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{{Technical data page}}
{{Technical data page}}
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 generator 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 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 13: Line 15:
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}
{{Mapping|legend=1| 1 1 1 | 0 4 9 }}


[[Optimal tuning]]s:
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/9 = 176.028
* [[WE]]: ~2 = 1199.5586{{c}}, ~10/9 = 176.0950{{c}}
* [[POTE]]:~2 = 1\1, ~10/9 = 176.160
: [[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, 75, 109 }}
{{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 and 11-limit bunya. Again, 41edo can be used as a tuning, making the two identical, which is also the case if we turn to the 2.3.5.11 subgroup temperament, dispensing with 7. However, 11-limit bunya, like 7-limit bunya, profits a little from a slightly sharper fifth, such as the 14<sup>1/26</sup> generator supplies, or even sharper yet, as for instance by the val {{val| 355 563 823 997 1230 }}, with a 52/355 generator.
[[245/243]] gives octacot, which splits the generator in halves. [[3125/3087]] gives dodecacot, which splits the generator in thirds. [[50/49]] gives weasel, which splits the period in halves.  
 
Since [[16/13]] is shy of (10/9)<sup>2</sup> by just [[325/324]], it is likewise natural to extend our winning streak with these temperaments by adding this to the list of commas. This gives us 13-limit monkey and 13-limit bunya. Once again, 41edo is recommended as a tuning for monkey, while bunya can with advantage tune the fifth sharper: 17\116 as a generator with a fifth a cent and a half sharp or 11\75 with a fifth two cents sharp.


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
As discussed above, tetracot works well for the 2.3.5.11.13 subgroup, in which it tempers out 100/99, 144/143 and 243/242.
The [[S-expression]]-based comma list of this temperament is {[[243/242|S9/S11]], [[100/99|S10]]}.
Subgroup: 2.3.5.11
Subgroup: 2.3.5.11


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 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 175.7765
* WE: ~2 = 1199.3274{{c}}, ~10/9 = 175.8862{{c}}
* POTE ~2 = 1\1, ~10/9 = 175.985
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8847{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 20ce, 27e, 34, 41, 75e }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e }}


Badness: 0.014706
Badness (Sintel): 0.459


==== 2.3.5.11.13 subgroup ====
==== 2.3.5.11.13 subgroup ====
Line 64: 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 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 175.8150
* WE: ~2 = 1198.6852{{c}}, ~10/9 = 176.0034{{c}}
* POTE ~2 = 1\1, ~10/9 = 176.196
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.0854{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 20ce, 27e, 34, 41, 75e, 109ef }}
{{Optimal ET sequence|legend=0| 7, 20ce, 27e, 34, 41, 75e, 109ef }}


Badness: 0.012311
Badness (Sintel): 0.489


=== 2.3.5.13 subgroup ===
== Monkey ==
[[Subgroup]]: 2.3.5.13
{{Main| Monkey }}
 
[[Comma list]]: 325/324, 512/507
 
{{Mapping|legend=1| 1 1 1 4 | 0 4 9 -2 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/9 = 176.0783
* [[CWE]]: ~2 = 1\1, ~10/9 = 176.2975
 
{{Optimal ET sequence|legend=1| 7, 20c, 27, 34, 245bff, 279bfff }}
 
[[Badness]]: 0.0165


== 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.
{{See also| Tetracot }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 98: Line 81:
{{Mapping|legend=1| 1 1 1 5 | 0 4 9 -15 }}
{{Mapping|legend=1| 1 1 1 5 | 0 4 9 -15 }}


Optimal tunings:
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~10/9 = 175.676
* [[WE]]: ~2 = 1200.7982{{c}}, ~10/9 = 175.7757{{c}}
* POTE:~2 = 1\1, ~10/9 = 175.659
: [[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 ET sequence|legend=1| 7, 34, 41 }}
{{Optimal ET sequence|legend=1| 7, 34, 41 }}


[[Badness]]: 0.073437
[[Badness]] (Sintel): 1.86


=== 11-limit ===
=== 11-limit ===
Line 113: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 175.598
* WE: ~2 = 1200.3988{{c}}, ~10/9 = 175.6287{{c}}
* POTE ~2 = 1\1, ~10/9 = 175.570
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.5750{{c}}


