Tetracot family: Difference between revisions
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Badness (Sintel): 1.32 | Badness (Sintel): 1.32 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
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Badness (Sintel): 1.33 | Badness (Sintel): 1.33 | ||
=== 19-limit === | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 843: | Line 843: | ||
Badness (Sintel): 1.28 | Badness (Sintel): 1.28 | ||
=== Weasly === | |||
{{Todo|review|unify precision}} | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 50/49, 99/98, 144/143, 243/242 | |||
Mapping: {{mapping| 2 2 2 3 4 8 | 0 4 9 9 10 -2 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 599.285{{c}}, ~10/9 = 175.641{{c}} | |||
* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.728{{c}} | |||
{{Optimal ET sequence|legend=0| 14c, 20cde, 34d, 48 }} | |||
Badness (Sintel): 1.72 | |||
==== 17-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 50/49, 85/84, 99/98, 144/143, 243/242 | |||
Mapping: {{mapping| 2 2 2 3 4 8 7 | 0 4 9 9 10 -2 4 }} | |||
Optimal tunings: | |||
* WE: ~7/5 = 599.494{{c}}, ~10/9 = 175.613{{c}} | |||
* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.681{{c}} | |||
{{Optimal ET sequence|legend=0| 14c, 20cde, 34d, 48 }} | |||
Badness (Sintel): 1.54 | |||
==== 19-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 50/49, 85/84, 99/98, 144/143, 190/189, 243/242 | |||
Mapping: {{mapping| 2 2 2 3 4 8 7 5 | 0 4 9 9 10 -2 4 12}} | |||
Optimal tunings: | |||
* WE: ~7/5 = 599.464{{c}}, ~10/9 = 175.523{{c}} | |||
* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.593{{c}} | |||
{{Optimal ET sequence|legend=0| 14c, 20cdehh, 34dh, 48 }} | |||
Badness (Sintel): 1.48 | |||
== Other subgroup extensions == | == Other subgroup extensions == | ||
Revision as of 22:39, 20 January 2026
- 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
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]]
- WE: ~2 = 1199.5586 ¢, ~10/9 = 176.0950 ¢
- error map: ⟨-0.441 +1.984 -1.900]
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.0965 ¢
- error map: ⟨0.000 +2.431 -1.445]
- 5-odd-limit: ~10/9 = [-1/9 0 1/9⟩
Optimal ET sequence: 7, 20c, 27, 34, 75, 109
Badness (Sintel): 1.14
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 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 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 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
Subgroup-val mapping: [⟨1 1 1 2], ⟨0 4 9 10]]
Optimal tunings:
- WE: ~2 = 1199.3274 ¢, ~10/9 = 175.8862 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.8847 ¢
Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e
Badness (Sintel): 0.459
2.3.5.11.13 subgroup
Subgroup: 2.3.5.11.13
Comma list: 100/99, 144/143, 243/242
Subgroup-val mapping: [⟨1 1 1 2 4], ⟨0 4 9 10 -2]]
Optimal tunings:
- WE: ~2 = 1198.6852 ¢, ~10/9 = 176.0034 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.0854 ¢
Optimal ET sequence: 7, 20ce, 27e, 34, 41, 75e, 109ef
Badness (Sintel): 0.489
2.3.5.13 subgroup
Subgroup: 2.3.5.13
Comma list: 325/324, 512/507
Subgroup-val mapping: [⟨1 1 1 4], ⟨0 4 9 -2]]
Optimal tunings:
- WE: ~2 = 1198.8502 ¢, ~10/9 = 176.2195 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.2975 ¢
Optimal ET sequence: 7, 20c, 27, 34, 245bff, 279bfff
Badness (Sintel): 0.551
Monkey
Subgroup: 2.3.5.7
Comma list: 875/864, 5120/5103
Mapping: [⟨1 1 1 5], ⟨0 4 9 -15]]
- WE: ~2 = 1200.7982 ¢, ~10/9 = 175.7757 ¢
