Tetracot family: Difference between revisions
Re-organize |
Re-organize. Split the monkey and bunya section for an intro to the subgroup extension |
||
| Line 23: | Line 23: | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[5-odd-limit]]: ~10/9 = {{monzo| -1/9 0 1/9 }} | * [[5-odd-limit]]: ~10/9 = {{monzo| -1/9 0 1/9 }} | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5 | ||
{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 75, 109 }} | {{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 75, 109 }} | ||
| Line 30: | Line 30: | ||
=== Overview to extensions === | === Overview to extensions === | ||
==== Subgroup extensions ==== | |||
Since the generator in all reasonable tunings is between 10/9 and [[11/10]], it is natural to extend tetracot to the [[11-limit]] by tempering out (10/9)/(11/10) = [[100/99]]. This gives the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] version of tetracot, dispensing with 7. For this, [[41edo]] can be used as a tuning. | |||
Since [[16/13]] is shy of (10/9)<sup>2</sup> by just [[325/324]], it is likewise natural to extend our winning streak by adding this to the list of commas. This gives us [[2.3.5.11.13 subgroup|2.3.5.11.13-subgroup]] tetracot, which tempers out 100/99, [[144/143]] and [[243/242]], with the [[S-expression]]-based comma list {[[243/242|S9/S11]], [[100/99|S10]], [[144/143|S12]]}. | |||
==== Full 7-limit extensions ==== | |||
The second comma of the comma list defines which 7-limit family member we are looking at. [[875/864]], the keema, gives monkey. [[225/224]] gives bunya. [[64/63]] gives modus. [[126/125]] gives wollemia. These all use the same generators as tetracot. | The second comma of the comma list defines which 7-limit family member we are looking at. [[875/864]], the keema, gives monkey. [[225/224]] gives bunya. [[64/63]] gives modus. [[126/125]] gives wollemia. These all use the same generators as tetracot. | ||
[[245/243]] gives octacot, which splits the generator in halves. [[3125/3087]] gives dodecacot, which splits the generator in thirds. [[50/49]] gives weasel, which splits the period in halves. | [[245/243]] gives octacot, which splits the generator in halves. [[3125/3087]] gives dodecacot, which splits the generator in thirds. [[50/49]] gives weasel, which splits the period in halves. | ||
=== 2.3.5.11 subgroup === | === 2.3.5.11 subgroup === | ||
| Line 78: | Line 72: | ||
== Monkey == | == Monkey == | ||
{{Main| Monkey }} | {{Main| Monkey }} | ||
Monkey tempers out the [[keema]]. The keema, 875/864, is the amount by which three [[6/5|just minor thirds]] fall short of [[7/4]], and tells us the ~7/4 of monkey is reached by three such minor thirds in succession. It can be described as the {{nowrap| 34 & 41 }} temperament. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to [[#Bunya|bunya]] with the same tuning. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 157: | Line 153: | ||
== Bunya == | == Bunya == | ||
{{Main| Bunya }} | {{Main| Bunya }} | ||
Bunya adds [[225/224]] to the list of commas and may be described as the {{nowrap| 34d & 41 }} temperament. [[41edo]] can again be used as a tuning, in which case it is the same as [[#Monkey|monkey]]. However, bunya profits a little from a slightly sharper fifth. An excellent generator is 14<sup>1/26</sup>, 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, or even sharper yet: 17\116 with a fifth a cent and a half sharp, or 11\75 with a fifth two cents sharp. [[Octave stretching]], if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 459: | Line 457: | ||
{{See also| Chords of 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 {{nowrap| 41 & 68 }}. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)<sup>1/18</sup>, 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. | Octacot cuts the Gordian knot of deciding between the [[#Monkey|monkey]] and [[#Bunya|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 {{nowrap| 41 & 68 }}. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)<sup>1/18</sup>, 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. | 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. | ||
Revision as of 14:53, 16 April 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
Subgroup extensions
Since the generator in all reasonable tunings is between 10/9 and 11/10, it is natural to extend tetracot to the 11-limit by tempering out (10/9)/(11/10) = 100/99. This gives the 2.3.5.11-subgroup version of tetracot, dispensing with 7. For this, 41edo can be used as a tuning.
Since 16/13 is shy of (10/9)2 by just 325/324, it is likewise natural to extend our winning streak by adding this to the list of commas. This gives us 2.3.5.11.13-subgroup tetracot, which tempers out 100/99, 144/143 and 243/242, with the S-expression-based comma list {S9/S11, S10, S12}.
Full 7-limit extensions
The second comma of the comma list defines which 7-limit family member we are looking at. 875/864, the keema, gives monkey. 225/224 gives bunya. 64/63 gives modus. 126/125 gives wollemia. These all use the same generators as tetracot.
245/243 gives octacot, which splits the generator in halves. 3125/3087 gives dodecacot, which splits the generator in thirds. 50/49 gives weasel, which splits the period in halves.
2.3.5.11 subgroup
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
Monkey
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 such 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.
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
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, bunya profits a little from a slightly sharper fifth. An excellent generator is 141/26, 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, or even sharper yet: 17\116 with a fifth a cent and a half sharp, or 11\75 with a fifth two cents sharp. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.
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
Tetracot (2.3.5.13)
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
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
Devisemi (2.3.5.7.19)
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