Tetracot family: Difference between revisions

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The parent of the '''tetracot family''' is '''tetracot''', the 5-limit temperament [[tempering out]] [[20000/19683]] = {{monzo| 5 -9 4 }}, the minimal diesis or tetracot comma. The dual of this comma is the wedgie {{multival| 4 9 5 }}, which tells us [[10/9]] is a generator, and that four of them give [[3/2]]. In fact, (10/9)<sup>4</sup> = 20000/19683 × 3/2. We also have (10/9)<sup>9</sup> = (20000/19683)<sup>2</sup> × 5/2. From this it is evident we should flatten the generator a bit, and [[34edo|34EDO]] does this and makes for a recommendable tuning. Another possibility is to use (5/2)<sup>(1/9)</sup> for a generator. The 13-note MOS gives enough space for eight triads, with the 20-note MOS supplying many more.
{{Technical data page}}
The parent of the '''tetracot family''' is [[tetracot]], the [[5-limit]] [[regular temperament|temperament]] [[tempering out]] the [[tetracot comma]] ([[ratio]]: 20000/19683, {{monzo|legend=1| 5 -9 4 }}).  
 
== Tetracot ==
{{Main| Tetracot }}
 
The [[generator]] of tetracot is [[~]][[10/9]], and that four of these give [[~]][[3/2]]. In fact, (10/9)<sup>4</sup> = (20000/19683)⋅(3/2). We also have (10/9)<sup>9</sup> = (20000/19683)<sup>2</sup>⋅(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)<sup>1/9</sup> 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).
The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).


== Tetracot ==
[[Subgroup]]: 2.3.5
Subgroup: 2.3.5
 
[[Comma list]]: 20000/19683
 
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5586{{c}}, ~10/9 = 176.0950{{c}}
: [[error map]]: {{val| -0.441 +1.984 -1.900 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 176.0965{{c}}
: error map: {{val| 0.000 +2.431 -1.445 }}
 
[[Minimax tuning]]:
* [[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 }}
 
[[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|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.
 
[[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


Comma list: 20000/19683
Subgroup-val mapping: {{mapping| 1 1 1 2 | 0 4 9 10 }}


[[POTE generator]]: ~10/9 = 176.160
Optimal tunings:  
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}


[[Mapping]]: [{{val| 1 1 1 }}, {{val| 0 4 9 }}]
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}


{{Val list|legend=1| 7, 20c, 27, 34, 75, 109, 470b, 579b }}
Badness (Sintel): 0.459


[[Badness]]: 0.048518
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13


=== Extensions ===
Comma list: 100/99, 144/143, 243/242
The second comma of the [[Normal lists|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 ====
Subgroup-val mapping: {{mapping| 1 1 1 2 4 | 0 4 9 10 -2 }}
'''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&amp;41 temperament, if the vals in question are taken to be [[patent val]]s, meaning that ''n''×log<sub>2</sub>(prime) rounded to the nearest integer gives the mapping. [[41edo|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 41&amp;75 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 7s 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.
Optimal tunings:
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}


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, {{multival| 4 9 -15 10 … }} and 11-limit bunya, {{multival| 4 9 26 10 … }}. Again, 41edo can be used as a tuning, making the two identical, which is also the case if we turn to the 2.3.5.11 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.
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}


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 {{multival| 4 9 -15 10 -2 … }} for 13-limit monkey and {{multival| 4 9 26 10 -2 … }} for 13-limit bunya. Once again, 41 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.
Badness (Sintel): 0.489


== Monkey ==
== Monkey ==
Subgroup: 2.3.5.7
{{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


[[Comma list]]: 875/864, 5120/5103
[[Comma list]]: 875/864, 5120/5103


[[Mapping]]: [{{val| 1 1 1 5 }}, {{val| 0 4 9 -15 }}]
{{Mapping|legend=1| 1 1 1 5 | 0 4 9 -15 }}


{{Multival|legend=1| 4 9 -15 5 -35 -60 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7982{{c}}, ~10/9 = 175.7757{{c}}
: [[error map]]: {{val| +0.798 +1.946 -3.534 -1.470 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.6622{{c}}
: error map: {{val| 0.000 +0.694 -5.354 -3.759 }}