- error map: ⟨+0.798 +1.946 -3.534 -1.470]
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.6622 ¢
- error map: ⟨0.000 +0.694 -5.354 -3.759]
Optimal ET sequence: 7, 34, 41
Badness (Sintel): 1.86
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:
- WE: ~2 = 1200.3988 ¢, ~10/9 = 175.6287 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.5750 ¢
Optimal ET sequence: 7, 34, 41
Badness (Sintel): 1.28
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:
- WE: ~2 = 1199.9206 ¢, ~10/9 = 175.6108 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.6217 ¢
Optimal ET sequence: 7, 34, 41
Badness (Sintel): 1.17
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 tunings:
- WE: ~2 = 1199.5029 ¢, ~10/9 = 175.6832 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.7558 ¢
Optimal ET sequence: 7, 34, 41
Badness (Sintel): 1.32
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 tunings:
- WE: ~2 = 1199.7318 ¢, ~10/9 = 175.6498 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.6901 ¢
Optimal ET sequence: 7, 34, 41
Badness (Sintel): 1.35
Bunya
Subgroup: 2.3.5.7
Comma list: 225/224, 15625/15309
Mapping: [⟨1 1 1 -1], ⟨0 4 9 26]]
- WE: ~2 = 1200.2991 ¢, ~10/9 = 175.7844 ¢
- error map: ⟨+0.299 +1.482 -3.955 +1.270]
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.7567 ¢
- error map: ⟨0.000 +1.072 -4.503 +0.849]
Optimal ET sequence: 7d, …, 34d, 41, 116, 157c, 198c
Badness (Sintel): 1.59
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:
- WE: ~2 = 1199.7481 ¢, ~10/9 = 175.7401 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.7637 ¢
Optimal ET sequence: 7d, …, 34d, 41, 116e
Badness (Sintel): 1.04
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:
- WE: ~2 = 1199.1044 ¢, ~10/9 = 175.7545 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.8526 ¢
Optimal ET sequence: 7d, 34d, 41, 116ef
Badness (Sintel): 1.03
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 tunings:
- WE: ~2 = 1198.7905 ¢, ~10/9 = 175.7757 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.9302 ¢
Optimal ET sequence: 34d, 41, 75e
Badness (Sintel): 1.19
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 tunings:
- WE: ~2 = 1198.7904 ¢, ~10/9 = 175.7755 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 175.9287 ¢
Optimal ET sequence: 34dh, 41, 75e
Badness (Sintel): 1.18
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]]
- WE: ~2 = 1196.7884 ¢, ~10/9 = 176.7292 ¢
- error map: ⟨-3.212 +1.750 +1.038 +4.494]
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1188 ¢
- error map: ⟨0.000 +6.520 +7.755 +14.224]
Optimal ET sequence: 7, 20c, 27, 61d, 88bcd, 149bccddd
Badness (Sintel): 1.73
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:
- WE: ~2 = 1196.4227 ¢, ~10/9 = 176.5252 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.9286 ¢
Optimal ET sequence: 7, 20ce, 27e, 34d, 61de
Badness (Sintel): 1.16
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:
- WE: ~2 = 1196.8686 ¢, ~10/9 = 176.4915 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.8735 ¢
Optimal ET sequence: 7, 20ce, 27e, 34d, 61de
Badness (Sintel): 0.984
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 tunings:
- WE: ~2 = 1196.8783 ¢, ~10/9 = 176.5241 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.8969 ¢
Optimal ET sequence: 7g, …, 27eg, 34d
Badness (Sintel): 1.10
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 tunings:
- WE: ~2 = 1196.6939 ¢, ~10/9 = 176.5426 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 176.9645 ¢
Optimal ET sequence: 7g, …, 27eg, 34dh, 61degh
Badness (Sintel): 1.09
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:
- WE: ~2 = 1198.5026 ¢, ~10/9 = 176.9786 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1589 ¢
Optimal ET sequence: 7, 20c, 27
Badness (Sintel): 2.09
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:
- WE: ~2 = 1198.5149 ¢, ~10/9 = 176.9778 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1681 ¢
Optimal ET sequence: 7, 20c, 27
Badness (Sintel): 1.61
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 tunings:
- WE: ~2 = 1197.4542 ¢, ~10/9 = 177.1828 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.5355 ¢
Badness (Sintel): 1.79
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 tunings:
- WE: ~2 = 1197.3233 ¢, ~10/9 = 177.2025 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.5878 ¢
Badness (Sintel): 1.70
Wollemia
Subgroup: 2.3.5.7
Comma list: 126/125, 2240/2187
Mapping: [⟨1 1 1 0], ⟨0 4 9 19]]
- WE: ~2 = 1197.6555 ¢, ~10/9 = 177.0104 ¢
- error map: ⟨-2.345 +3.742 +4.435 -5.628]
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1667 ¢
- error map: ⟨0.000 +6.712 +8.186 -2.659]
Optimal ET sequence: 7d, 20cd, 27, 61, 88bc, 115bc
Badness (Sintel): 1.78
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:
- WE: ~2 = 1196.6462 ¢, ~10/9 = 176.9174 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1370 ¢
Optimal ET sequence: 7d, 20cde, 27e
Badness (Sintel): 1.24
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:
- WE: ~2 = 1197.4576 ¢, ~10/9 = 176.8557 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.0949 ¢
Optimal ET sequence: 7d, 20cde, 27e
Badness (Sintel): 1.29
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 tunings:
- WE: ~2 = 1197.4770 ¢, ~10/9 = 176.7733 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.0123 ¢
Optimal ET sequence: 7dg, 27eg
Badness (Sintel): 1.25
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 tunings:
- WE: ~2 = 1197.4380 ¢, ~10/9 = 176.8774 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/9 = 177.1216 ¢
Optimal ET sequence: 7dgh, 27eg
Badness (Sintel): 1.28
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]]
- WE: ~2 = 1199.6782 ¢, ~21/20 = 88.0528 ¢
- error map: ⟨-0.322 +2.145 -1.686 -0.889]
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0525 ¢
- error map: ⟨0.000 +2.465 -1.369 -0.248]
Optimal ET sequence: 14c, 27, 41, 68, 109
Badness (Sintel): 0.857
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:
- WE: ~2 = 1199.6025 ¢, ~21/20 = 87.9460 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.9453 ¢
Optimal ET sequence: 14c, 27e, 41, 109e
Badness (Sintel): 0.796
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:
- WE: ~2 = 1198.8609 ¢, ~21/20 = 87.0219 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0557 ¢
Optimal ET sequence: 14c, 27e, 41, 68e, 109ef
Badness (Sintel): 0.962
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:
- WE: ~2 = 1198.4494 ¢, ~21/20 = 87.9878 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0324 ¢
Optimal ET sequence: 14c, 27eg, 41, 68egg
Badness (Sintel): 1.07
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:
- WE: ~2 = 1198.5995 ¢, ~20/19 = 88.0081 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.0471 ¢
Optimal ET sequence: 14c, 27eg, 41, 68egg
Badness (Sintel): 1.01
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:
- WE: ~2 = 1199.4441 ¢, ~21/20 = 88.1380 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.1375 ¢
Optimal ET sequence: 14cf, 27e, 41f
Badness (Sintel): 1.14
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 tunings:
- WE: ~2 = 1198.4257 ¢, ~21/20 = 88.1636 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.1642 ¢
Optimal ET sequence: 14cf, 27eg
Badness (Sintel): 1.19
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 tunings:
- WE: ~2 = 1198.5748 ¢, ~20/19 = 88.1631 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.1637 ¢
Optimal ET sequence: 14cf, 27eg
Badness (Sintel): 1.09
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:
- WE: ~2 = 1200.5116 ¢, ~21/20 = 87.7346 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.7257 ¢
Optimal ET sequence: 14cf, 27eff, 41
Badness (Sintel): 1.17
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 tunings:
- WE: ~2 = 1199.6667 ¢, ~21/20 = 87.7494 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.7559 ¢
Optimal ET sequence: 14cf, 27effg, 41
Badness (Sintel): 1.26
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 tunings:
- WE: ~2 = 1199.9909 ¢, ~20/19 = 87.7474 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/19 = 87.7476 ¢
Optimal ET sequence: 14cf, 27effg, 41
Badness (Sintel): 1.19
Dificot
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 243/242, 245/242, 343/338
Mapping: [⟨1 -7 -17 -9 -18 -14], ⟨0 16 36 22 40 33]]
- mapping generators: ~2, ~13/9
Optimal tunings:
- WE: ~2 = 1199.1496 ¢, ~13/9 = 643.5328 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/9 = 643.9567 ¢
Optimal ET sequence: 13cdeef, 28ccdef, 41
Badness (Sintel): 2.14
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:
- WE: ~2 = 1199.8843 ¢, ~21/20 = 88.0261 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0329 ¢
Optimal ET sequence: 27, 41, 68, 109, 150, 259
Badness (Sintel): 1.31
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:
- WE: ~2 = 1199.5060 ¢, ~21/20 = 88.0388 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.0697 ¢
Optimal ET sequence: 27, 41, 68, 109f
Badness (Sintel): 1.29
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:
- WE: ~2 = 1199.3845 ¢, ~21/20 = 88.0589 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 88.1027 ¢
Optimal ET sequence: 27, 41, 68, 109f
Badness (Sintel): 1.37
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:
- WE: ~2 = 1199.4449 ¢, ~20/19 = 88.0723 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.1107 ¢
Optimal ET sequence: 27, 41, 68, 109f
Badness (Sintel): 1.25
Dodecacot
Subgroup: 2.3.5.7
Comma list: 3125/3087, 10976/10935
Mapping: [⟨1 1 1 1], ⟨0 12 27 37]]
- mapping generators: ~2, ~28/27
- WE: ~2 = 1199.6912 ¢, ~28/27 = 58.6600 ¢
- error map: ⟨-0.309 +1.657 -2.802 +1.287]
- CWE: ~2 = 1200.0000 ¢, ~28/27 = 58.6624 ¢
- error map: ⟨0.000 +1.993 -2.430 +1.681]
Optimal ET sequence: 20cd, 41, 143d, 184, 225
Badness (Sintel): 3.03
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:
- WE: ~2 = 1199.3125 ¢, ~28/27 = 58.6317 ¢
- CWE: ~2 = 1200.0000 ¢, ~28/27 = 58.6360 ¢
Optimal ET sequence: 20cde, 41
Badness (Sintel): 1.97
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:
- WE: ~2 = 1199.0713 ¢, ~28/27 = 58.5932 ¢
- CWE: ~2 = 1200.0000 ¢, ~28/27 = 58.5982 ¢
Optimal ET sequence: 20cdef, 41
Badness (Sintel): 1.80
Weasel
Weasel, named by Mike Battaglia in 2012[2] and also known as byhearted[note 1], tempers out 50/49 and splits the octave in halves; its ploidacot is diploid tetracot.
