Tetracot family: Difference between revisions

Revision as of 14:11, 20 November 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The parent of the tetracot family is tetracot, the 5-limit temperament tempering out the tetracot comma (ratio: 20000/19683, monzo: [5 -9 4⟩).

Tetracot

Main article: Tetracot

The generator of tetracot is ~10/9, and that four of these 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).

Subgroup: 2.3.5

Comma list: 20000/19683

Mapping: [⟨1 1 1], ⟨0 4 9]]

Optimal tunings:

CTE: ~2 = 1200.000 ¢, ~10/9 = 176.028 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 176.160 ¢

Minimax tuning:

5-odd-limit: ~10/9 = [-1/9 0 1/9⟩

unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 7, 20c, 27, 34, 75, 109

Badness (Smith): 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 14^1/26 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 and 11-limit bunya. 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 14^1/26 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 13-limit monkey and 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 = 1200.0000 ¢, ~10/9 = 175.7765 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 175.985 ¢

Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e

Badness (Smith): 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 = 1200.0000 ¢, ~10/9 = 175.8150 ¢
POTE ~2 = 1200.000 ¢, ~10/9 = 176.196 ¢

Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e, 109ef

Badness (Smith): 0.012311

2.3.5.13 subgroup

Subgroup: 2.3.5.13

Comma list: 325/324, 512/507

Sval mapping: [⟨1 1 1 4], ⟨0 4 9 -2]]

Optimal tunings:

CTE: ~2 = 1200.0000 ¢, ~10/9 = 176.0783 ¢
CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.2975 ¢

Optimal ET sequence: 7, 20c, 27, 34, 245bff, 279bfff

Badness (Smith): 0.0165

Monkey

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 875/864, 5120/5103

Mapping: [⟨1 1 1 5], ⟨0 4 9 -15]]

Optimal tunings:

CTE: ~2 = 1200.000 ¢, ~10/9 = 175.676 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 175.659 ¢

Optimal ET sequence: 7, 34, 41

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.598 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 175.570 ¢

Optimal ET sequence: 7, 34, 41

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.618 ¢
POTE ~2 = 1200.000 ¢, ~10/9 = 175.622 ¢

Optimal ET sequence: 7, 34, 41

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.754 ¢

Optimal ET sequence: 7, 34, 41

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.697 ¢

Optimal ET sequence: 7, 34, 41

Badness (Smith): 0.022158

Bunya

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 225/224, 15625/15309

Mapping: [⟨1 1 1 -1], ⟨0 4 9 26]]

Optimal tunings:

CTE: ~2 = 1200.000 ¢, ~10/9 = 175.785 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 175.741 ¢

Optimal ET sequence: 7d, …, 34d, 41, 116, 157c, 198c

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.738 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 175.777 ¢

Optimal ET sequence: 7d, …, 34d, 41, 116e

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.748 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 175.886 ¢

Optimal ET sequence: 7d, 34d, 41, 116ef

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.811 ¢

Optimal ET sequence: 34d, 41, 75e

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 175.802 ¢

Optimal ET sequence: 34dh, 41, 75e

Badness (Smith): 0.019460

Modus

See also: Tetracot

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 = 1200.000 ¢, ~10/9 = 176.818 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 177.203 ¢

Optimal ET sequence: 7, 20c, 27, 34d, 61d, 95dd

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.446 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 177.053 ¢

Optimal ET sequence: 7, 20ce, 27e, 34d

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.471 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 176.953 ¢

Optimal ET sequence: 7, 20ce, 27e, 34d

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.453 ¢

Optimal ET sequence: 7g, …, 27eg, 34d

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.538 ¢

Optimal ET sequence: 7g, …, 27eg, 34dh

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.990 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 177.200 ¢

Optimal ET sequence: 7, 20c, 27, 34de, 61dee

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 177.017 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 177.197 ¢

