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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 }}).  
The name comes from members of the Araucaria family of conifers, which have four cotyledons (though sometimes these are fused).


== Tetracot ==
== Tetracot ==
{{Main| 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).


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 12: Line 15:
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 176.028
[[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 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 41, 75, 334bc, 409bc, 484bc, 559bbc }}
{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 75, 109 }}


[[Badness]]: 0.048518
[[Badness]] (Sintel): 1.14


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at.
==== Subgroup extensions ====
* [[875/864]], the keema, gives monkey;
Since the generator in all reasonable tunings is between 10/9 and [[11/10]], it is natural to extend tetracot to the [[11-limit]] by tempering out (10/9)/(11/10) = [[100/99]]. This gives the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] version of tetracot, dispensing with 7. For this, [[41edo]] can be used as a tuning.  
* 179200/177147 (or equivalently [[225/224]]) gives bunya;
* [[245/243]] gives octacot, which splits the generator in half.


==== Monkey and bunya ====
Since [[16/13]] is shy of (10/9)<sup>2</sup> by just [[325/324]], it is likewise natural to extend our winning streak by adding this to the list of commas. This gives us [[2.3.5.11.13 subgroup|2.3.5.11.13-subgroup]] tetracot, which tempers out 100/99, [[144/143]] and [[243/242]], with the [[S-expression]]-based comma list {[[243/242|S9/S11]], [[100/99|S10]], [[144/143|S12]]}.  
''Monkey'' tempers out the keema. The keema, 875/864, is the amount by which three just minor thirds fall short of 7/4, and tells us the ~7/4 of monkey is reached by three minor thirds in succession. It can be described as the 34 &amp; 41 temperament. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to bunya with the same tuning.


''Bunya'' adds 225/224 to the list of commas and may be described as the 34d &amp; 41 temperament. 41edo can again be used as a tuning, in which case it is the same as monkey. However an excellent alternative is 14<sup>1/26</sup> as a generator, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper, but still less than a cent sharp, fifth. Octave stretching, if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.
==== Full 7-limit extensions ====
The second comma of the comma list defines which 7-limit family member we are looking at. [[875/864]], the keema, gives monkey. [[225/224]] gives bunya. [[64/63]] gives modus. [[126/125]] gives wollemia. These all use the same generators as tetracot.  


Since the generator in all cases is between 10/9 and 11/10, it is natural to extend these temperaments to the 11-limit by tempering out (10/9)/(11/10) = [[100/99]]. This gives 11-limit monkey, {{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 subgroup temperament, dispensing with 7. However, 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 14<sup>1/26</sup> generator supplies, or even sharper yet, as for instance by the val {{val| 355 563 823 997 1230 }}, with a 52/355 generator.
[[245/243]] gives octacot, which splits the generator in halves. [[3125/3087]] gives dodecacot, which splits the generator in thirds. [[50/49]] gives weasel, which splits the period in halves.  
 
Since [[16/13]] is shy of (10/9)<sup>2</sup> by just [[325/324]], it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us {{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.


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
{{Main| Tetracot }}
{{See also| No-sevens subgroup temperaments #Tetracot }}
As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which tempering out 100/99, 144/143 and 243/242.
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


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


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


Gencom: [2 10/9; 100/99 243/242]
Optimal tunings:  
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 175.985
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}


{{Optimal ET sequence|legend=1| 7, 27e, 34, 41, 75e }}
Badness (Sintel): 0.459


==== 2.3.5.11.13 subgroup ====
==== 2.3.5.11.13 subgroup ====
Line 60: Line 60:
Comma list: 100/99, 144/143, 243/242
Comma list: 100/99, 144/143, 243/242


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


Gencom: [2 10/9; 100/99 144/143 243/242]
Optimal tunings:  
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 176.196
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}


{{Optimal ET sequence|legend=1| 7, 27e, 34, 41, 75e }}
Badness (Sintel): 0.489


== Monkey ==
== Monkey ==
{{See also| Tetracot }}
{{Main| Monkey }}
 
Monkey tempers out the [[keema]]. The keema, 875/864, is the amount by which three [[6/5|just minor thirds]] fall short of [[7/4]], and tells us the ~7/4 of monkey is reached by three such minor thirds in succession. It can be described as the {{nowrap| 34 & 41 }} temperament. [[41edo]] is an excellent tuning for monkey, and has the effect of making monkey identical to [[#Bunya|bunya]] with the same tuning.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 77: Line 81:
{{Mapping|legend=1| 1 1 1 5 | 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 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 175.676
{{Optimal ET sequence|legend=1| 7, 34, 41 }}