[[POTE generator]]: ~10/9 = 175.659
{{Optimal ET sequence|legend=1| 7, 34, 41 }}


{{Val list|legend=1| 7, 27d, 34, 41, 321ccdd }}
[[Badness]] (Sintel): 1.86
 
[[Badness]]: 0.073437


=== 11-limit ===
=== 11-limit ===
Line 51: Line 96:
Comma list: 100/99, 243/242, 385/384
Comma list: 100/99, 243/242, 385/384


Mapping: [{{val| 1 1 1 5 2 }}, {{val| 0 4 9 -15 10 }}]
Mapping: {{mapping| 1 1 1 5 2 | 0 4 9 -15 10 }}


POTE generator: ~10/9 = 175.570
Optimal tunings:  
* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}


Optimal GPV sequence: {{Val list| 7, 27de, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}


Badness: 0.038836
Badness (Sintel): 1.28


=== 13-limit ===
=== 13-limit ===
Line 64: Line 111:
Comma list: 100/99, 105/104, 144/143, 243/242
Comma list: 100/99, 105/104, 144/143, 243/242


Mapping: [{{val| 1 1 1 5 2 4 }}, {{val| 0 4 9 -15 10 -2 }}]
Mapping: {{mapping| 1 1 1 5 2 4 | 0 4 9 -15 10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9206{{c}}, ~10/9 = 175.6108{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6217{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 5 2 4 6 | 0 4 9 -15 10 -2 -13 }}
 
Optimal tunings:
* WE: ~2 = 1199.5029{{c}}, ~10/9 = 175.6832{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7558{{c}}
 
{{Optimal ET sequence|legend=0| 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


POTE generator: ~10/9 = 175.622
Mapping: {{mapping| 1 1 1 5 2 4 6 6 | 0 4 9 -15 10 -2 -13 -12 }}


Optimal GPV sequence: {{Val list| 7, 27de, 34, 41 }}
Optimal tunings:  
* WE: ~2 = 1199.7318{{c}}, ~10/9 = 175.6498{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6901{{c}}


Badness: 0.028410
{{Optimal ET sequence|legend=0| 7, 34, 41 }}
 
Badness (Sintel): 1.35


== Bunya ==
== Bunya ==
Subgroup: 2.3.5.7
{{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


[[Comma list]]: 225/224, 15625/15309
[[Comma list]]: 225/224, 15625/15309


[[Mapping]]: [{{val| 1 1 1 -1 }}, {{val| 0 4 9 26 }}]
{{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}
 
{{Multival|legend=1| 4 9 26 5 30 35 }}


[[POTE generator]]: ~10/9 = 175.741
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2991{{c}}, ~10/9 = 175.7844{{c}}
: [[error map]]: {{val| +0.299 +1.482 -3.955 +1.270 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 175.7567{{c}}
: error map: {{val| 0.000 +1.072 -4.503 +0.849 }}


{{Val list|legend=1| 34d, 41, 116, 157c, 198c }}
{{Optimal ET sequence|legend=1| 7d, …, 34d, 41, 116, 157c, 198c }}


[[Badness]]: 0.062897
[[Badness]] (Sintel): 1.59


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 100/99, 225/224, 1344/1331
Comma list: 100/99, 225/224, 243/242


Mapping: [{{val| 1 1 1 -1 2 }}, {{val| 0 4 9 26 10 }}]
Mapping: {{mapping| 1 1 1 -1 2 | 0 4 9 26 10 }}


POTE generator: ~10/9 = 175.777
Optimal tunings:  
* WE: ~2 = 1199.7481{{c}}, ~10/9 = 175.7401{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7637{{c}}


Optimal GPV sequence: {{Val list| 34d, 41, 116e, 157ce }}
{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}


Badness: 0.031332
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Line 105: Line 192:
Comma list: 100/99, 144/143, 225/224, 243/242
Comma list: 100/99, 144/143, 225/224, 243/242


Mapping: [{{val| 1 1 1 -1 2 4 }}, {{val| 0 4 9 26 10 -2 }}]
Mapping: {{mapping| 1 1 1 -1 2 4 | 0 4 9 26 10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.1044{{c}}, ~10/9 = 175.7545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8526{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 -1 2 4 6 | 0 4 9 26 10 -2 -13 }}
 