Optimal ET sequence: 7, 20c, 27, 34de, 61dee

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 177.378 ¢

Optimal ET sequence: 7, 20c, 27g

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 177.505 ¢

Optimal ET sequence: 7, 20c, 27g

Badness (Smith): 0.028026

Wollemia

See also: Tetracot

Subgroup: 2.3.5.7

Comma list: 126/125, 2240/2187

Mapping: [⟨1 1 1 0], ⟨0 4 9 19]]

Optimal tunings:

CTE: ~2 = 1200.000 ¢, ~10/9 = 176.900 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 177.357 ¢

Optimal ET sequence: 7d, 20cd, 27, 34, 61, 95d, 156bcd

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.704 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 177.413 ¢

Optimal ET sequence: 7d, 20cde, 27e, 34, 95dee

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.716 ¢
POTE: ~2 = 1200.000 ¢, ~10/9 = 177.231 ¢

Optimal ET sequence: 7d, 20cde, 27e, 34, 95dee

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.641 ¢

Optimal ET sequence: 27eg, 34

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 176.749 ¢

Optimal ET sequence: 27eg, 34, 95deegh

Badness (Smith): 0.021108

Octacot

See also: Chords of 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. 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]]

Optimal tunings:

CTE: ~2 = 1200.000 ¢, ~21/20 = 88.023 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 88.076 ¢

Optimal ET sequence: 14c, 27, 41, 68, 109, 150, 259, 409bc, 668bbcc

Badness (Smith): 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 = 1200.000 ¢, ~21/20 = 87.910 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 87.975 ¢

Optimal ET sequence: 14c, 27e, 41, 150ee, 191ee, 232cee

Badness (Smith): 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 = 1200.000 ¢, ~21/20 = 87.926 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 88.106 ¢

Optimal ET sequence: 14c, 27e, 41, 150eef, 191eeff, 232ceeff

Badness (Smith): 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 = 1200.000 ¢, ~18/17 = 87.842 ¢
POTE: ~2 = 1200.000 ¢, ~18/17 = 88.102 ¢

Optimal ET sequence: 14c, 27eg, 41

Badness (Smith): 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 = 1200.000 ¢, ~18/17 = 87.866 ¢
POTE: ~2 = 1200.000 ¢, ~18/17 = 88.111 ¢

Optimal ET sequence: 14c, 27eg, 41

Badness (Smith): 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 = 1200.000 ¢, ~21/20 = 88.090 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 88.179 ¢

Optimal ET sequence: 14cf, 27e, 41f, 68ef, 109eff

Badness (Smith): 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 = 1200.000 ¢, ~18/17 = 88.011 ¢

Optimal ET sequence: 14cf, 27eg, 41f, 109effgg

Badness (Smith): 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 = 1200.000 ¢, ~18/17 = 88.017 ¢

Optimal ET sequence: 14cf, 27eg, 41f, 109effgg

Badness (Smith): 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 = 1200.000 ¢, ~21/20 = 87.770 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 87.697 ¢

Optimal ET sequence: 14cf, 27eff, 41

Badness (Smith): 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 = 1200.000 ¢, ~18/17 = 87.728 ¢

Optimal ET sequence: 14cf, 27effg, 41

Badness (Smith): 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 = 1200.000 ¢, ~18/17 = 87.750 ¢

Optimal ET sequence: 14cf, 27effg, 41

Badness (Smith): 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 = 1200.000 ¢, ~13/9 = 643.916 ¢
POTE: ~2 = 1200.000 ¢, ~13/9 = 643.989 ¢

Optimal ET sequence: 13cdeef, 28ccdef, 41

Badness (Smith): 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 = 1200.000 ¢, ~21/20 = 88.026 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 88.035 ¢

Optimal ET sequence: 27, 41, 68, 109, 150, 259

Badness (Smith): 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 = 1200.000 ¢, ~21/20 = 88.041 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 88.075 ¢

Optimal ET sequence: 27, 41, 68, 109f

Badness (Smith): 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 = 1200.000 ¢, ~21/20 = 88.093 ¢
POTE: ~2 = 1200.000 ¢, ~21/20 = 88.104 ¢