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


Badness: 0.038836
Badness (Sintel): 1.28


=== 13-limit ===
=== 13-limit ===
Line 128: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 175.618
* WE: ~2 = 1199.9206{{c}}, ~10/9 = 175.6108{{c}}
* POTE ~2 = 1\1, ~10/9 = 175.622
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.6217{{c}}


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


Badness: 0.028410
Badness (Sintel): 1.17


=== 17-limit ===
=== 17-limit ===
Line 143: 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, 34, 41 }}
{{Optimal ET sequence|legend=0| 7, 34, 41 }}


Badness: 0.025936
Badness (Sintel): 1.32


=== 19-limit ===
=== 19-limit ===
Line 156: 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, 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 171: Line 162:
{{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}
{{Mapping|legend=1| 1 1 1 -1 | 0 4 9 26 }}


Optimal tunings:
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~10/9 = 175.785
* [[WE]]: ~2 = 1200.2991{{c}}, ~10/9 = 175.7844{{c}}
* POTE:~2 = 1\1, ~10/9 = 175.741
: [[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 ET sequence|legend=1| 7d, …, 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 ===
Line 186: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 175.738
* WE: ~2 = 1199.7481{{c}}, ~10/9 = 175.7401{{c}}
* POTE ~2 = 1\1, ~10/9 = 175.777
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.7637{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, …, 34d, 41, 116e }}
{{Optimal ET sequence|legend=0| 7d, …, 34d, 41, 116e }}


Badness: 0.031332
Badness (Sintel): 1.04


=== 13-limit ===
=== 13-limit ===
Line 201: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 175.748
* WE: ~2 = 1199.1044{{c}}, ~10/9 = 175.7545{{c}}
* POTE ~2 = 1\1, ~10/9 = 175.886
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 175.8526{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7d, 34d, 41, 116ef }}
{{Optimal ET sequence|legend=0| 7d, 34d, 41, 116ef }}


Badness: 0.024886
Badness (Sintel): 1.03


=== 17-limit ===
=== 17-limit ===
Line 216: 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 }}
{{Optimal ET sequence|legend=0| 34d, 41, 75e }}


Badness: 0.023404
Badness (Sintel): 1.19


=== 19-limit ===
=== 19-limit ===
Line 229: 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, 75e }}
{{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 246: 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 tunings:
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~10/9 = 176.818
* [[WE]]: ~2 = 1196.7884{{c}}, ~10/9 = 176.7292{{c}}
* POTE:~2 = 1\1, ~10/9 = 177.203
: [[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 261: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 176.446
* WE: ~2 = 1196.4227{{c}}, ~10/9 = 176.5252{{c}}
* POTE ~2 = 1\1, ~10/9 = 177.053
* 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 276: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 176.471
* WE: ~2 = 1196.8686{{c}}, ~10/9 = 176.4915{{c}}
* POTE ~2 = 1\1, ~10/9 = 176.953
* 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 291: 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 304: 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}}
Optimal ET sequence: {{Optimal ET sequence| 7g, …, 27eg, 34dh }}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 176.9645{{c}}


Badness: 0.017941
{{Optimal ET sequence|legend=0| 7g, …, 27eg, 34dh, 61degh }}


; Music
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 ===
Line 324: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 176.990
* WE: ~2 = 1198.5026{{c}}, ~10/9 = 176.9786{{c}}
* POTE ~2 = 1\1, ~10/9 = 177.200
* 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 339: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 177.017
* WE: ~2 = 1198.5149{{c}}, ~10/9 = 176.9778{{c}}
* POTE ~2 = 1\1, ~10/9 = 177.197
* 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 354: 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 367: 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 382: Line 386:
{{Mapping|legend=1| 1 1 1 0 | 0 4 9 19 }}
{{Mapping|legend=1| 1 1 1 0 | 0 4 9 19 }}


