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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]] 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
[[Comma list]]: 20000/19683


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


[[POTE generator]]: ~10/9 = 176.160
[[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]]:  
[[Minimax tuning]]:  
* 5-odd-limit: ~10/9 = {{monzo| -1/9 0 1/9 }}
* [[5-odd-limit]]: ~10/9 = {{monzo| -1/9 0 1/9 }}
: Eigenmonzos (unchanged intervals): 2, 5
: [[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.


{{Val list|legend=1| 7, 20c, 27, 34, 75, 109, 470b, 579b }}
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]]}.


[[Badness]]: 0.048518
==== 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.  


=== Extensions ===
[[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.  
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 ====
=== 2.3.5.11 subgroup ===
'''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]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.
Subgroup: 2.3.5.11


'''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.
Comma list: 100/99, 243/242


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.
Subgroup-val mapping: {{mapping| 1 1 1 2 | 0 4 9 10 }}


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, 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.
Optimal tunings:
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}


=== Subgroup temperament ===
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}
{{see also| No-sevens subgroup temperaments #Tetracot }}


The tetracot temperament works well for the 2.3.5.11 subgroup, in which tempering out 100/99 and 243/242. In this temperament, 3/2 is divided into four equal parts, which represents both 10/9 and 11/10.
Badness (Sintel): 0.459


Subgroup: 2.3.5.11
==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13


[[Comma list]]: 100/99, 243/242
Comma list: 100/99, 144/143, 243/242


[[Gencom]]: [2 10/9; 100/99 243/242]
Subgroup-val mapping: {{mapping| 1 1 1 2 4 | 0 4 9 10 -2 }}


[[Mapping|Sval mapping]]: [{{val|1 1 1 2}}, {{val|0 4 9 10}}]
Optimal tunings:  
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}


[[Tp tuning|POL2 generator]]: ~10/9 = 175.985
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}


{{Val list|legend=1| 7, 27e, 34, 41, 75e }}
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 72: 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 85: 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 }}


POTE generator: ~10/9 = 175.622
Optimal tunings:  
* WE: ~2 = 1199.9206{{c}}, ~10/9 = 175.6108{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6217{{c}}


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


Badness: 0.028410
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
 
Mapping: {{mapping| 1 1 1 5 2 4 6 6 | 0 4 9 -15 10 -2 -13 -12 }}
 
Optimal tunings:
* WE: ~2 = 1199.7318{{c}}, ~10/9 = 175.6498{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6901{{c}}
 
{{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 126: 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}}


POTE generator: ~10/9 = 175.886
{{Optimal ET sequence|legend=0| 34d, 41, 75e }}


Optimal GPV sequence: {{Val list| 34d, 41, 75e, 116ef }}
Badness (Sintel): 1.19


Badness: 0.024886
=== 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 }}
 
Optimal tunings:
* WE: ~2 = 1198.7904{{c}}, ~10/9 = 175.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.9287{{c}}
 
{{Optimal ET sequence|legend=0| 34dh, 41, 75e }}
 
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 152: 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 165: 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 }}


POTE generator: ~10/9 = 176.953
Optimal tunings:  
* WE: ~2 = 1196.8686{{c}}, ~10/9 = 176.4915{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.8735{{c}}


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


Badness: 0.023806
Badness (Sintel): 0.984


; Musical examples
==== 17-limit ====
* [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]
Subgroup: 2.3.5.7.11.13.17
* [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]]
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
 
Mapping: {{mapping| 1 1 1 4 2 4 1 5 | 0 4 9 -8 10 -2 21 -5 }}
 
Optimal tunings:
* WE: ~2 = 1196.6939{{c}}, ~10/9 = 176.5426{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9645{{c}}
 
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh, 61degh }}
 
Badness (Sintel): 1.09


=== 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 185: 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 198: 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


POTE generator: ~10/9 = 177.197
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Optimal GPV sequence: {{Val list| 7, 20c, 27, 61dee, 88bcdee }}
Comma list: 52/51, 55/54, 64/63, 66/65, 143/140


Badness: 0.039043
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
 
Comma list: 52/51, 55/54, 64/63, 66/65, 77/76, 143/140
 
Mapping: {{mapping| 1 1 1 4 3 4 5 5 | 0 4 9 -8 3 -2 -6 -5 }}
 
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 226: 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: [{{val| 1 1 1 0 2 4 }}, {{val| 0 4 9 19 10 -2 }}]
Mapping: {{mapping| 1 1 1 0 2 4 | 0 4 9 19 10 -2 }}


POTE generator: ~10/9 = 177.231
Optimal tunings:  
* WE: ~2 = 1197.4576{{c}}, ~10/9 = 176.8557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.0949{{c}}


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


Badness: 0.031219
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: {{mapping| 1 1 1 0 2 4 1 1 | 0 4 9 19 10 -2 21 22 }}
 
Optimal tunings:
* WE: ~2 = 1197.4380{{c}}, ~10/9 = 176.8774{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1216{{c}}
 
{{Optimal ET sequence|legend=0| 7dgh, 27eg }}
 
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]] 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 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 273: 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 286: 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 299: 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 312: 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 325: 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
 
Mapping: {{mapping| 1 1 1 2 2 2 3 | 0 8 18 11 20 23 15 }}
 
Optimal tunings:
* WE: ~2 = 1198.4257{{c}}, ~21/20 = 88.1636{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1642{{c}}
 
{{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 }}


POTE generator: ~21/20 = 88.179
Optimal tunings:  
* WE: ~2 = 1198.5748{{c}}, ~20/19 = 88.1631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1637{{c}}


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


Badness: 0.027601
Badness (Sintel): 1.09


==== Octopod ====
==== Octopod ====
Line 338: 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


POTE generator: ~21/20 = 87.697
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 8 18 11 20 37 15 }}


Optimal GPV sequence: {{Val list| 41, 137cd, 178cd }}
Optimal tunings:  
* WE: ~2 = 1199.6667{{c}}, ~21/20 = 87.7494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7559{{c}}


Badness: 0.028326
{{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 }}
 
Optimal tunings:
* WE: ~2 = 1199.9909{{c}}, ~20/19 = 87.7474{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 87.7476{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}
 
Badness (Sintel): 1.19


==== Dificot ====
==== Dificot ====
Line 351: 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 364: 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 377: 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 390: 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 403: 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 }}
[[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 }}


[[POTE generator]]: ~28/27 = 58.675
{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}


{{Val list|legend=1| 41, 143d, 184, 225, 409bcd }}
[[Badness]] (Sintel): 3.03
 
[[Badness]]: 0.119761


=== 11-limit ===
=== 11-limit ===
Line 431: 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 444: 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
 
Comma list: 50/49, 78/77, 85/84, 99/98, 243/242
 
Mapping: {{mapping| 2 2 2 3 4 3 7 | 0 4 9 9 10 15 4 }}
 
Optimal tunings:
* WE: ~7/5 = 599.7509{{c}}, ~10/9 = 175.6684{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~10/9 = 175.6839{{c}}
 
{{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 }}


POTE generator: ~27/26 = 58.639
Badness (Sintel): 0.508


Optimal GPV sequence: {{Val list| 20cdef, 41 }}
== Notes ==
<references group="note"/>


Badness: 0.043645
== References ==
<references/>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Tetracot]]
[[Category:Tetracot family| ]] <!-- main article -->
[[Category:Tetracot family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
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