Tetracot family: Difference between revisions

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[[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
: [[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 }}
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=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[normal forms|normal comma list]] defines which 7-limit family member we are looking at.
==== Subgroup extensions ====
* [[875/864]], the keema, gives monkey;
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.  
* 179200/177147 (or equivalently [[225/224]]) gives bunya;
* [[245/243]] gives octacot, which splits the generator in half.


==== Monkey and bunya ====
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]]}.  
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 {{nowrap| 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 {{nowrap| 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<sup>1/26</sup> 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.
==== 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.  


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<sup>1/26</sup> generator supplies, or even sharper yet, as for instance by the val {{val| 355 563 823 997 1230 }}, with a 52/355 generator.
[[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.  
 
Since [[16/13]] is shy of (10/9)<sup>2</sup> 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 ===
=== 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 {[[243/242|S9/S11]], [[100/99|S10]]}.
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


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Badness (Sintel): 0.489
Badness (Sintel): 0.489


=== 2.3.5.13 subgroup ===
== Monkey ==
Subgroup: 2.3.5.13
{{Main| Monkey }}
 
Comma list: 325/324, 512/507
 
Subgroup-val mapping: {{mapping| 1 1 1 4 | 0 4 9 -2 }}
 
Optimal tunings:
* WE: ~2 = 1198.8502{{c}}, ~10/9 = 176.2195{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.2975{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20c, 27, 34, 245bff, 279bfff }}


Badness (Sintel): 0.551
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.


== Monkey ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Bunya ==
== 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


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== Modus ==
== Modus ==
Modus was named by [[Mike Battaglia]] in 2012 for its fantastic modmos structures<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_102416.html#102467 Yahoo! Tuning Group | ''Guaranteed meantone successor'']</ref>.
{{Main| Modus }}
 
Modus tempers out [[64/63]] as well as [[4375/4374]], and may be described as the {{nowrap| 27 & 34d }} temperament. While less accurate than [[#Monkey|monkey]] or [[#Bunya|bunya]], it is nonetheless very useful because it is simpler and because of the harmonic puns it possesses. [[27edo]], [[34edo]] and [[61edo]] can all be used as tunings.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Wollemia ==
== Wollemia ==
{{Main| Wollemia }}
Wollemia tempers out [[126/125]] as well as [[2240/2187]], and may be described as the {{nowrap| 27 & 34 }} temperament. [[27edo]] may be recommended as a tuning, in which case it is identical to modus with the same tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{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 splits the difference between the [[#Monkey|monkey]] and [[#Bunya|bunya]] mappings for 7 by cutting the generator in half. 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.
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=== 13-limit ===
=== 13-limit ===
The canonical mapping finds 13/8 at +15 generators rather than using the regular tetracot mapping, in order to find [[15/13]] as being half of [[4/3]].
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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=== Weasly ===
=== Weasly ===
{{Todo|review|unify precision}}
{{Todo|review|unify precision}}
The alternative extension uses the same mapping of 13 as in tetracot, though many other intervals of 13 take more generators to reach as a result.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 50/49, 85/84, 99/98, 144/143, 243/242
Comma list: 50/49, 85/84, 99/98, 144/143, 243/242
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==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 50/49, 85/84, 99/98, 144/143, 190/189, 243/242
Comma list: 50/49, 85/84, 99/98, 144/143, 190/189, 243/242
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* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.593{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~10/9 = 175.593{{c}}


{{Optimal ET sequence|legend=0| 14c, 20cdehh, 34dh, 48 }}
{{Optimal ET sequence|legend=0| 14c, 34dh, 48 }}


Badness (Sintel): 1.48
Badness (Sintel): 1.48


== Other subgroup extensions ==
== Other subgroup extensions ==
=== Tetracot (2.3.5.13) ===
Subgroup: 2.3.5.13
Comma list: 325/324, 512/507
Subgroup-val mapping: {{mapping| 1 1 1 4 | 0 4 9 -2 }}
Optimal tunings:
* WE: ~2 = 1198.8502{{c}}, ~10/9 = 176.2195{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.2975{{c}}
{{Optimal ET sequence|legend=0| 7, 20c, 27, 34, 245bff, 279bfff }}
Badness (Sintel): 0.551
=== Devisemi (2.3.5.19) ===
=== Devisemi (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19
[[Subgroup]]: 2.3.5.19
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{{Mapping|legend=3| 1 1 1 0 0 0 0 3 | 0 8 18 0 0 0 0 17 }}
{{Mapping|legend=3| 1 1 1 0 0 0 0 3 | 0 8 18 0 0 0 0 17 }}
: mapping generators: ~2, ~20/19
: mapping generators: ~2, ~20/19


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[[Badness]] (Sintel): 1.30
[[Badness]] (Sintel): 1.30


==== 2.3.5.7.19 subgroup ====
=== Devisemi (2.3.5.7.19) ===
Subgroup: 2.3.5.7.19
Subgroup: 2.3.5.7.19


Latest revision as of 03:14, 10 June 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

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:

  • 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]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5

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

Main article: 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]]

Optimal tunings:

  • 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

Main article: 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]]

Optimal tunings:

  • 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

Main article: Modus

Modus tempers out 64/63 as well as 4375/4374, and may be described as the 27 & 34d temperament. While less accurate than monkey or bunya, it is nonetheless very useful because it is simpler and because of the harmonic puns it possesses. 27edo, 34edo and 61edo can all be used as tunings.

Subgroup: 2.3.5.7

Comma list: 64/63, 4375/4374

Mapping: [⟨1 1 1 4], ⟨0 4 9 -8]]

Optimal tunings:

  • 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 ¢

Optimal ET sequence: 7, 20c

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 ¢

Optimal ET sequence: 7, 20c

Badness (Sintel): 1.70

Wollemia

Main article: Wollemia

Wollemia tempers out 126/125 as well as 2240/2187, and may be described as the 27 & 34 temperament. 27edo may be recommended as a tuning, in which case it is identical to modus with the same tuning.

Subgroup: 2.3.5.7

Comma list: 126/125, 2240/2187

Mapping: [⟨1 1 1 0], ⟨0 4 9 19]]

Optimal tunings:

  • 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

See also: Chords of octacot

Octacot splits the difference between the monkey and bunya mappings for 7 by cutting the generator in half. 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:

  • 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

Optimal tunings:

  • 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

See also: No-fives subgroup temperaments § Byhearted

Weasel, named by Mike Battaglia in 2012[1] 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

Optimal tunings:

  • 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

The canonical mapping finds 13/8 at +15 generators rather than using the regular tetracot mapping, in order to find 15/13 as being half of 4/3.

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

The alternative extension uses the same mapping of 13 as in tetracot, though many other intervals of 13 take more generators to reach as a result.

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: 2.3.5.7.11.13.17

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: 2.3.5.7.11.13.17.19

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, 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

Optimal tunings:

  • 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

Notes

  1. ↑ Alias by Xenllium.

References

  1. ↑ Yahoo! Tuning Group | This temperament should have a name
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