Optimal tunings:
* WE: ~2 = 1198.7905{{c}}, ~10/9 = 175.7757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9302{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 -1 2 4 6 0 | 0 4 9 26 10 -2 -13 29 }}


POTE generator: ~10/9 = 175.886
Optimal tunings:  
* WE: ~2 = 1198.7904{{c}}, ~10/9 = 175.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9287{{c}}


Optimal GPV sequence: {{Val list| 34d, 41, 75e, 116ef }}
{{Optimal ET sequence|legend=0| 34dh, 41, 75e }}


Badness: 0.024886
Badness (Sintel): 1.18


== Modus ==
== Modus ==
Subgroup: 2.3.5.7
{{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


[[Comma list]]: 64/63, 4375/4374
[[Comma list]]: 64/63, 4375/4374


[[Mapping]]: [{{val| 1 1 1 4 }}, {{val| 0 4 9 -8 }}]
{{Mapping|legend=1| 1 1 1 4 | 0 4 9 -8 }}


[[POTE generator]]: ~10/9 = 177.203
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1196.7884{{c}}, ~10/9 = 176.7292{{c}}
: [[error map]]: {{val| -3.212 +1.750 +1.038 +4.494 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1188{{c}}
: error map: {{val| 0.000 +6.520 +7.755 +14.224 }}


{{Val list|legend=1| 7, 20c, 27, 61d, 88bcd }}
{{Optimal ET sequence|legend=1| 7, 20c, 27, 61d, 88bcd, 149bccddd }}


[[Badness]]: 0.068184
[[Badness]] (Sintel): 1.73


=== 11-limit ===
=== 11-limit ===
Line 131: Line 258:
Comma list: 64/63, 100/99, 243/242
Comma list: 64/63, 100/99, 243/242


Mapping: [{{val| 1 1 1 4 2 }}, {{val| 0 4 9 -8 10 }}]
Mapping: {{mapping| 1 1 1 4 2 | 0 4 9 -8 10 }}


POTE generator: ~10/9 = 177.053
Optimal tunings:  
* WE: ~2 = 1196.4227{{c}}, ~10/9 = 176.5252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9286{{c}}


Optimal GPV sequence: {{Val list| 7, 20ce, 27e, 34d, 61de }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}


Badness: 0.035149
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 144: Line 273:
Comma list: 64/63, 78/77, 100/99, 144/143
Comma list: 64/63, 78/77, 100/99, 144/143


Mapping: [{{val| 1 1 1 4 2 4 }}, {{val| 0 4 9 -8 10 -2 }}]
Mapping: {{mapping| 1 1 1 4 2 4 | 0 4 9 -8 10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1196.8686{{c}}, ~10/9 = 176.4915{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8735{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 4 2 4 1 | 0 4 9 -8 10 -2 21 }}
 
Optimal tunings:
* WE: ~2 = 1196.8783{{c}}, ~10/9 = 176.5241{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8969{{c}}
 
{{Optimal ET sequence|legend=0| 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


POTE generator: ~10/9 = 176.953
Mapping: {{mapping| 1 1 1 4 2 4 1 5 | 0 4 9 -8 10 -2 21 -5 }}


Optimal GPV sequence: {{Val list| 7, 20ce, 27e, 34d, 61de }}
Optimal tunings:  
* WE: ~2 = 1196.6939{{c}}, ~10/9 = 176.5426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9645{{c}}


Badness: 0.023806
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh, 61degh }}


; Musical examples
Badness (Sintel): 1.09
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Perc-Sitar.mp3 Tetracot Perc-Sitar] by [http://soundcloud.com/dustin-schallert/tetracot-perc-sitar Dustin Schallert]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Jam.mp3 Tetracot Jam] by [http://soundcloud.com/dustin-schallert/tetracot-jam Dustin Schallert]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Schallert/Tetracot%20Pump.mp3 Tetracot Pump] by [http://soundcloud.com/dustin-schallert/tetracot-pump Dustin Schallert] all in [[27edo]]


=== Ponens ===
=== 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.
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
Subgroup: 2.3.5.7.11
Line 164: Line 320:
Comma list: 55/54, 64/63, 363/350
Comma list: 55/54, 64/63, 363/350


Mapping: [{{val| 1 1 1 4 3 }}, {{val| 0 4 9 -8 3 }}]
Mapping: {{mapping| 1 1 1 4 3 | 0 4 9 -8 3 }}