Optimal tunings:
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~10/9 = 176.900
* [[WE]]: ~2 = 1197.6555{{c}}, ~10/9 = 177.0104{{c}}
* POTE:~2 = 1\1, ~10/9 = 177.357
: [[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 397: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 176.704
* WE: ~2 = 1196.6462{{c}}, ~10/9 = 176.9174{{c}}
* POTE ~2 = 1\1, ~10/9 = 177.413
* 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 412: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~10/9 = 176.716
* WE: ~2 = 1197.4576{{c}}, ~10/9 = 176.8557{{c}}
* POTE ~2 = 1\1, ~10/9 = 177.231
* 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 427: 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 440: 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. 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.
Line 459: Line 469:
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}
{{Mapping|legend=1| 1 1 1 2 | 0 8 18 11 }}


Optimal tunings:
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~21/20 = 88.023
* [[WE]]: ~2 = 1199.6782{{c}}, ~21/20 = 88.0528{{c}}
* 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, 150, 259, 409bc, 668bbcc }}
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}


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


=== 11-limit ===
=== 11-limit ===
Line 474: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~21/20 = 87.910
* WE: ~2 = 1199.6025{{c}}, ~21/20 = 87.9460{{c}}
* POTE ~2 = 1\1, ~21/20 = 87.975
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.9453{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150ee, 191ee, 232cee }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 109e }}


Badness: 0.024078
Badness (Sintel): 0.796


==== 13-limit ====
==== 13-limit ====
Line 489: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~21/20 = 87.926
* WE: ~2 = 1198.8609{{c}}, ~21/20 = 87.0219{{c}}
* POTE ~2 = 1\1, ~21/20 = 88.106
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0557{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, 27e, 41, 150eef, 191eeff, 232ceeff }}
{{Optimal ET sequence|legend=0| 14c, 27e, 41, 68e, 109ef }}


Badness: 0.023276
Badness (Sintel): 0.962


===== 17-limit =====
===== 17-limit =====
Line 504: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~18/17 = 87.842
* WE: ~2 = 1198.4494{{c}}, ~21/20 = 87.9878{{c}}
* POTE ~2 = 1\1, ~18/17 = 88.102
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0324{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}


Badness: 0.021088
Badness (Sintel): 1.07


===== 19-limit =====
===== 19-limit =====
Line 519: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~18/17 = 87.866
* WE: ~2 = 1198.5995{{c}}, ~20/19 = 88.0081{{c}}
* POTE ~2 = 1\1, ~18/17 = 88.111
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.0471{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14c, 27eg, 41 }}
{{Optimal ET sequence|legend=0| 14c, 27eg, 41, 68egg }}


Badness: 0.016652
Badness (Sintel): 1.01


==== Octocat ====
==== Octocat ====
Line 534: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~21/20 = 88.090
* WE: ~2 = 1199.4441{{c}}, ~21/20 = 88.1380{{c}}
* POTE ~2 = 1\1, ~21/20 = 88.179
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1375{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14cf, 27e, 41f, 68ef, 109eff }}
{{Optimal ET sequence|legend=0| 14cf, 27e, 41f }}


Badness: 0.027601
Badness (Sintel): 1.14


===== 17-limit =====
===== 17-limit =====
Line 549: Line 561:
Mapping: {{mapping| 1 1 1 2 2 2 3 | 0 8 18 11 20 23 15 }}
Mapping: {{mapping| 1 1 1 2 2 2 3 | 0 8 18 11 20 23 15 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 88.011
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: {{Optimal ET sequence| 14cf, 27eg, 41f, 109effgg }}
{{Optimal ET sequence|legend=0| 14cf, 27eg }}


Badness: 0.023345
Badness (Sintel): 1.19


===== 19-limit =====
===== 19-limit =====
Line 562: Line 576:
Mapping: {{mapping| 1 1 1 2 2 2 3 3 | 0 8 18 11 20 23 15 17 }}
Mapping: {{mapping| 1 1 1 2 2 2 3 3 | 0 8 18 11 20 23 15 17 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 88.017
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: {{Optimal ET sequence| 14cf, 27eg, 41f, 109effgg }}
{{Optimal ET sequence|legend=0| 14cf, 27eg }}


Badness: 0.017916
Badness (Sintel): 1.09


==== Octopod ====
==== Octopod ====
Line 575: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~21/20 = 87.770
* WE: ~2 = 1200.5116{{c}}, ~21/20 = 87.7346{{c}}
* POTE ~2 = 1\1, ~21/20 = 87.697
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 87.7257{{c}}


Optimal ET sequence: {{Optimal ET sequence| 14cf, 27eff, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27eff, 41 }}