Optimal ET sequence: 27, 41, 68, 109f

Badness (Smith): 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 = 1200.000 ¢, ~19/18 = 88.093 ¢
POTE: ~2 = 1200.000 ¢, ~19/18 = 88.113 ¢

Optimal ET sequence: 27, 41, 68, 109f

Badness (Smith): 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]]

Optimal tunings:

CTE: ~2 = 1200.000 ¢, ~28/27 = 58.648 ¢
POTE: ~2 = 1200.000 ¢, ~28/27 = 58.675 ¢

Optimal ET sequence: 41, 143d, 184, 225, 266c

Badness (Smith): 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 = 1200.000 ¢, ~28/27 = 58.602 ¢
POTE: ~2 = 1200.000 ¢, ~28/27 = 58.665 ¢

Optimal ET sequence: 20cde, 41

Badness (Smith): 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 = 1200.000 ¢, ~27/26 = 58.551 ¢
POTE: ~2 = 1200.000 ¢, ~27/26 = 58.639 ¢

Optimal ET sequence: 20cdef, 21cdef, 41

Badness: 0.043645

Byhearted

See also: 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

Optimal tuning (CTE): ~7/5 = 600.000 ¢, ~10/9 = 175.472 ¢

Optimal ET sequence: 14c, 34d, 48, 82d, 130cdd

Badness (Smith): 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 = 600.000 ¢, ~10/9 = 175.401 ¢

Optimal ET sequence: 14c, 34d, 48, 82d, 130cdd

Badness (Smith): 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 = 600.000 ¢, ~10/9 = 175.586 ¢

Optimal ET sequence: 14cf, 34d, 48f, 82d

Badness (Smith): 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 = 600.000 ¢, ~10/9 = 175.596 ¢

Optimal ET sequence: 14cf, 34d, 48f, 82d

Badness (Smith): 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 = 600.000 ¢, ~10/9 = 175.498 ¢

Optimal ET sequence: 14cf, 34dh, 48f, 82dh

Badness (Smith): 0.021029

References

↑ Yahoo! Tuning Group | Guaranteed meantone successor

[1] Yahoo! Tuning Group | Guaranteed meantone successor

[1]