{{Optimal ET sequence|legend=1| 7, 27d, 34, 41, 239cd, 280ccdd, 321ccdd }}
[[Badness]] (Sintel): 1.86
 
[[Badness]]: 0.073437


=== 11-limit ===
=== 11-limit ===
Line 92: Line 98:
Mapping: {{mapping| 1 1 1 5 2 | 0 4 9 -15 10 }}
Mapping: {{mapping| 1 1 1 5 2 | 0 4 9 -15 10 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.598
Optimal tunings:
* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}


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


Badness: 0.038836
Badness (Sintel): 1.28


=== 13-limit ===
=== 13-limit ===
Line 105: Line 113:
Mapping: {{mapping| 1 1 1 5 2 4 | 0 4 9 -15 10 -2 }}
Mapping: {{mapping| 1 1 1 5 2 4 | 0 4 9 -15 10 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.618
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: {{Optimal ET sequence| 7, 27de, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}


Badness: 0.028410
Badness (Sintel): 1.17


=== 17-limit ===
=== 17-limit ===
Line 118: Line 128:
Mapping: {{mapping| 1 1 1 5 2 4 6 | 0 4 9 -15 10 -2 -13 }}
Mapping: {{mapping| 1 1 1 5 2 4 6 | 0 4 9 -15 10 -2 -13 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.754
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: {{Optimal ET sequence| 7, …, 27de, 34, 41, 116defg, 157cdeffgg }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}


Badness: 0.025936
Badness (Sintel): 1.32


=== 19-limit ===
=== 19-limit ===
Line 131: Line 143:
Mapping: {{mapping| 1 1 1 5 2 4 6 6 | 0 4 9 -15 10 -2 -13 -12 }}
Mapping: {{mapping| 1 1 1 5 2 4 6 6 | 0 4 9 -15 10 -2 -13 -12 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.697
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: {{Optimal ET sequence| 7, 27deh, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}


Badness: 0.022158
Badness (Sintel): 1.35


== Bunya ==
== Bunya ==
{{See also| Tetracot }}
{{Main| Bunya }}
 
Bunya adds [[225/224]] to the list of commas and may be described as the {{nowrap| 34d & 41 }} temperament. [[41edo]] can again be used as a tuning, in which case it is the same as [[#Monkey|monkey]]. However, bunya profits a little from a slightly sharper fifth. An excellent generator is 14<sup>1/26</sup>, giving just ~7's and an improved value for ~5, at the cost of a slightly sharper but still less-than-a-cent-sharp fifth, or even sharper yet: 17\116 with a fifth a cent and a half sharp, or 11\75 with a fifth two cents sharp. [[Octave stretching]], if employed, also serves to distinguish bunya from monkey, as its octaves should be stretched considerably less.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 146: Line 162:
{{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}
{{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}


{{Multival|legend=1| 4 9 26 5 30 35 }}
[[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 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 175.785
{{Optimal ET sequence|legend=1| 7d, …, 34d, 41, 116, 157c, 198c }}


{{Optimal ET sequence|legend=1| 34d, 41, 116, 157c, 355cc }}
[[Badness]] (Sintel): 1.59
 
[[Badness]]: 0.062897


=== 11-limit ===
=== 11-limit ===
Line 161: Line 179:
Mapping: {{mapping| 1 1 1 -1 2 | 0 4 9 26 10 }}
Mapping: {{mapping| 1 1 1 -1 2 | 0 4 9 26 10 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.738
Optimal tunings:
* WE: ~2 = 1199.7481{{c}}, ~10/9 = 175.7401{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7637{{c}}


Optimal ET sequence: {{Optimal ET sequence| 34d, 41, 116e, 157ce, 198cee, 239cee }}
{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}


Badness: 0.031332
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Line 174: Line 194:
Mapping: {{mapping| 1 1 1 -1 2 4 | 0 4 9 26 10 -2 }}
Mapping: {{mapping| 1 1 1 -1 2 4 | 0 4 9 26 10 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.748
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: {{Optimal ET sequence| 34d, 41, 116ef, 157ceff, 198ceeff }}
{{Optimal ET sequence|legend=0| 7d, 34d, 41, 116ef }}


