wedgie. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.
Subgroup: 2.3.5.7
Comma list: 245/243, 2401/2400
Mapping: [⟨1 1 1 2], ⟨0 8 18 11]]
Wedgie: ⟨⟨ 8 18 11 10 -5 -25 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 88.023
- POTE:~2 = 1\1, ~21/20 = 88.076
Optimal ET sequence: 14c, 27, 41, 68, 109, 150, 259, 409bc, 668bbcc
Badness: 0.033845
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 243/242, 245/242
Mapping: [⟨1 1 1 2 2], ⟨0 8 18 11 20]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 87.910
- POTE ~2 = 1\1, ~21/20 = 87.975
Optimal ET sequence: 14c, 27e, 41, 150ee, 191ee, 232cee
Badness: 0.024078
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 144/143, 196/195, 243/242
Mapping: [⟨1 1 1 2 2 4], ⟨0 8 18 11 20 -4]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 87.926
- POTE ~2 = 1\1, ~21/20 = 88.106
Optimal ET sequence: 14c, 27e, 41, 150eef, 191eeff, 232ceeff
Badness: 0.023276
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 120/119, 144/143, 154/153, 189/187
Mapping: [⟨1 1 1 2 2 4 3], ⟨0 8 18 11 20 -4 15]]
Optimal tunings:
- CTE: ~2 = 1\1, ~18/17 = 87.842
- POTE ~2 = 1\1, ~18/17 = 88.102
Optimal ET sequence: 14c, 27eg, 41
Badness: 0.021088
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 120/119, 133/132, 144/143, 154/153, 189/187
Mapping: [⟨1 1 1 2 2 4 3 3], ⟨0 8 18 11 20 -4 15 17]]
Optimal tunings:
- CTE: ~2 = 1\1, ~18/17 = 87.866
- POTE ~2 = 1\1, ~18/17 = 88.111
Optimal ET sequence: 14c, 27eg, 41
Badness: 0.016652
Octocat
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 91/90, 100/99, 245/242
Mapping: [⟨1 1 1 2 2 2], ⟨0 8 18 11 20 23]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 88.090
- POTE ~2 = 1\1, ~21/20 = 88.179
Optimal ET sequence: 14cf, 27e, 41f, 68ef, 109eff
Badness: 0.027601
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 52/51, 78/77, 91/90, 100/99, 189/187
Mapping: [⟨1 1 1 2 2 2 3], ⟨0 8 18 11 20 23 15]]
Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 88.011
Optimal ET sequence: 14cf, 27eg, 41f, 109effgg
Badness: 0.023345
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 52/51, 78/77, 91/90, 100/99, 133/132, 189/187
Mapping: [⟨1 1 1 2 2 2 3 3], ⟨0 8 18 11 20 23 15 17]]
Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 88.017
Optimal ET sequence: 14cf, 27eg, 41f, 109effgg
Badness: 0.017916
Octopod
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 243/242, 245/242
Mapping: [⟨1 1 1 2 2 1], ⟨0 8 18 11 20 37]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 87.770
- POTE ~2 = 1\1, ~21/20 = 87.697
Optimal ET sequence: 14cf, 27eff, 41
Badness: 0.028326
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 120/119, 154/153, 243/242
Mapping: [⟨1 1 1 2 2 1 3], ⟨0 8 18 11 20 37 15]]
Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 87.728
Optimal ET sequence: 14cf, 27effg, 41
Badness: 0.024660
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 120/119, 133/132, 154/153, 209/208
Mapping: [⟨1 1 1 2 2 1 3 3], ⟨0 8 18 11 20 37 15 17]]
Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 87.750
Optimal ET sequence: 14cf, 27effg, 41
Badness: 0.019504
Dificot
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 243/242, 245/242, 343/338
Mapping: [⟨1 9 19 13 22 19], ⟨0 -16 -36 -22 -40 -33]]
Optimal tunings:
- CTE: ~2 = 1\1, ~13/9 = 643.916
- POTE ~2 = 1\1, ~13/9 = 643.989
Optimal ET sequence: 13cdeef, 28ccdef, 41
Badness: 0.051876
October
Subgroup: 2.3.5.7.11
Comma list: 245/243, 385/384, 1375/1372
Mapping: [⟨1 1 1 2 5], ⟨0 8 18 11 -21]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 88.026
- POTE ~2 = 1\1, ~21/20 = 88.035
Optimal ET sequence: 27, 41, 68, 109, 150, 259