POTE generator: ~10/9 = 177.200
Optimal tunings:  
* WE: ~2 = 1198.5026{{c}}, ~10/9 = 176.9786{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1589{{c}}


Optimal GPV sequence: {{Val list| 7, 20c, 27, 61dee, 88bcdee }}
{{Optimal ET sequence|legend=0| 7, 20c, 27 }}


Badness: 0.063077
Badness (Sintel): 2.09


==== 13-limit ====
==== 13-limit ====
Line 177: Line 335:
Comma list: 55/54, 64/63, 66/65, 143/140
Comma list: 55/54, 64/63, 66/65, 143/140


Mapping: [{{val| 1 1 1 4 3 4 }}, {{val| 0 4 9 -8 3 -2 }}]
Mapping: {{mapping| 1 1 1 4 3 4 | 0 4 9 -8 3 -2 }}
 
Optimal tunings:
* WE: ~2 = 1198.5149{{c}}, ~10/9 = 176.9778{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1681{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 4 3 4 5 | 0 4 9 -8 3 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1197.4542{{c}}, ~10/9 = 177.1828{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5355{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20c }}
 
Badness (Sintel): 1.79
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


POTE generator: ~10/9 = 177.197
Comma list: 52/51, 55/54, 64/63, 66/65, 77/76, 143/140


Optimal GPV sequence: {{Val list| 7, 20c, 27, 61dee, 88bcdee }}
Mapping: {{mapping| 1 1 1 4 3 4 5 5 | 0 4 9 -8 3 -2 -6 -5 }}


Badness: 0.039043
Optimal tunings:
* WE: ~2 = 1197.3233{{c}}, ~10/9 = 177.2025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.5878{{c}}
 
{{Optimal ET sequence|legend=0| 7, 20c }}
 
Badness (Sintel): 1.70


== Wollemia ==
== Wollemia ==
Subgroup: 2.3.5.7
{{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


[[Comma list]]: 126/125, 2240/2187
[[Comma list]]: 126/125, 2240/2187


[[Mapping]]: [{{val| 1 1 1 0 }}, {{val| 0 4 9 19 }}]
{{Mapping|legend=1| 1 1 1 0 | 0 4 9 19 }}
 
{{Multival|legend=1| 4 9 19 5 19 19 }}


[[POTE generator]]: ~10/9 = 177.357
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.6555{{c}}, ~10/9 = 177.0104{{c}}
: [[error map]]: {{val| -2.345 +3.742 +4.435 -5.628 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 177.1667{{c}}
: error map: {{val| 0.000 +6.712 +8.186 -2.659 }}


{{Val list|legend=1| 27, 61, 88bc, 115bc }}
{{Optimal ET sequence|legend=1| 7d, 20cd, 27, 61, 88bc, 115bc }}


[[Badness]]: 0.070522
[[Badness]] (Sintel): 1.78


=== 11-limit ===
=== 11-limit ===
Line 205: Line 401:
Comma list: 56/55, 100/99, 243/242
Comma list: 56/55, 100/99, 243/242


Mapping: [{{val| 1 1 1 0 2 }}, {{val| 0 4 9 19 10 }}]
Mapping: {{mapping| 1 1 1 0 2 | 0 4 9 19 10 }}


POTE generator: ~10/9 = 177.413
Optimal tunings:  
* WE: ~2 = 1196.6462{{c}}, ~10/9 = 176.9174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1370{{c}}


Optimal GPV sequence: {{Val list| 27e, 34, 61e }}
{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}


Badness: 0.037551
Badness (Sintel): 1.24


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 91/90, 100/99, 352/351
Comma list: 56/55, 91/90, 100/99, 243/242
 
Mapping: {{mapping| 1 1 1 0 2 4 | 0 4 9 19 10 -2 }}
 
Optimal tunings:
* WE: ~2 = 1197.4576{{c}}, ~10/9 = 176.8557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0949{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 0 2 4 1 | 0 4 9 19 10 -2 21 }}
 
Optimal tunings:
* WE: ~2 = 1197.4770{{c}}, ~10/9 = 176.7733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0123{{c}}
 