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
- WE: ~7/5 = 599.6934 ¢, ~10/9 = 175.5626 ¢
- error map: ⟨-0.613 -0.318 -6.864 +10.318]
- CWE: ~7/5 = 1200.0000 ¢, ~10/9 = 175.5632 ¢
- error map: ⟨0.000 +0.298 -6.245 +11.243]
Optimal ET sequence: 14c, 34d, 48
Badness (Sintel): 2.82
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 tunings:
- WE: ~7/5 = 599.6525 ¢, ~10/9 = 175.5103 ¢
- CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.5086 ¢
Optimal ET sequence: 14c, 34d, 48
Badness (Sintel): 1.45
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 tunings:
- WE: ~7/5 = 599.4539 ¢, ~10/9 = 175.7393 ¢
- CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.7502 ¢
Optimal ET sequence: 14cf, 20cdef, 34d
Badness (Sintel): 1.32
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 tunings:
- WE: ~7/5 = 599.7509 ¢, ~10/9 = 175.6684 ¢
- CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.6839 ¢
Optimal ET sequence: 14cf, 20cdef, 34d
Badness (Sintel): 1.33
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 tunings:
- WE: ~7/5 = 599.6682 ¢, ~10/9 = 175.5994 ¢
- CWE: ~7/5 = 600.0000 ¢, ~10/9 = 175.6190 ¢
Optimal ET sequence: 14cf, 20cdefhh, 34dh, 48f
Badness (Sintel): 1.28
Weasly
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 99/98, 144/143, 243/242
Mapping: [⟨2 2 2 3 4 8], ⟨0 4 9 9 10 -2]]
Optimal tunings:
- WE: ~7/5 = 599.285 ¢, ~10/9 = 175.641 ¢
- CWE: ~7/5 = 600.000 ¢, ~10/9 = 175.728 ¢
Optimal ET sequence: 14c, 20cde, 34d, 48
Badness (Sintel): 1.72
17-limit
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 85/84, 99/98, 144/143, 243/242
Mapping: [⟨2 2 2 3 4 8 7], ⟨0 4 9 9 10 -2 4]]
Optimal tunings:
- WE: ~7/5 = 599.494 ¢, ~10/9 = 175.613 ¢
- CWE: ~7/5 = 600.000 ¢, ~10/9 = 175.681 ¢
Optimal ET sequence: 14c, 20cde, 34d, 48
Badness (Sintel): 1.54
19-limit
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 85/84, 99/98, 144/143, 190/189, 243/242
Mapping: [⟨2 2 2 3 4 8 7 5], ⟨0 4 9 9 10 -2 4 12]]
Optimal tunings:
- WE: ~7/5 = 599.464 ¢, ~10/9 = 175.523 ¢
- CWE: ~7/5 = 600.000 ¢, ~10/9 = 175.593 ¢
Optimal ET sequence: 14c, 20cdehh, 34dh, 48
Badness (Sintel): 1.48
Other subgroup extensions
Devisemi (2.3.5.19)
Subgroup: 2.3.5.19
Comma list: 361/360, 20000/19683
Subgroup-val mapping: [⟨1 1 1 3], ⟨0 8 18 17]]
Gencom mapping: [⟨1 1 1 0 0 0 0 3], ⟨0 8 18 0 0 0 0 17]]
- mapping generators: ~2, ~20/19
- WE: ~2 = 1199.6900 ¢, ~20/19 = 88.0541 ¢
- error map: ⟨-0.310 +2.168 -1.649 -1.523]
- CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.0538 ¢
- error map: ⟨0.000 +2.475 -1.345 -0.598]
Optimal ET sequence: 14c, 27, 41, 68, 109
Badness (Sintel): 1.30
2.3.5.7.19 subgroup
Subgroup: 2.3.5.7.19
Comma list: 190/189, 245/243, 361/360
Subgroup-val mapping: [⟨1 1 1 2 3], ⟨0 8 18 11 17]]
Gencom mapping: [⟨1 1 1 2 0 0 0 3], ⟨0 8 18 11 0 0 0 17]]
Optimal tunings:
- WE: ~2 = 1199.7591 ¢, ~20/19 = 88.0570 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/19 = 88.0564 ¢
Optimal ET sequence: 14c, 27, 41, 68, 109
Badness (Sintel): 0.508