Badness: 0.024886
Badness (Sintel): 1.03


=== 17-limit ===
=== 17-limit ===
Line 187: Line 209:
Mapping: {{mapping| 1 1 1 -1 2 4 6 | 0 4 9 26 10 -2 -13 }}
Mapping: {{mapping| 1 1 1 -1 2 4 6 | 0 4 9 26 10 -2 -13 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.811
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: {{Optimal ET sequence| 34d, 41, 75e, 116efg, 157ceffgg }}
{{Optimal ET sequence|legend=0| 34d, 41, 75e }}


Badness: 0.023404
Badness (Sintel): 1.19


=== 19-limit ===
=== 19-limit ===
Line 200: Line 224:
Mapping: {{mapping| 1 1 1 -1 2 4 6 0 | 0 4 9 26 10 -2 -13 29 }}
Mapping: {{mapping| 1 1 1 -1 2 4 6 0 | 0 4 9 26 10 -2 -13 29 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 175.802
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: {{Optimal ET sequence| 34dh, 41, 116efg, 157ceffgg }}
{{Optimal ET sequence|legend=0| 34dh, 41, 75e }}


Badness: 0.019460
Badness (Sintel): 1.18


== Modus ==
== Modus ==
{{See also| Tetracot }}
{{Main| Modus }}


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 217: Line 243:
{{Mapping|legend=1| 1 1 1 4 | 0 4 9 -8 }}
{{Mapping|legend=1| 1 1 1 4 | 0 4 9 -8 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 176.818
[[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 }}


{{Optimal ET sequence|legend=1| 7, 20c, 27, 34d, 61d, 95dd }}
{{Optimal ET sequence|legend=1| 7, 20c, 27, 61d, 88bcd, 149bccddd }}


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


=== 11-limit ===
=== 11-limit ===
Line 230: Line 260:
Mapping: {{mapping| 1 1 1 4 2 | 0 4 9 -8 10 }}
Mapping: {{mapping| 1 1 1 4 2 | 0 4 9 -8 10 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.446
Optimal tunings:
* WE: ~2 = 1196.4227{{c}}, ~10/9 = 176.5252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9286{{c}}


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


Badness: 0.035149
Badness (Sintel): 1.16


==== 13-limit ====
==== 13-limit ====
Line 243: Line 275:
Mapping: {{mapping| 1 1 1 4 2 4 | 0 4 9 -8 10 -2 }}
Mapping: {{mapping| 1 1 1 4 2 4 | 0 4 9 -8 10 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.471
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: {{Optimal ET sequence| 7, 20ce, 27e, 34d }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34d, 61de }}


Badness: 0.023806
Badness (Sintel): 0.984


==== 17-limit ====
==== 17-limit ====
Line 256: Line 290:
Mapping: {{mapping| 1 1 1 4 2 4 1 | 0 4 9 -8 10 -2 21 }}
Mapping: {{mapping| 1 1 1 4 2 4 1 | 0 4 9 -8 10 -2 21 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.453
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: {{Optimal ET sequence| 7g, …, 27eg, 34d }}
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34d }}


Badness: 0.021501
Badness (Sintel): 1.10


==== 19-limit ====
==== 19-limit ====
Line 269: Line 305:
Mapping: {{mapping| 1 1 1 4 2 4 1 5 | 0 4 9 -8 10 -2 21 -5 }}
Mapping: {{mapping| 1 1 1 4 2 4 1 5 | 0 4 9 -8 10 -2 21 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.538
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: {{Optimal ET sequence| 7g, …, 27eg, 34dh }}
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh, 61degh }}


Badness: 0.017941
Badness (Sintel): 1.09
 
; Music
* [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 ===
Line 289: Line 322:
Mapping: {{mapping| 1 1 1 4 3 | 0 4 9 -8 3 }}
Mapping: {{mapping| 1 1 1 4 3 | 0 4 9 -8 3 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.990
Optimal tunings:
* WE: ~2 = 1198.5026{{c}}, ~10/9 = 176.9786{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1589{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 20c, 27, 34de, 61dee }}
{{Optimal ET sequence|legend=0| 7, 20c, 27 }}


Badness: 0.063077
Badness (Sintel): 2.09


==== 13-limit ====
==== 13-limit ====
Line 302: Line 337:
Mapping: {{mapping| 1 1 1 4 3 4 | 0 4 9 -8 3 -2 }}
Mapping: {{mapping| 1 1 1 4 3 4 | 0 4 9 -8 3 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.017
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: {{Optimal ET sequence| 7, 20c, 27, 34de, 61dee }}
{{Optimal ET sequence|legend=0| 7, 20c, 27 }}