Badness: 0.039643
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 245/243, 275/273, 385/384
Mapping: [⟨1 1 1 2 5 4], ⟨0 8 18 11 -21 -4]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 88.041
- POTE ~2 = 1\1, ~21/20 = 88.075
Optimal ET sequence: 27, 41, 68, 109f
Badness: 0.031136
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 154/153, 170/169, 196/195, 245/243, 256/255
Mapping: [⟨1 1 1 2 5 4 6], ⟨0 8 18 11 -21 -4 -26]]
Optimal tunings:
- CTE: ~2 = 1\1, ~21/20 = 88.093
- POTE ~2 = 1\1, ~21/20 = 88.104
Optimal ET sequence: 27, 41, 68, 109f
Badness: 0.026833
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 154/153, 170/169, 190/189, 196/195, 209/208, 245/243
Mapping: [⟨1 1 1 2 5 4 6 3], ⟨0 8 18 11 -21 -4 -26 17]]
Optimal tunings:
- CTE: ~2 = 1\1, ~19/18 = 88.093
- POTE ~2 = 1\1, ~19/18 = 88.113
Optimal ET sequence: 27, 41, 68, 109f
Badness: 0.020511
Devisemi (2.3.5.7.19)
2.3.5.19 subgroup
Subgroup: 2.3.5.19
Comma list: 361/360, 20000/19683
Gencom: [2 19/18; 361/360 20000/19683]
Gencom mapping: [⟨1 1 1 0 0 0 0 3], ⟨0 8 18 0 0 0 0 17]]
Sval mapping: [⟨1 1 1 3], ⟨0 8 18 17]]
POL2 generator: ~19/18 = 88.077
Optimal ET sequence: 14c, 27, 41, 68, 109
RMS error: 0.5701 cents
2.3.5.7.19 subgroup
Subgroup: 2.3.5.7.19
Comma list: 190/189, 245/243, 361/360
Gencom: [2 19/18; 190/189 245/243 361/360]
Gencom mapping: [⟨1 1 1 2 0 0 0 3], ⟨0 8 18 11 0 0 0 17]]
Sval mapping: [⟨1 1 1 2 3], ⟨0 8 18 11 17]]
POL2 generator: ~19/18 = 88.075
Optimal ET sequence: 14c, 27, 41, 68, 109
RMS error: 0.5780 cents
Dodecacot
Subgroup: 2.3.5.7
Comma list: 3125/3087, 10976/10935
Mapping: [⟨1 1 1 1], ⟨0 12 27 37]]
Wedgie: ⟨⟨ 12 27 37 15 25 10 ]]
Optimal tunings:
- CTE: ~2 = 1\1, ~28/27 = 58.648
- POTE:~2 = 1\1, ~28/27 = 58.675
Optimal ET sequence: 41, 143d, 184, 225, 266c
Badness: 0.119761
11-limit
Subgroup: 2.3.5.7.11
Comma list: 100/99, 243/242, 1375/1372
Mapping: [⟨1 1 1 1 2], ⟨0 12 27 37 30]]
Optimal tunings:
- CTE: ~2 = 1\1, ~28/27 = 58.602
- POTE ~2 = 1\1, ~28/27 = 58.665
Optimal ET sequence: 20cde, 41
Badness: 0.059528
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 196/195, 243/242, 275/273
Mapping: [⟨1 1 1 1 2 2], ⟨0 12 27 37 30 35]]
Optimal tunings:
- CTE: ~2 = 1\1, ~27/26 = 58.551
- POTE ~2 = 1\1, ~27/26 = 58.639
Optimal ET sequence: 20cdef, 21cdef, 41
Badness: 0.043645
Byhearted
- See also: Jubilismic clan and No-fives subgroup temperaments #Byhearted
Subgroup: 2.3.5.7
Comma list: 50/49, 19683/19208
Mapping: [⟨2 2 2 3], ⟨0 4 9 9]]
- Mapping generators: ~7/5, ~10/9
Wedgie: ⟨⟨ 8 18 18 10 6 -9 ]]
Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.472
Optimal ET sequence: 14c, 34d, 48, 82d, 130cdd
Badness: 0.111574
11-limit
Subgroup: 2.3.5.7.11
Comma list: 50/49, 99/98, 243/242
Mapping: [⟨2 2 2 3 4], ⟨0 4 9 9 10]]
Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.401
Optimal ET sequence: 14c, 34d, 48, 82d, 130cdd
Badness: 0.043923
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 78/77, 99/98, 243/242
Mapping: [⟨2 2 2 3 4 3], ⟨0 4 9 9 10 15]]
Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.586
Optimal ET sequence: 14cf, 34d, 48f, 82d
Badness: 0.031948
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 78/77, 85/84, 99/98, 243/242
Mapping: [⟨2 2 2 3 4 3 7], ⟨0 4 9 9 10 15 4]]
Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.596
Optimal ET sequence: 14cf, 34d, 48f, 82d
Badness: 0.026119
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 50/49, 78/77, 85/84, 99/98, 135/133, 243/242
Mapping: [⟨2 2 2 3 4 3 7 5], ⟨0 4 9 9 10 15 4 12]]
Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.498
Optimal ET sequence: 14cf, 34dh, 48f, 82dh
Badness: 0.021029