@@ Line 16: / Line 16: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~10/9 = 176.028
+* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.028{{c}}
-* [[POTE]]:~2 = 1\1, ~10/9 = 176.160
+* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.160{{c}}
 [[Minimax tuning]]:
@@ Line 25: / Line 25: @@
 {{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 75, 109 }}
-[[Badness]]: 0.048518
+[[Badness]] (Smith): 0.048518
 === Overview to extensions ===
@@ Line 54: / Line 54: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 175.7765
+* CTE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7765{{c}}
-* POTE ~2 = 1\1, ~10/9 = 175.985
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.985{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20ce, 27e, 34, 41, 75e }}
+{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}
-Badness: 0.014706
+Badness (Smith): 0.014706
 ==== 2.3.5.11.13 subgroup ====
@@ Line 69: / Line 69: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 175.8150
+* CTE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8150{{c}}
-* POTE ~2 = 1\1, ~10/9 = 176.196
+* POTE ~2 = 1200.000{{c}}, ~10/9 = 176.196{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20ce, 27e, 34, 41, 75e, 109ef }}
+{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}
-Badness: 0.012311
+Badness (Smith): 0.012311
 === 2.3.5.13 subgroup ===
-[[Subgroup]]: 2.3.5.13
+Subgroup: 2.3.5.13
-[[Comma list]]: 325/324, 512/507
+Comma list: 325/324, 512/507
-{{Mapping|legend=1| 1 1 1 4 | 0 4 9 -2 }}
+Sval mapping: {{mapping| 1 1 1 4 | 0 4 9 -2 }}
-[[Optimal tuning]]s:
+Optimal tunings:
-* [[CTE]]: ~2 = 1\1, ~10/9 = 176.0783
+* CTE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0783{{c}}
-* [[CWE]]: ~2 = 1\1, ~10/9 = 176.2975
+* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.2975{{c}}
-{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 245bff, 279bfff }}
+{{Optimal ET sequence|legend=0| 7, 20c, 27, 34, 245bff, 279bfff }}
-[[Badness]]: 0.0165
+Badness (Smith): 0.0165
 == Monkey ==
@@ Line 100: / Line 100: @@
 {{Mapping|legend=1| 1 1 1 5 | 0 4 9 -15 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1\1, ~10/9 = 175.676
+* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.676{{c}}
-* POTE:~2 = 1\1, ~10/9 = 175.659
+* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.659{{c}}
 {{Optimal ET sequence|legend=1| 7, 34, 41 }}
-[[Badness]]: 0.073437
+[[Badness]] (Smith): 0.073437
 === 11-limit ===
@@ Line 116: / Line 116: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 175.598
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.598{{c}}
-* POTE ~2 = 1\1, ~10/9 = 175.570
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.570{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 34, 41 }}
+{{Optimal ET sequence|legend=0| 7, 34, 41 }}
-Badness: 0.038836
+Badness (Smith): 0.038836
 === 13-limit ===
@@ Line 131: / Line 131: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 175.618
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.618{{c}}
-* POTE ~2 = 1\1, ~10/9 = 175.622
+* POTE ~2 = 1200.000{{c}}, ~10/9 = 175.622{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 34, 41 }}
+{{Optimal ET sequence|legend=0| 7, 34, 41 }}
-Badness: 0.028410
+Badness (Smith): 0.028410
 === 17-limit ===
@@ Line 145: / Line 145: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 175.754{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 34, 41 }}
+{{Optimal ET sequence|legend=0| 7, 34, 41 }}
-Badness: 0.025936
+Badness (Smith): 0.025936
 === 19-limit ===
@@ Line 158: / Line 158: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 175.697{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 34, 41 }}
+{{Optimal ET sequence|legend=0| 7, 34, 41 }}
-Badness: 0.022158
+Badness (Smith): 0.022158
 == Bunya ==
@@ Line 173: / Line 173: @@
 {{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1\1, ~10/9 = 175.785
+* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.785{{c}}
-* POTE:~2 = 1\1, ~10/9 = 175.741
+* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 175.741{{c}}
 {{Optimal ET sequence|legend=1| 7d, …, 34d, 41, 116, 157c, 198c }}
-[[Badness]]: 0.062897
+[[Badness]] (Smith): 0.062897
 === 11-limit ===