Badness: 0.028326
Badness (Sintel): 1.17


===== 17-limit =====
===== 17-limit =====
Line 590: Line 606:
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 8 18 11 20 37 15 }}
Mapping: {{mapping| 1 1 1 2 2 1 3 | 0 8 18 11 20 37 15 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 87.728
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: {{Optimal ET sequence| 14cf, 27effg, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}


Badness: 0.024660
Badness (Sintel): 1.26


===== 19-limit =====
===== 19-limit =====
Line 603: Line 621:
Mapping: {{mapping| 1 1 1 2 2 1 3 3 | 0 8 18 11 20 37 15 17 }}
Mapping: {{mapping| 1 1 1 2 2 1 3 3 | 0 8 18 11 20 37 15 17 }}


Optimal tuning (CTE): ~2 = 1\1, ~18/17 = 87.750
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: {{Optimal ET sequence| 14cf, 27effg, 41 }}
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41 }}


Badness: 0.019504
Badness (Sintel): 1.19


==== Dificot ====
==== Dificot ====
Line 614: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~13/9 = 643.916
* WE: ~2 = 1199.1496{{c}}, ~13/9 = 643.5328{{c}}
* POTE ~2 = 1\1, ~13/9 = 643.989
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 643.9567{{c}}


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


Badness: 0.051876
Badness (Sintel): 2.14


=== October ===
=== October ===
Line 631: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~21/20 = 88.026
* WE: ~2 = 1199.8843{{c}}, ~21/20 = 88.0261{{c}}
* POTE ~2 = 1\1, ~21/20 = 88.035
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0329{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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 646: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~21/20 = 88.041
* WE: ~2 = 1199.5060{{c}}, ~21/20 = 88.0388{{c}}
* POTE ~2 = 1\1, ~21/20 = 88.075
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.0697{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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 661: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~21/20 = 88.093
* WE: ~2 = 1199.3845{{c}}, ~21/20 = 88.0589{{c}}
* POTE ~2 = 1\1, ~21/20 = 88.104
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 88.1027{{c}}


Optimal ET sequence: {{Optimal ET sequence| 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 676: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~19/18 = 88.093
* WE: ~2 = 1199.4449{{c}}, ~20/19 = 88.0723{{c}}
* POTE ~2 = 1\1, ~19/18 = 88.113
* CWE: ~2 = 1200.0000{{c}}, ~20/19 = 88.1107{{c}}
 
Optimal ET sequence: {{Optimal ET sequence| 27, 41, 68, 109f }}
 
Badness: 0.020511
 
=== Devisemi (2.3.5.7.19) ===
 
==== 2.3.5.19 subgroup ====
Subgroup: 2.3.5.19
 
[[Comma list]]: 361/360, 20000/19683
 
[[Gencom]]: [2 19/18; 361/360 20000/19683]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 0 0 0 3}}, {{val|0 8 18 0 0 0 0 17}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 1 3}}, {{val|0 8 18 17}}]
 
[[Tp tuning|POL2 generator]]: ~19/18 = 88.077
 
{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5701 cents
 
==== 2.3.5.7.19 subgroup ====
Subgroup: 2.3.5.7.19
 
[[Comma list]]: 190/189, 245/243, 361/360
 
[[Gencom]]: [2 19/18; 190/189 245/243 361/360]
 
[[Gencom|Gencom mapping]]: [{{val|1 1 1 2 0 0 0 3}}, {{val|0 8 18 11 0 0 0 17}}]
 
[[Mapping|Sval mapping]]: [{{val|1 1 1 2 3}}, {{val|0 8 18 11 17}}]


[[Tp tuning|POL2 generator]]: ~19/18 = 88.075
{{Optimal ET sequence|legend=0| 27, 41, 68, 109f }}


{{Optimal ET sequence|legend=1| 14c, 27, 41, 68, 109 }}
Badness (Sintel): 1.25
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5780 cents


== Dodecacot ==
== Dodecacot ==
Line 726: 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


Optimal tunings:
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~28/27 = 58.648
* [[WE]]: ~2 = 1199.6912{{c}}, ~28/27 = 58.6600{{c}}
* POTE:~2 = 1\1, ~28/27 = 58.675
: [[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 ET sequence|legend=1| 41, 143d, 184, 225, 266c }}
{{Optimal ET sequence|legend=1| 20cd, 41, 143d, 184, 225 }}