{{Optimal ET sequence|legend=0| 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: [{{val| 1 1 1 0 2 4 }}, {{val| 0 4 9 19 10 -2 }}]
Mapping: {{mapping| 1 1 1 0 2 4 1 1 | 0 4 9 19 10 -2 21 22 }}


POTE generator: ~10/9 = 177.231
Optimal tunings:  
* WE: ~2 = 1197.4380{{c}}, ~10/9 = 176.8774{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1216{{c}}


Optimal GPV sequence: {{Val list| 27e, 34, 61e }}
{{Optimal ET sequence|legend=0| 7dgh, 27eg }}


Badness: 0.031219
Badness (Sintel): 1.28


== Octacot ==
== Octacot ==
{{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 41&amp;68. [[68edo|68EDO]] or [[109edo|109EDO]] can be used as tunings, as can (5/2)<sup>(1/18)</sup>, which gives just major thirds. Another tuning is [[150edo|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 two temperament with octaves rather than rank one 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, giving {{multival| 8 18 11 20 -4 … }} as the octave part of the wedgie. Generators of 3\41, 8\109 and 11\150 (88 cents) are all good choices for the 7, 11 and 13 limits.
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 2401/2400
[[Comma list]]: 245/243, 2401/2400


[[Mapping]]: [{{val| 1 1 1 2 }}, {{val| 0 8 18 11 }}]
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}
 
{{Multival|legend=1| 8 18 11 10 -5 -25 }}


[[POTE generator]]: ~21/20 = 88.076
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6782{{c}}, ~21/20 = 88.0528{{c}}
: [[error map]]: {{val| -0.322 +2.145 -1.686 -0.889 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 88.0525{{c}}
: error map: {{val| 0.000 +2.465 -1.369 -0.248 }}


{{Val list|legend=1| 14c, 27, 41, 68, 109 }}
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}


[[Badness]]: 0.033845
[[Badness]] (Sintel): 0.857


=== 11-limit ===
=== 11-limit ===
Line 252: Line 484:
Comma list: 100/99, 243/242, 245/242
Comma list: 100/99, 243/242, 245/242


Mapping: [{{val| 1 1 1 2 2 }}, {{val| 0 8 18 11 20 }}]
Mapping: {{mapping| 1 1 1 2 2 | 0 8 18 11 20 }}


POTE generator: ~21/20 = 87.975
Optimal tunings:  
* WE: ~2 = 1199.6025{{c}}, ~21/20 = 87.9460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.9453{{c}}


Optimal GPV sequence: {{Val list| 27e, 41, 109e, 150e, 191e }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 109e }}


Badness: 0.024078
Badness (Sintel): 0.796


==== 13-limit ====
==== 13-limit ====
Line 265: Line 499:
Comma list: 100/99, 144/143, 196/195, 243/242
Comma list: 100/99, 144/143, 196/195, 243/242


Mapping: [{{val| 1 1 1 2 2 4 }}, {{val| 0 8 18 11 20 -4 }}]
Mapping: {{mapping| 1 1 1 2 2 4 | 0 8 18 11 20 -4 }}


POTE generator: ~21/20 = 88.106
Optimal tunings:  
* WE: ~2 = 1198.8609{{c}}, ~21/20 = 87.0219{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0557{{c}}


Optimal GPV sequence: {{Val list| 27e, 41, 68e, 109ef }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 68e, 109ef }}


Badness: 0.023276
Badness (Sintel): 0.962


===== 17-limit =====
===== 17-limit =====
Line 278: Line 514:
Comma list: 100/99, 120/119, 144/143, 154/153, 189/187
Comma list: 100/99, 120/119, 144/143, 154/153, 189/187


Mapping: [{{val| 1 1 1 2 2 4 3 }}, {{val| 0 8 18 11 20 -4 15 }}]
Mapping: {{mapping| 1 1 1 2 2 4 3 | 0 8 18 11 20 -4 15 }}


POTE generator: ~18/17 = 88.102
Optimal tunings:  
* WE: ~2 = 1198.4494{{c}}, ~21/20 = 87.9878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0324{{c}}


Optimal GPV sequence: {{Val list| 14c, 27eg, 41, 68egg, 109efgg }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}


Badness: 0.021088
Badness (Sintel): 1.07


===== 19-limit =====
===== 19-limit =====
Line 291: Line 529:
Comma list: 100/99, 120/119, 133/132, 144/143, 154/153, 189/187
Comma list: 100/99, 120/119, 133/132, 144/143, 154/153, 189/187