@@ Line 189: / Line 189: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 175.738
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.738{{c}}
-* POTE ~2 = 1\1, ~10/9 = 175.777
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.777{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7d, …, 34d, 41, 116e }}
+{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}
-Badness: 0.031332
+Badness (Smith): 0.031332
 === 13-limit ===
@@ Line 204: / Line 204: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 175.748
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 175.748{{c}}
-* POTE ~2 = 1\1, ~10/9 = 175.886
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 175.886{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7d, 34d, 41, 116ef }}
+{{Optimal ET sequence|legend=0| 7d, 34d, 41, 116ef }}
-Badness: 0.024886
+Badness (Smith): 0.024886
 === 17-limit ===
@@ Line 218: / Line 218: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 175.811{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 34d, 41, 75e }}
+{{Optimal ET sequence|legend=0| 34d, 41, 75e }}
-Badness: 0.023404
+Badness (Smith): 0.023404
 === 19-limit ===
@@ Line 231: / Line 231: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 175.802{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 34dh, 41, 75e }}
+{{Optimal ET sequence|legend=0| 34dh, 41, 75e }}
-Badness: 0.019460
+Badness (Smith): 0.019460
 == Modus ==
@@ Line 248: / Line 248: @@
 {{Mapping|legend=1| 1 1 1 4 | 0 4 9 -8 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1\1, ~10/9 = 176.818
+* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.818{{c}}
-* POTE: ~2 = 1\1, ~10/9 = 177.203
+* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 177.203{{c}}
 {{Optimal ET sequence|legend=1| 7, 20c, 27, 34d, 61d, 95dd }}
-[[Badness]]: 0.068184
+[[Badness]] (Smith): 0.068184
 === 11-limit ===
@@ Line 264: / Line 264: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 176.446
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.446{{c}}
-* POTE ~2 = 1\1, ~10/9 = 177.053
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.053{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20ce, 27e, 34d }}
+{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d }}
-Badness: 0.035149
+Badness (Smith): 0.035149
 ==== 13-limit ====
@@ Line 279: / Line 279: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 176.471
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.471{{c}}
-* POTE ~2 = 1\1, ~10/9 = 176.953
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 176.953{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20ce, 27e, 34d }}
+{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d }}
-Badness: 0.023806
+Badness (Smith): 0.023806
 ==== 17-limit ====
@@ Line 293: / Line 293: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 176.453{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7g, …, 27eg, 34d }}
+{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34d }}
-Badness: 0.021501
+Badness (Smith): 0.021501
 ==== 19-limit ====
@@ Line 306: / Line 306: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 176.538{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7g, …, 27eg, 34dh }}
+{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh }}
-Badness: 0.017941
+Badness (Smith): 0.017941
 ; Music
@@ Line 327: / Line 327: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 176.990
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.990{{c}}
-* POTE ~2 = 1\1, ~10/9 = 177.200
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.200{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20c, 27, 34de, 61dee }}
+{{Optimal ET sequence|legend=0| 7, 20c, 27, 34de, 61dee }}
-Badness: 0.063077
+Badness (Smith): 0.063077
 ==== 13-limit ====
@@ Line 342: / Line 342: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 177.017
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 177.017{{c}}
-* POTE ~2 = 1\1, ~10/9 = 177.197
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.197{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20c, 27, 34de, 61dee }}
+{{Optimal ET sequence|legend=0| 7, 20c, 27, 34de, 61dee }}
-Badness: 0.039043
+Badness (Smith): 0.039043
 ==== 17-limit ====
