Canou family: Difference between revisions

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+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 }}

Revision as of 17:40, 14 August 2022

The canou family of rank-3 temperaments tempers out the canousma, 4802000/4782969 = [4 -14 3 4⟩, a 7-limit comma measuring about 6.9 cents.

Canou

Main article: Canou temperament

The canou temperament features a period of an octave and generators of 3/2 and 81/70. The 81/70-generator is about 255 cents. Two of them interestingly make 980/729 at about 510 cents, an audibly off perfect fourth. Three make 14/9; four make 9/5. It therefore also features splitting the septimal diesis, 49/48, into three equal parts, making two distinct interseptimal intervals related to the 35th harmonic.

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.

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

Mapping: [⟨1 0 0 -1], ⟨0 1 2 2], ⟨0 0 -4 3]]

Lattice basis:

3/2 length = 0.8110, 81/70 length = 0.5135
Angle (3/2, 81/70) = 73.88 deg

Optimal tuning (CTE): ~3/2 = 702.3175, ~81/70 = 254.6220

Minimax tuning:

Eigenmonzo basis: 2.5.7
  • 9-odd-limit: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
Eigenmonzo basis: 2.7/5

Template:Val list

Badness: 1.122 × 10-3

Complexity spectrum: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5

2.3.5.7.17 subgroup

Subgroup: 2.3.5.7.17

Comma list: 1225/1224, 295936/295245

Mapping: [⟨1 0 0 -1 -5], ⟨0 1 2 2 6], ⟨0 0 -4 3 -2]]

Optimal tuning (CTE): ~3/2 = 702.3458, ~81/70 = 254.6233

Optimal GPV sequence: Template:Val list

Badness: 0.775 × 10-3

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 1216/1215, 1225/1224, 1445/1444

Mapping: [⟨1 0 0 -1 -5 -6], ⟨0 1 2 2 6 7], ⟨0 0 -4 3 -2 -4]]

Optimal tuning (CTE): ~3/2 = 702.3233, ~81/70 = 254.6279

Optimal GPV sequence: Template:Val list

Badness: 0.548 × 10-3

Synca

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

Subgroup: 2.3.5.7.11

Comma list: 19712/19683, 42875/42768

Mapping: [⟨1 0 0 -1 -7], ⟨0 1 2 2 7], ⟨0 0 -4 3 -3]]

Optimal tuning (CTE): ~3/2 = 702.2115, ~81/70 = 254.6215

Template:Val list

Badness: 2.042 × 10-3

Complexity spectrum: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 19712/19683, 42875/42768

Mapping: [⟨1 0 0 -1 -7 -13], ⟨0 1 2 2 7 10], ⟨0 0 -4 3 -3 4]]

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

Optimal GPV sequence: Template:Val list

Badness: 2.555 × 10-3

Canta

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

Comma list: 896/891, 472392/471625

Mapping: [⟨1 0 0 -1 6], ⟨0 1 2 2 -2], ⟨0 0 4 -3 -3]]

Optimal tuning (CTE): ~3/2 = 702.8093, ~64/55 = 254.3378

Template:Val list

Badness: 4.523 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 472392/471625

Mapping: [⟨1 0 0 -1 6 11], ⟨0 1 2 2 -2 -5], ⟨0 0 4 -3 -3 -3]]

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

Optimal GPV sequence: Template:Val list

Badness: 4.781 × 10-3

Semicanou

Semicanou adds 9801/9800, the kalisma, to the comma list, and may be described as 80 & 94 & 118. It splits the octave into two equal parts, each representing 99/70~140/99. Note that 99/70 = (81/70)(11/9), this extension is more than natural.

The other comma necessary to define it is 14641/14580, the semicanousma, which is the difference between 121/120 and 243/242. By flattening the 11th harmonic by one cent, it identifies 20/11 by three 11/9's stacked, so an octave can be divided into 11/9-11/9-11/9-11/10.

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

Comma list: 9801/9800, 14641/14580

Mapping: [⟨2 0 0 -2 1], ⟨0 1 2 2 2], ⟨0 0 -4 3 -1]]

Mapping generators: ~99/70, ~3, ~81/70

Optimal tuning (CTE): ~3/2 = 702.4262, ~81/70 = 254.6191

Template:Val list

Badness: 2.197 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 14641/14580

Mapping: [⟨2 0 0 -2 1 -11], ⟨0 1 2 2 2 5], ⟨0 0 -4 3 -1 6]]

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

Optimal GPV sequence: Template:Val list

Badness: 2.974 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1089/1088, 1225/1224, 14641/14580

Mapping: [⟨2 0 0 -2 1 -11 -10], ⟨0 1 2 2 2 5 6], ⟨0 0 -4 3 -1 6 -2]]

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

Optimal GPV sequence: Template:Val list

Badness: 2.421 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1089/1088, 1216/1215, 1225/1224, 1445/1444

Mapping: [⟨2 0 0 -2 1 -11 -10 -12], ⟨0 1 2 2 2 5 6 7], ⟨0 0 -4 3 -1 6 -2 -4]]

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

Optimal GPV sequence: Template:Val list

Badness: 2.177 × 10-3

Semicanoumint

This extension was named semicanou in the earlier materials. It adds 352/351, the minthma, to the comma list, so that the flat ~11/9 simultaneously represents ~39/32.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 9801/9800, 14641/14580

Mapping: [⟨2 0 0 -2 1 11], ⟨0 1 2 2 2 -1], ⟨0 0 -4 3 -1 -1]]

Optimal tuning (CTE): ~3/2 = 702.5374, ~81/70 = 254.6819

Optimal GPV sequence: Template:Val list

Badness: 2.701 × 10-3

Semicanouwolf

This extension was named gentsemicanou in the earlier materials. It adds 351/350, the ratwolfsma, as wells as 364/363, the gentle comma, to the comma list. Since 351/350 = (81/70)/(15/13), the 81/70-generator simultaneously represents 15/13, adding a lot of fun to the scale.

Not supported by many patent vals, 80edo easily makes the optimal. Yet 104edo in 104c val and 118edo in 118f val are worth mentioning, and the temperament may be described as 80 & 104c & 118f.

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 11011/10935

Mapping: [⟨2 0 0 -2 1 0], ⟨0 1 2 2 2 3], ⟨0 0 -4 3 -1 -5]]

Optimal tuning (CTE): ~3/2 = 702.7417, ~15/13 = 254.3382

Optimal GPV sequence: Template:Val list

Badness: 3.511 × 10-3

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