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Canou family: Difference between revisions

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Improve explanations esp. tunings
+2.3.5.7.17.19 subgroup and replace POTE with CTE
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For tunings, a basic option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament.  
For tunings, a basic option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament.  


Subgroup: 2.3.5.7
It has a neat extension to the 2.3.5.7.17.19 subgroup with virtually no additional errors. The [[comma basis]] is {1216/1215, 1225/1224, 1445/1444}. Otherwise, 11- and 13-limit extensions are somewhat less ideal.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[4802000/4782969]]
[[Comma list]]: [[4802000/4782969]]
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: Angle (3/2, 81/70) = 73.88 deg
: Angle (3/2, 81/70) = 73.88 deg


[[POTE generator]]s: ~3/2 = 702.3728, ~81/70 = 254.6253
Optimal tuning ([[CTE]]): ~3/2 = 702.3175, ~81/70 = 254.6220


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: 3 +c/14, 5 and 7 just
* [[7-odd-limit]]: 3 +c/14, 5 and 7 just
: [[Eigenmonzo]]s: 2, 5, 7
: [[Eigenmonzo basis]]: 2.5.7
* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
: [[Eigenmonzo]]s: 2, 7/5
: [[Eigenmonzo basis]]: 2.7/5


{{Val list|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}
{{Val list|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}
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[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5
[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5


=== Overview to extensions ===
=== 2.3.5.7.17 subgroup ===
Canou has a neat extension to the 2.3.5.7.17.19 subgroup with virtually no additional errors. The [[comma basis]] is {1216/1215, 1225/1224, 1445/1444}. Otherwise, 11- and 13-limit extensions are somewhat less ideal.
Subgroup: 2.3.5.7.17
 
Comma list: 1225/1224, 295936/295245
 
Mapping: [{{val| 1 0 0 -1 -5 }}, {{val| 0 1 2 2 6 }}, {{val| 0 0 -4 3 -2 }}]
 
Optimal tuning (CTE): ~3/2 = 702.3458, ~81/70 = 254.6233
 
Optimal GPV sequence: {{val list| 94, 99, 193, 217, 292, 311, 410, 1131, 1541b }}
 
Badness: 0.775 × 10<sup>-3</sup>
 
=== 2.3.5.7.17.19 subgroup ===
Subgroup: 2.3.5.7.17.19
 
Comma list: 1216/1215, 1225/1224, 1445/1444
 
Mapping: [{{val| 1 0 0 -1 -5 -6 }}, {{val| 0 1 2 2 6 7 }}, {{val| 0 0 -4 3 -2 -4 }}]
 
Optimal tuning (CTE): ~3/2 = 702.3233, ~81/70 = 254.6279
 
Optimal GPV sequence: {{val list| 94, 99, 118, 193, 217, 292h, 311, 410, 721 }}
 
Badness: 0.548 × 10<sup>-3</sup>


== Synca ==
== Synca ==
Synca, for symbiotic canou, adds the [[symbiotic comma]] and the [[wilschisma]] to the comma list.  
Synca, for symbiotic canou, adds the [[symbiotic comma]] and the [[wilschisma]] to the comma list.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 19712/19683, 42875/42768
[[Comma list]]: 19712/19683, 42875/42768
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[[Mapping]]: [{{val| 1 0 0 -1 -7 }}, {{val| 0 1 2 2 7 }}, {{val| 0 0 -4 3 -3 }}]
[[Mapping]]: [{{val| 1 0 0 -1 -7 }}, {{val| 0 1 2 2 7 }}, {{val| 0 0 -4 3 -3 }}]


[[POTE generator]]s: ~3/2 = 702.2549, ~81/70 = 254.6291
Optimal tuning ([[CTE]]): ~3/2 = 702.2115, ~81/70 = 254.6215


{{Val list|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
{{Val list|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
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Mapping: [{{val| 1 0 0 -1 -7 -13 }}, {{val| 0 1 2 2 7 10 }}, {{val| 0 0 -4 3 -3 4 }}]
Mapping: [{{val| 1 0 0 -1 -7 -13 }}, {{val| 0 1 2 2 7 10 }}, {{val| 0 0 -4 3 -3 4 }}]


POTE generators: ~3/2 = 702.1807, ~81/70 = 254.6239
Optimal tuning (CTE): ~3/2 = 702.2075, ~81/70 = 254.6183


Optimal GPV sequence: {{val list| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
Optimal GPV sequence: {{val list| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
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By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named ''gentcanta'' in earlier materials.  
By adding [[896/891]], the pentacircle comma, [[33/32]] is equated with 28/27, so the scale is filled with this 33/32~28/27 mixture. This may be described as 75e & 80 & 99e, and 80edo makes the optimal. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named ''gentcanta'' in earlier materials.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 896/891, 472392/471625
[[Comma list]]: 896/891, 472392/471625
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[[Mapping]]: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]
[[Mapping]]: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]


[[POTE generator]]s: ~3/2 = 703.7418, ~64/55 = 254.6133
Optimal tuning ([[CTE]]): ~3/2 = 702.8093, ~64/55 = 254.3378