@@ Line 356: / Line 356: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 177.378{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20c, 27g }}
+{{Optimal ET sequence|legend=0| 7, 20c, 27g }}
-Badness: 0.035200
+Badness (Smith): 0.035200
 ==== 19-limit ====
@@ Line 369: / Line 369: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 177.505{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7, 20c, 27g }}
+{{Optimal ET sequence|legend=0| 7, 20c, 27g }}
-Badness: 0.028026
+Badness (Smith): 0.028026
 == Wollemia ==
@@ Line 384: / Line 384: @@
 {{Mapping|legend=1| 1 1 1 0 | 0 4 9 19 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1\1, ~10/9 = 176.900
+* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 176.900{{c}}
-* POTE:~2 = 1\1, ~10/9 = 177.357
+* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 177.357{{c}}
 {{Optimal ET sequence|legend=1| 7d, 20cd, 27, 34, 61, 95d, 156bcd }}
-[[Badness]]: 0.070522
+[[Badness]] (Smith): 0.070522
 === 11-limit ===
@@ Line 400: / Line 400: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 176.704
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.704{{c}}
-* POTE ~2 = 1\1, ~10/9 = 177.413
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.413{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7d, 20cde, 27e, 34, 95dee }}
+{{Optimal ET sequence|legend=0| 7d, 20cde, 27e, 34, 95dee }}
-Badness: 0.037551
+Badness (Smith): 0.037551
 === 13-limit ===
@@ Line 415: / Line 415: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~10/9 = 176.716
+* CTE: ~2 = 1200.000{{c}}, ~10/9 = 176.716{{c}}
-* POTE ~2 = 1\1, ~10/9 = 177.231
+* POTE: ~2 = 1200.000{{c}}, ~10/9 = 177.231{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7d, 20cde, 27e, 34, 95dee }}
+{{Optimal ET sequence|legend=0| 7d, 20cde, 27e, 34, 95dee }}
-Badness: 0.031219
+Badness (Smith): 0.031219
 === 17-limit ===
@@ Line 429: / Line 429: @@
 Mapping: {{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
+Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 176.641{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27eg, 34 }}
+{{Optimal ET sequence|legend=0| 27eg, 34 }}
-Badness: 0.024471
+Badness (Smith): 0.024471
 === 19-limit ===
@@ Line 442: / Line 442: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~10/9 = 176.749{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27eg, 34, 95deegh }}
+{{Optimal ET sequence|legend=0| 27eg, 34, 95deegh }}
-Badness: 0.021108
+Badness (Smith): 0.021108
 == Octacot ==
@@ Line 461: / Line 461: @@
 {{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1\1, ~21/20 = 88.023
+* [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.023{{c}}
-* POTE: ~2 = 1\1, ~21/20 = 88.076
+* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 88.076{{c}}
 {{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109, 150, 259, 409bc, 668bbcc }}
-[[Badness]]: 0.033845
+[[Badness]] (Smith): 0.033845
 === 11-limit ===
@@ Line 477: / Line 477: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~21/20 = 87.910
+* CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.910{{c}}
-* POTE: ~2 = 1\1, ~21/20 = 87.975
+* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.975{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150ee, 191ee, 232cee }}
+{{Optimal ET sequence|legend=0| 14c, 27e, 41, 150ee, 191ee, 232cee }}
-Badness: 0.024078
+Badness (Smith): 0.024078
 ==== 13-limit ====
@@ Line 492: / Line 492: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~21/20 = 87.926
+* CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.926{{c}}
-* POTE: ~2 = 1\1, ~21/20 = 88.106
+* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.106{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150eef, 191eeff, 232ceeff }}
+{{Optimal ET sequence|legend=0| 14c, 27e, 41, 150eef, 191eeff, 232ceeff }}
-Badness: 0.023276
+Badness (Smith): 0.023276
 ===== 17-limit =====
@@ Line 507: / Line 507: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~18/17 = 87.842
+* CTE: ~2 = 1200.000{{c}}, ~18/17 = 87.842{{c}}
-* POTE: ~2 = 1\1, ~18/17 = 88.102
+* POTE: ~2 = 1200.000{{c}}, ~18/17 = 88.102{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}
+{{Optimal ET sequence|legend=0| 14c, 27eg, 41 }}
-Badness: 0.021088
+Badness (Smith): 0.021088