Badness: 0.039043
Badness (Sintel): 1.61


==== 17-limit ====
==== 17-limit ====
Line 315: Line 352:
Mapping: {{mapping| 1 1 1 4 3 4 5 | 0 4 9 -8 3 -2 -6 }}
Mapping: {{mapping| 1 1 1 4 3 4 5 | 0 4 9 -8 3 -2 -6 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.378
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: {{Optimal ET sequence| 7, 20c, 27g }}
{{Optimal ET sequence|legend=0| 7, 20c }}


Badness: 0.035200
Badness (Sintel): 1.79


==== 19-limit ====
==== 19-limit ====
Line 328: Line 367:
Mapping: {{mapping| 1 1 1 4 3 4 5 5 | 0 4 9 -8 3 -2 -6 -5 }}
Mapping: {{mapping| 1 1 1 4 3 4 5 5 | 0 4 9 -8 3 -2 -6 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 177.505
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: {{Optimal ET sequence| 7, 20c, 27g }}
{{Optimal ET sequence|legend=0| 7, 20c }}


Badness: 0.028026
Badness (Sintel): 1.70


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 343: Line 386:
{{Mapping|legend=1| 1 1 1 0 | 0 4 9 19 }}
{{Mapping|legend=1| 1 1 1 0 | 0 4 9 19 }}


{{Multival|legend=1| 4 9 19 5 19 19 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1197.6555{{c}}, ~10/9 = 177.0104{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 176.900
: [[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 }}


{{Optimal ET sequence|legend=1| 7d, 20cd, 27, 34, 61, 95d, 156bcd }}
{{Optimal ET sequence|legend=1| 7d, 20cd, 27, 61, 88bc, 115bc }}


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


=== 11-limit ===
=== 11-limit ===
Line 358: Line 403:
Mapping: {{mapping| 1 1 1 0 2 | 0 4 9 19 10 }}
Mapping: {{mapping| 1 1 1 0 2 | 0 4 9 19 10 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.704
Optimal tunings:
* WE: ~2 = 1196.6462{{c}}, ~10/9 = 176.9174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 177.1370{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, 20cde, 27e, 34, 95dee }}
{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}


Badness: 0.037551
Badness (Sintel): 1.24


=== 13-limit ===
=== 13-limit ===
Line 371: Line 418:
Mapping: {{mapping| 1 1 1 0 2 4 | 0 4 9 19 10 -2 }}
Mapping: {{mapping| 1 1 1 0 2 4 | 0 4 9 19 10 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.716
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: {{Optimal ET sequence| 7d, 20cde, 27e, 34, 95dee }}
{{Optimal ET sequence|legend=0| 7d, 20cde, 27e }}


Badness: 0.031219
Badness (Sintel): 1.29


=== 17-limit ===
=== 17-limit ===
Line 384: Line 433:
Mapping: {{mapping| 1 1 1 0 2 4 1 | 0 4 9 19 10 -2 21 }}
Mapping: {{mapping| 1 1 1 0 2 4 1 | 0 4 9 19 10 -2 21 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.641
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: {{Optimal ET sequence| 27eg, 34 }}
{{Optimal ET sequence|legend=0| 7dg, 27eg }}


Badness: 0.024471
Badness (Sintel): 1.25


=== 19-limit ===
=== 19-limit ===
Line 397: Line 448:
Mapping: {{mapping| 1 1 1 0 2 4 1 1 | 0 4 9 19 10 -2 21 22 }}
Mapping: {{mapping| 1 1 1 0 2 4 1 1 | 0 4 9 19 10 -2 21 22 }}


Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 176.749
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: {{Optimal ET sequence| 27eg, 34, 95deegh }}
{{Optimal ET sequence|legend=0| 7dgh, 27eg }}


Badness: 0.021108
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-2 temperament with octaves rather than rank-1 without them.
Octacot splits the difference between the [[#Monkey|monkey]] and [[#Bunya|bunya]] mappings for 7 by cutting the generator in half. It adds [[245/243]] to the normal comma list, and also tempers out [[2401/2400]]. It may also be described as {{nowrap| 41 & 68 }}. [[68edo]] or [[109edo]] can be used as tunings, as can (5/2)<sup>1/18</sup>, which gives just major thirds. Another tuning is [[150edo]], which has a generator, 11\150, of exactly 88 cents. This relates octacot to the [[88cET]] non-octave temperament, which like [[Carlos Alpha]] arguably makes more sense viewed as part of a rank-2 temperament with octaves rather than rank-1 without them.