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


=== 11-limit ===
=== 11-limit ===
Line 742: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~28/27 = 58.602
* WE: ~2 = 1199.3125{{c}}, ~28/27 = 58.6317{{c}}
* POTE ~2 = 1\1, ~28/27 = 58.665
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.6360{{c}}


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


Badness: 0.059528
Badness (Sintel): 1.97


=== 13-limit ===
=== 13-limit ===
Line 757: 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 tunings:
Optimal tunings:  
* CTE: ~2 = 1\1, ~27/26 = 58.551
* WE: ~2 = 1199.0713{{c}}, ~28/27 = 58.5932{{c}}
* POTE ~2 = 1\1, ~27/26 = 58.639
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 58.5982{{c}}
 
{{Optimal ET sequence|legend=0| 20cdef, 41 }}


Optimal ET sequence: {{Optimal ET sequence| 20cdef, 21cdef, 41 }}
Badness (Sintel): 1.80


Badness: 0.043645
== Weasel ==
{{See also| No-fives subgroup temperaments #Byhearted }}


== 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.
: ''See also: [[Jubilismic clan]] and [[No-fives subgroup temperaments #Skwares|No-fives subgroup temperaments #Byhearted]]''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 773: Line 763:


{{Mapping|legend=1| 2 2 2 3 | 0 4 9 9 }}
{{Mapping|legend=1| 2 2 2 3 | 0 4 9 9 }}
: mapping generators: ~7/5, ~10/9


: Mapping generators: ~7/5, ~10/9
[[Optimal tuning]]s:  
 
* [[WE]]: ~7/5 = 599.6934{{c}}, ~10/9 = 175.5626{{c}}
Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.472
: [[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, 82d, 130cdd }}
{{Optimal ET sequence|legend=1| 14c, 34d, 48 }}


[[Badness]]: 0.111574
[[Badness]] (Sintel): 2.82


=== 11-limit ===
=== 11-limit ===
Line 789: Line 782:
Mapping: {{mapping| 2 2 2 3 4 | 0 4 9 9 10 }}
Mapping: {{mapping| 2 2 2 3 4 | 0 4 9 9 10 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.401
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: {{Optimal ET sequence| 14c, 34d, 48, 82d, 130cdd }}
{{Optimal ET sequence|legend=0| 14c, 34d, 48 }}


Badness: 0.043923
Badness (Sintel): 1.45


=== 13-limit ===
=== 13-limit ===
The canonical mapping finds 13/8 at +15 generators rather than using the regular tetracot mapping, in order to find [[15/13]] as being half of [[4/3]].
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 802: Line 799:
Mapping: {{mapping| 2 2 2 3 4 3 | 0 4 9 9 10 15 }}
Mapping: {{mapping| 2 2 2 3 4 3 | 0 4 9 9 10 15 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.586
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: {{Optimal ET sequence| 14cf, 34d, 48f, 82d }}
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}


Badness: 0.031948
Badness (Sintel): 1.32


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


Line 815: Line 814:
Mapping: {{mapping| 2 2 2 3 4 3 7 | 0 4 9 9 10 15 4 }}
Mapping: {{mapping| 2 2 2 3 4 3 7 | 0 4 9 9 10 15 4 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.596
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: {{Optimal ET sequence| 14cf, 34d, 48f, 82d }}
{{Optimal ET sequence|legend=0| 14cf, 20cdef, 34d }}


Badness: 0.026119
Badness (Sintel): 1.33


=== 19-limit ===
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 828: Line 829:
Mapping: {{mapping| 2 2 2 3 4 3 7 5 | 0 4 9 9 10 15 4 12 }}
Mapping: {{mapping| 2 2 2 3 4 3 7 5 | 0 4 9 9 10 15 4 12 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~10/9 = 175.498
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


Optimal ET sequence: {{Optimal ET sequence| 14cf, 34dh, 48f, 82dh }}
Mapping: {{mapping| 2 2 2 3 4 8 7 5 | 0 4 9 9 10 -2 4 12}}


Badness: 0.021029
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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