Mapping: [{{val| 1 1 1 2 2 4 3 3 }}, {{val| 0 8 18 11 20 -4 15 17 }}]
Mapping: {{mapping| 1 1 1 2 2 4 3 3 | 0 8 18 11 20 -4 15 17 }}


POTE generator: ~18/17 = 88.111
Optimal tunings:  
* WE: ~2 = 1198.5995{{c}}, ~20/19 = 88.0081{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0471{{c}}


Optimal GPV sequence: {{Val list| 14c, 27eg, 41, 68egg, 109efgg }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}


Badness: 0.016652
Badness (Sintel): 1.01


==== Octocat ====
==== Octocat ====
Line 304: Line 544:
Comma list: 78/77, 91/90, 100/99, 245/242
Comma list: 78/77, 91/90, 100/99, 245/242


Mapping: [{{val| 1 1 1 2 2 2 }}, {{val| 0 8 18 11 20 23 }}]
Mapping: {{mapping| 1 1 1 2 2 2 | 0 8 18 11 20 23 }}
 
Optimal tunings:
* WE: ~2 = 1199.4441{{c}}, ~21/20 = 88.1380{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1375{{c}}
 
{{Optimal ET sequence|legend=0| 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


POTE generator: ~21/20 = 88.179
Mapping: {{mapping| 1 1 1 2 2 2 3 | 0 8 18 11 20 23 15 }}


Optimal GPV sequence: {{Val list| 27e, 41f, 68ef }}
Optimal tunings:  
* WE: ~2 = 1198.4257{{c}}, ~21/20 = 88.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1642{{c}}


Badness: 0.027601
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 2 2 2 3 3 | 0 8 18 11 20 23 15 17 }}
 
Optimal tunings:
* WE: ~2 = 1198.5748{{c}}, ~20/19 = 88.1631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1637{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27eg }}
 
Badness (Sintel): 1.09


==== Octopod ====
==== Octopod ====
Line 317: Line 589:
Comma list: 100/99, 105/104, 243/242, 245/242
Comma list: 100/99, 105/104, 243/242, 245/242


Mapping: [{{val| 1 1 1 2 2 1 }}, {{val| 0 8 18 11 20 37 }}]
Mapping: {{mapping| 1 1 1 2 2 1 | 0 8 18 11 20 37 }}
 
Optimal tunings:
* WE: ~2 = 1200.5116{{c}}, ~21/20 = 87.7346{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7257{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 2 2 1 3 | 0 8 18 11 20 37 15 }}
 
Optimal tunings:
* WE: ~2 = 1199.6667{{c}}, ~21/20 = 87.7494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7559{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 1 2 2 1 3 3 | 0 8 18 11 20 37 15 17 }}


POTE generator: ~21/20 = 87.697
Optimal tunings:  
* WE: ~2 = 1199.9909{{c}}, ~20/19 = 87.7474{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 87.7476{{c}}


Optimal GPV sequence: {{Val list| 41, 137cd, 178cd }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}


Badness: 0.028326
Badness (Sintel): 1.19


==== Dificot ====
==== Dificot ====
Line 330: Line 634:
Comma list: 100/99, 243/242, 245/242, 343/338
Comma list: 100/99, 243/242, 245/242, 343/338


Mapping: [{{val| 1 9 19 13 22 19 }}, {{val| 0 -16 -36 -22 -40 -33 }}]
Mapping: {{mapping| 1 -7 -17 -9 -18 -14 | 0 16 36 22 40 33 }}
: mapping generators: ~2, ~13/9


POTE generator: ~13/9 = 643.989
Optimal tunings:  
* WE: ~2 = 1199.1496{{c}}, ~13/9 = 643.5328{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 643.9567{{c}}


Optimal GPV sequence: {{Val list| 13cdeef, 28ccdef, 41 }}
{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}


Badness: 0.051876
Badness (Sintel): 2.14


=== October ===
=== October ===
Line 343: Line 650:
Comma list: 245/243, 385/384, 1375/1372
Comma list: 245/243, 385/384, 1375/1372


Mapping: [{{val|1 1 1 2 5}}, {{val|0 8 18 11 -21}}]
Mapping: {{mapping| 1 1 1 2 5 | 0 8 18 11 -21 }}