{{Val list|legend=1| 75e, 80, 99e, 179e }}
{{Val list|legend=1| 75e, 80, 99e, 179e }}
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Mapping: [{{val| 1 0 0 -1 6 11 }}, {{val| 0 1 2 2 -2 -5 }}, {{val| 0 0 4 -3 -3 -3 }}]
Mapping: [{{val| 1 0 0 -1 6 11 }}, {{val| 0 1 2 2 -2 -5 }}, {{val| 0 0 4 -3 -3 -3 }}]


POTE generators: ~3/2 = 703.8695, ~64/55 = 254.6321
Optimal tuning (CTE): ~3/2 = 703.6228, ~64/55 = 254.3447


Optimal GPV sequence: {{val list| 75e, 80, 99ef, 179ef }}
Optimal GPV sequence: {{val list| 75e, 80, 99ef, 179ef }}
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Natural extensions arise up to the 19-limit, and 410edo provides a satisfactory tuning solution to any of them.  
Natural extensions arise up to the 19-limit, and 410edo provides a satisfactory tuning solution to any of them.  


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 9801/9800, 14641/14580
[[Comma list]]: 9801/9800, 14641/14580
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Mapping generators: ~99/70, ~3, ~81/70
Mapping generators: ~99/70, ~3, ~81/70


[[POTE generator]]s: ~3/2 = 702.3850, ~81/70 = 254.6168
Optimal tuning ([[CTE]]): ~3/2 = 702.4262, ~81/70 = 254.6191


{{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
{{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
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Mapping: [{{val| 2 0 0 -2 1 -11 }}, {{val| 0 1 2 2 2 5 }}, {{val| 0 0 -4 3 -1 6 }}]
Mapping: [{{val| 2 0 0 -2 1 -11 }}, {{val| 0 1 2 2 2 5 }}, {{val| 0 0 -4 3 -1 6 }}]


POTE generators: ~3/2 = 702.5046, ~81/70 = 254.6501
Optimal tuning (CTE): ~3/2 = 702.4802, ~81/70 = 254.6526


Optimal GPV sequence: {{val list| 80f, 94, 118f, 198, 410 }}
Optimal GPV sequence: {{val list| 80f, 94, 118f, 198, 410 }}
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Mapping: [{{val| 2 0 0 -2 1 -11 -10 }}, {{val| 0 1 2 2 2 5 6 }}, {{val| 0 0 -4 3 -1 6 -2 }}]
Mapping: [{{val| 2 0 0 -2 1 -11 -10 }}, {{val| 0 1 2 2 2 5 6 }}, {{val| 0 0 -4 3 -1 6 -2 }}]


POTE generators: ~3/2 = 702.4241, ~81/70 = 254.6672
Optimal tuning (CTE): ~3/2 = 702.4415, ~81/70 = 254.6663


Optimal GPV sequence: {{val list| 94, 118f, 198g, 212g, 292, 410 }}
Optimal GPV sequence: {{val list| 94, 118f, 198g, 212g, 292, 410 }}
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Mapping: [{{val| 2 0 0 -2 1 -11 -10 -12 }}, {{val| 0 1 2 2 2 5 6 7 }}, {{val| 0 0 -4 3 -1 6 -2 -4 }}]
Mapping: [{{val| 2 0 0 -2 1 -11 -10 -12 }}, {{val| 0 1 2 2 2 5 6 7 }}, {{val| 0 0 -4 3 -1 6 -2 -4 }}]


POTE generators: ~3/2 = 702.3551, ~81/70 = 254.6866
Optimal tuning (CTE): ~3/2 = 702.4030, ~81/70 = 254.6870


Optimal GPV sequence: {{val list| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
Optimal GPV sequence: {{val list| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
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Mapping: [{{val| 2 0 0 -2 1 11 }}, {{val| 0 1 2 2 2 -1 }}, {{val| 0 0 -4 3 -1 -1 }}]
Mapping: [{{val| 2 0 0 -2 1 11 }}, {{val| 0 1 2 2 2 -1 }}, {{val| 0 0 -4 3 -1 -1 }}]


POTE generators: ~3/2 = 702.8788, ~81/70 = 254.6664 or ~11/9 = 345.3336
Optimal tuning (CTE): ~3/2 = 702.5374, ~81/70 = 254.6819


Optimal GPV sequence: {{val list| 80, 94, 118, 174d, 198, 490f }}
Optimal GPV sequence: {{val list| 80, 94, 118, 174d, 198, 490f }}
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Mapping: [{{val| 2 0 0 -2 1 0 }}, {{val| 0 1 2 2 2 3 }}, {{val| 0 0 -4 3 -1 -5 }}]
Mapping: [{{val| 2 0 0 -2 1 0 }}, {{val| 0 1 2 2 2 3 }}, {{val| 0 0 -4 3 -1 -5 }}]


POTE generators: ~3/2 = 702.7876, ~15/13 = 254.3411 or ~11/9 = 345.6789
Optimal tuning (CTE): ~3/2 = 702.7417, ~15/13 = 254.3382


Optimal GPV sequence: {{val list| 80, 104c, 118f, 198f, 420cff }}
Optimal GPV sequence: {{val list| 80, 104c, 118f, 198f, 420cff }}
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