 ===== 19-limit =====
@@ Line 522: / Line 522: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~18/17 = 87.866
+* CTE: ~2 = 1200.000{{c}}, ~18/17 = 87.866{{c}}
-* POTE: ~2 = 1\1, ~18/17 = 88.111
+* POTE: ~2 = 1200.000{{c}}, ~18/17 = 88.111{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}
+{{Optimal ET sequence|legend=0| 14c, 27eg, 41 }}
-Badness: 0.016652
+Badness (Smith): 0.016652
 ==== Octocat ====
@@ Line 537: / Line 537: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~21/20 = 88.090
+* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.090{{c}}
-* POTE: ~2 = 1\1, ~21/20 = 88.179
+* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.179{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27e, 41f, 68ef, 109eff }}
+{{Optimal ET sequence|legend=0| 14cf, 27e, 41f, 68ef, 109eff }}
-Badness: 0.027601
+Badness (Smith): 0.027601
 ===== 17-limit =====
@@ Line 551: / Line 551: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 88.011{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27eg, 41f, 109effgg }}
+{{Optimal ET sequence|legend=0| 14cf, 27eg, 41f, 109effgg }}
-Badness: 0.023345
+Badness (Smith): 0.023345
 ===== 19-limit =====
@@ Line 564: / Line 564: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 88.017{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27eg, 41f, 109effgg }}
+{{Optimal ET sequence|legend=0| 14cf, 27eg, 41f, 109effgg }}
-Badness: 0.017916
+Badness (Smith): 0.017916
 ==== Octopod ====
@@ Line 578: / Line 578: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~21/20 = 87.770
+* CTE: ~2 = 1200.000{{c}}, ~21/20 = 87.770{{c}}
-* POTE ~2 = 1\1, ~21/20 = 87.697
+* POTE: ~2 = 1200.000{{c}}, ~21/20 = 87.697{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27eff, 41 }}
+{{Optimal ET sequence|legend=0| 14cf, 27eff, 41 }}
-Badness: 0.028326
+Badness (Smith): 0.028326
 ===== 17-limit =====
@@ Line 592: / Line 592: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 87.728{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27effg, 41 }}
+{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
-Badness: 0.024660
+Badness (Smith): 0.024660
 ===== 19-limit =====
@@ Line 605: / Line 605: @@
 Mapping: {{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 tuning (CTE): ~2 = 1200.000{{c}}, ~18/17 = 87.750{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 27effg, 41 }}
+{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
-Badness: 0.019504
+Badness (Smith): 0.019504
 ==== Dificot ====
@@ Line 619: / Line 619: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~13/9 = 643.916
+* CTE: ~2 = 1200.000{{c}}, ~13/9 = 643.916{{c}}
-* POTE ~2 = 1\1, ~13/9 = 643.989
+* POTE: ~2 = 1200.000{{c}}, ~13/9 = 643.989{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 13cdeef, 28ccdef, 41 }}
+{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}
-Badness: 0.051876
+Badness (Smith): 0.051876
 === October ===
@@ Line 634: / Line 634: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~21/20 = 88.026
+* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.026{{c}}
-* POTE ~2 = 1\1, ~21/20 = 88.035
+* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.035{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27, 41, 68, 109, 150, 259 }}
+{{Optimal ET sequence|legend=0| 27, 41, 68, 109, 150, 259 }}
-Badness: 0.039643
+Badness (Smith): 0.039643
 ==== 13-limit ====
@@ Line 649: / Line 649: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~21/20 = 88.041
+* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.041{{c}}
-* POTE ~2 = 1\1, ~21/20 = 88.075
+* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.075{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27, 41, 68, 109f }}
+{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
-Badness: 0.031136
+Badness (Smith): 0.031136
 ==== 17-limit ====
@@ Line 664: / Line 664: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~21/20 = 88.093
+* CTE: ~2 = 1200.000{{c}}, ~21/20 = 88.093{{c}}
-* POTE ~2 = 1\1, ~21/20 = 88.104
+* POTE: ~2 = 1200.000{{c}}, ~21/20 = 88.104{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27, 41, 68, 109f }}
+{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
-Badness: 0.026833
+Badness (Smith): 0.026833
 ==== 19-limit ====