POTE generator: ~21/20 = 88.035
Optimal tunings:  
* WE: ~2 = 1199.8843{{c}}, ~21/20 = 88.0261{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0329{{c}}


Optimal GPV sequence: {{Val list| 27, 41, 68, 109, 150, 259 }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109, 150, 259 }}


Badness: 0.039643
Badness (Sintel): 1.31


==== 13-limit ====
==== 13-limit ====
Line 356: Line 665:
Comma list: 196/195, 245/243, 275/273, 385/384
Comma list: 196/195, 245/243, 275/273, 385/384


Mapping: [{{val|1 1 1 2 5 4}}, {{val|0 8 18 11 -21 -4}}]
Mapping: {{mapping| 1 1 1 2 5 4 | 0 8 18 11 -21 -4 }}


POTE generator: ~21/20 = 88.075
Optimal tunings:  
* WE: ~2 = 1199.5060{{c}}, ~21/20 = 88.0388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0697{{c}}


Optimal GPV sequence: {{Val list| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


Badness: 0.031136
Badness (Sintel): 1.29


==== 17-limit ====
==== 17-limit ====
Line 369: Line 680:
Comma list: 154/153, 170/169, 196/195, 245/243, 256/255
Comma list: 154/153, 170/169, 196/195, 245/243, 256/255


Mapping: [{{val|1 1 1 2 5 4 6}}, {{val|0 8 18 11 -21 -4 -26}}]
Mapping: {{mapping| 1 1 1 2 5 4 6 | 0 8 18 11 -21 -4 -26 }}


POTE generator: ~21/20 = 88.104
Optimal tunings:  
* WE: ~2 = 1199.3845{{c}}, ~21/20 = 88.0589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1027{{c}}


Optimal GPV sequence: {{Val list| 27, 41, 68, 109f }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


Badness: 0.026833
Badness (Sintel): 1.37


==== 19-limit ====
==== 19-limit ====
Line 382: Line 695:
Comma list: 154/153, 170/169, 190/189, 196/195, 209/208, 245/243
Comma list: 154/153, 170/169, 190/189, 196/195, 209/208, 245/243


Mapping: [{{val|1 1 1 2 5 4 6 3}}, {{val|0 8 18 11 -21 -4 -26 17}}]
Mapping: {{mapping| 1 1 1 2 5 4 6 3 | 0 8 18 11 -21 -4 -26 17 }}


POTE generator: ~19/18 = 88.113
Optimal tunings:  
* WE: ~2 = 1199.4449{{c}}, ~20/19 = 88.0723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1107{{c}}


Optimal GPV sequence: {{Val list| 27, 41, 68, 109f, 177ffg }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


Badness: 0.020511
Badness (Sintel): 1.25


== Dodecacot ==
== Dodecacot ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 3125/3087, 10976/10935
[[Comma list]]: 3125/3087, 10976/10935


[[Mapping]]: [{{val| 1 1 1 1 }}, {{val| 0 12 27 37 }}]
{{Mapping|legend=1| 1 1 1 1 | 0 12 27 37 }}
 
: mapping generators: ~2, ~28/27
{{Multival|legend=1| 12 27 37 15 25 10 }}


[[POTE generator]]: ~28/27 = 58.675
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6912{{c}}, ~28/27 = 58.6600{{c}}
: [[error map]]: {{val| -0.309 +1.657 -2.802 +1.287 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 58.6624{{c}}
: error map: {{val| 0.000 +1.993 -2.430 +1.681 }}


{{Val list|legend=1| 41, 143d, 184, 225, 409bcd }}
{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}


[[Badness]]: 0.119761
[[Badness]] (Sintel): 3.03


=== 11-limit ===
=== 11-limit ===
Line 410: Line 728:
Comma list: 100/99, 243/242, 1375/1372
Comma list: 100/99, 243/242, 1375/1372


Mapping: [{{val| 1 1 1 1 2 }}, {{val| 0 12 27 37 30 }}]
Mapping: {{mapping| 1 1 1 1 2 | 0 12 27 37 30 }}


POTE generator: ~28/27 = 58.665
Optimal tunings:  
* WE: ~2 = 1199.3125{{c}}, ~28/27 = 58.6317{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.6360{{c}}