Once again and for the same reasons, it is natural to add 100/99 and 325/324 to the list of commas, 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
Line 416: Line 469:
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}


{{Multival|legend=1| 8 18 11 10 -5 -25 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.6782{{c}}, ~21/20 = 88.0528{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 88.076
: [[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 }}


{{Optimal ET sequence|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 431: Line 486:
Mapping: {{mapping| 1 1 1 2 2 | 0 8 18 11 20 }}
Mapping: {{mapping| 1 1 1 2 2 | 0 8 18 11 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 444: Line 501:
Mapping: {{mapping| 1 1 1 2 2 4 | 0 8 18 11 20 -4 }}
Mapping: {{mapping| 1 1 1 2 2 4 | 0 8 18 11 20 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 457: Line 516:
Mapping: {{mapping| 1 1 1 2 2 4 3 | 0 8 18 11 20 -4 15 }}
Mapping: {{mapping| 1 1 1 2 2 4 3 | 0 8 18 11 20 -4 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 470: Line 531:
Mapping: {{mapping| 1 1 1 2 2 4 3 3 | 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 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 14c, 27eg, 41, 68egg, 109efgg }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}


Badness: 0.016652
Badness (Sintel): 1.01


==== Octocat ====
==== Octocat ====
Line 483: Line 546:
Mapping: {{mapping| 1 1 1 2 2 2 | 0 8 18 11 20 23 }}
Mapping: {{mapping| 1 1 1 2 2 2 | 0 8 18 11 20 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 88.179
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


{{Optimal ET sequence|legend=1| 27e, 41f, 68ef }}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.027601
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 496: Line 591:
Mapping: {{mapping| 1 1 1 2 2 1 | 0 8 18 11 20 37 }}
Mapping: {{mapping| 1 1 1 2 2 1 | 0 8 18 11 20 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 87.697
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 }}


{{Optimal ET sequence|legend=1| 41, 137cd, 178cd }}
Badness (Sintel): 1.17


Badness: 0.028326
===== 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 }}
 
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 507: Line 634:
Comma list: 100/99, 243/242, 245/242, 343/338
Comma list: 100/99, 243/242, 245/242, 343/338


Mapping: {{mapping| 1 9 19 13 22 19 | 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


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 13cdeef, 28ccdef, 41 }}
{{Optimal ET sequence|legend=0| 13cdeef, 28ccdef, 41 }}


Badness: 0.051876
Badness (Sintel): 2.14


=== October ===
=== October ===
Line 522: Line 652:
Mapping: {{mapping| 1 1 1 2 5 | 0 8 18 11 -21 }}
Mapping: {{mapping| 1 1 1 2 5 | 0 8 18 11 -21 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 535: Line 667:
Mapping: {{mapping| 1 1 1 2 5 4 | 0 8 18 11 -21 -4 }}
Mapping: {{mapping| 1 1 1 2 5 4 | 0 8 18 11 -21 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 548: Line 682:
Mapping: {{mapping| 1 1 1 2 5 4 6 | 0 8 18 11 -21 -4 -26 }}
Mapping: {{mapping| 1 1 1 2 5 4 6 | 0 8 18 11 -21 -4 -26 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 561: Line 697:
Mapping: {{mapping| 1 1 1 2 5 4 6 3 | 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 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 27, 41, 68, 109f, 177ffg }}
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


Badness: 0.020511
Badness (Sintel): 1.25


== Dodecacot ==
== Dodecacot ==
Line 573: Line 711:


{{Mapping|legend=1| 1 1 1 1 | 0 12 27 37 }}
{{Mapping|legend=1| 1 1 1 1 | 0 12 27 37 }}
: mapping generators: ~2, ~28/27
: 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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 58.675
{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}


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


=== 11-limit ===
=== 11-limit ===
Line 591: Line 730:
Mapping: {{mapping| 1 1 1 1 2 | 0 12 27 37 30 }}
Mapping: {{mapping| 1 1 1 1 2 | 0 12 27 37 30 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 20cde, 41 }}
{{Optimal ET sequence|legend=0| 20cde, 41 }}


Badness: 0.059528
Badness (Sintel): 1.97


=== 13-limit ===
=== 13-limit ===
Line 604: Line 745:
Mapping: {{mapping| 1 1 1 1 2 2 | 0 12 27 37 30 35 }}
Mapping: {{mapping| 1 1 1 1 2 2 | 0 12 27 37 30 35 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 58.639
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


{{Optimal ET sequence|legend=1| 20cdef, 41 }}
Mapping: {{mapping| 2 2 2 3 4 8 | 0 4 9 9 10 -2 }}


Badness: 0.043645
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 family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Tetracot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Listen]]
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