@@ Line 679: / Line 679: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~19/18 = 88.093
+* CTE: ~2 = 1200.000{{c}}, ~19/18 = 88.093{{c}}
-* POTE ~2 = 1\1, ~19/18 = 88.113
+* POTE: ~2 = 1200.000{{c}}, ~19/18 = 88.113{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 27, 41, 68, 109f }}
+{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}
-Badness: 0.020511
+Badness (Smith): 0.020511
 === Devisemi (2.3.5.7.19) ===
@@ Line 728: / Line 728: @@
 {{Mapping|legend=1| 1 1 1 1 | 0 12 27 37 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1\1, ~28/27 = 58.648
+* [[CTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.648{{c}}
-* POTE: ~2 = 1\1, ~28/27 = 58.675
+* [[POTE]]: ~2 = 1200.000{{c}}, ~28/27 = 58.675{{c}}
 {{Optimal ET sequence|legend=1| 41, 143d, 184, 225, 266c }}
-[[Badness]]: 0.119761
+[[Badness]] (Smith): 0.119761
 === 11-limit ===
@@ Line 744: / Line 744: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~28/27 = 58.602
+* CTE: ~2 = 1200.000{{c}}, ~28/27 = 58.602{{c}}
-* POTE: ~2 = 1\1, ~28/27 = 58.665
+* POTE: ~2 = 1200.000{{c}}, ~28/27 = 58.665{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 20cde, 41 }}
+{{Optimal ET sequence|legend=0| 20cde, 41 }}
-Badness: 0.059528
+Badness (Smith): 0.059528
 === 13-limit ===
@@ Line 759: / Line 759: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~27/26 = 58.551
+* CTE: ~2 = 1200.000{{c}}, ~27/26 = 58.551{{c}}
-* POTE: ~2 = 1\1, ~27/26 = 58.639
+* POTE: ~2 = 1200.000{{c}}, ~27/26 = 58.639{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 20cdef, 21cdef, 41 }}
+{{Optimal ET sequence|legend=0| 20cdef, 21cdef, 41 }}
 Badness: 0.043645
@@ Line 776: / Line 776: @@
 : mapping generators: ~7/5, ~10/9
-Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.472
+[[Optimal tuning]] ([[CTE]]): ~7/5 = 600.000{{c}}, ~10/9 = 175.472{{c}}
 {{Optimal ET sequence|legend=1| 14c, 34d, 48, 82d, 130cdd }}
-[[Badness]]: 0.111574
+[[Badness]] (Smith): 0.111574
 === 11-limit ===
@@ Line 789: / Line 789: @@
 Mapping: {{mapping| 2 2 2 3 4 | 0 4 9 9 10 }}
-Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.401
+Optimal tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.401{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14c, 34d, 48, 82d, 130cdd }}
+{{Optimal ET sequence|legend=0| 14c, 34d, 48, 82d, 130cdd }}
-Badness: 0.043923
+Badness (Smith): 0.043923
 === 13-limit ===
@@ Line 802: / Line 802: @@
 Mapping: {{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 tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.586{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 34d, 48f, 82d }}
+{{Optimal ET sequence|legend=0| 14cf, 34d, 48f, 82d }}
-Badness: 0.031948
+Badness (Smith): 0.031948
 === 17-limit ===
@@ Line 815: / Line 815: @@
 Mapping: {{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 tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.596{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 34d, 48f, 82d }}
+{{Optimal ET sequence|legend=0| 14cf, 34d, 48f, 82d }}
-Badness: 0.026119
+Badness (Smith): 0.026119
 === 19-limit ===
@@ Line 828: / Line 828: @@
 Mapping: {{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 tuning (CTE): ~7/5 = 600.000{{c}}, ~10/9 = 175.498{{c}}
+{{Optimal ET sequence|legend=0| 14cf, 34dh, 48f, 82dh }}
-Optimal ET sequence: {{Optimal ET sequence| 14cf, 34dh, 48f, 82dh }}
+Badness (Smith): 0.021029
-Badness: 0.021029
+== References ==
 [[Category:Temperament families]]

Tetracot family: Difference between revisions

Revision as of 14:11, 20 November 2025

Tetracot

Overview to extensions

Monkey and bunya

2.3.5.11 subgroup

2.3.5.11.13 subgroup

2.3.5.13 subgroup

Monkey

11-limit

13-limit

17-limit

19-limit

Bunya

11-limit

13-limit

17-limit

19-limit

Modus

11-limit

13-limit

17-limit

19-limit

Ponens

13-limit

17-limit

19-limit

Wollemia

11-limit

13-limit

17-limit

19-limit

Octacot

11-limit

13-limit

17-limit

19-limit

Octocat

17-limit

19-limit

Octopod

17-limit

19-limit

Dificot

October

13-limit

17-limit

19-limit

Devisemi (2.3.5.7.19)

2.3.5.19 subgroup

2.3.5.7.19 subgroup

Dodecacot

11-limit

13-limit

Byhearted

11-limit

13-limit

17-limit

19-limit

References

Navigation menu

Search