Optimal GPV sequence: {{Val list| 20cde, 41 }}
{{Optimal ET sequence|legend=0| 20cde, 41 }}


Badness: 0.059528
Badness (Sintel): 1.97


=== 13-limit ===
=== 13-limit ===
Line 423: Line 743:
Comma list: 100/99, 196/195, 243/242, 275/273
Comma list: 100/99, 196/195, 243/242, 275/273


Mapping: [{{val| 1 1 1 1 2 2 }}, {{val| 0 12 27 37 30 35 }}]
Mapping: {{mapping| 1 1 1 1 2 2 | 0 12 27 37 30 35 }}
 
Optimal tunings:
* WE: ~2 = 1199.0713{{c}}, ~28/27 = 58.5932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.5982{{c}}
 
{{Optimal ET sequence|legend=0| 20cdef, 41 }}
 
Badness (Sintel): 1.80
 
== Weasel ==
{{See also| No-fives subgroup temperaments #Byhearted }}
 
Weasel, named by [[Mike Battaglia]] in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104304.html Yahoo! Tuning Group | ''This temperament should have a name'']</ref> and also known as ''byhearted''<ref group="note">Alias by [[Xenllium]]. </ref>, 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|legend=1| 2 2 2 3 | 0 4 9 9 }}
: mapping generators: ~7/5, ~10/9
 
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6934{{c}}, ~10/9 = 175.5626{{c}}
: [[error map]]: {{val| -0.613 -0.318 -6.864 +10.318 }}
* [[CWE]]: ~7/5 = 1200.0000{{c}}, ~10/9 = 175.5632{{c}}
: error map: {{val| 0.000 +0.298 -6.245 +11.243 }}
 
{{Optimal ET sequence|legend=1| 14c, 34d, 48 }}
 
[[Badness]] (Sintel): 2.82
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 99/98, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 | 0 4 9 9 10 }}
 
Optimal tunings:
* WE: ~7/5 = 599.6525{{c}}, ~10/9 = 175.5103{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.5086{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 2 2 3 4 3 | 0 4 9 9 10 15 }}
 
Optimal tunings:
* WE: ~7/5 = 599.4539{{c}}, ~10/9 = 175.7393{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.7502{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}
 
Badness (Sintel): 1.32
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~27/26 = 58.639
Comma list: 50/49, 78/77, 85/84, 99/98, 243/242


Optimal GPV sequence: {{Val list| 20cdef, 41 }}
Mapping: {{mapping| 2 2 2 3 4 3 7 | 0 4 9 9 10 15 4 }}


Badness: 0.043645
Optimal tunings:  
* WE: ~7/5 = 599.7509{{c}}, ~10/9 = 175.6684{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6839{{c}}


[[Category:Regular temperament theory]]
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 2 2 3 4 3 7 5 | 0 4 9 9 10 15 4 12 }}
 
Optimal tunings:
* WE: ~7/5 = 599.6682{{c}}, ~10/9 = 175.5994{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6190{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 20cdefhh, 34dh, 48f }}
 
Badness (Sintel): 1.28
 
=== Weasly ===
{{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
 
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: 2.3.5.7.11.13.17
 
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: 2.3.5.7.11.13.17.19
 
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, 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: {{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) ===
[[Subgroup]]: 2.3.5.19
 
[[Comma list]]: 361/360, 20000/19683
 
{{Mapping|legend=2| 1 1 1 3 | 0 8 18 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
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6900{{c}}, ~20/19 = 88.0541{{c}}
: [[error map]]: {{val| -0.310 +2.168 -1.649 -1.523 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/19 = 88.0538{{c}}
: error map: {{val| 0.000 +2.475 -1.345 -0.598 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 1 1 2 3 | 0 8 18 11 17 }}
 
Gencom mapping: {{mapping| 1 1 1 2 0 0 0 3 | 0 8 18 11 0 0 0 17 }}
 
Optimal tunings:
* WE: ~2 = 1199.7591{{c}}, ~20/19 = 88.0570{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0564{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 27, 41, 68, 109 }}
 
Badness (Sintel): 0.508
 
== Notes ==
<references group="note"/>
 
== References ==
<references/>
 
[[Category:Tetracot family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Tetracot]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Tetracot family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Listen]]

Latest revision as of 12:24, 14 August 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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