Canou family: Difference between revisions
→Canou: about tuning options |
+synca (94 & 99e & 118) |
||
| Line 11: | Line 11: | ||
For tunings, a basic option would be [[99edo]]. Others such as [[80edo]], [[94edo]], and [[118edo]] are possible; [[19edo]] (perferably with stretched octaves) also provides a good trivial case, whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament. | For tunings, a basic option would be [[99edo]]. Others such as [[80edo]], [[94edo]], and [[118edo]] are possible; [[19edo]] (perferably with stretched octaves) also provides a good trivial case, whereas the [[optimal patent val]] goes up to [[1131edo]], relating it to the [[amicable]] temperament. | ||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 4802000/4782969 | [[Comma list]]: 4802000/4782969 | ||
[[Mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 1 2 2 }}, {{val| 0 0 -4 3 }}] | [[Mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 1 2 2 }}, {{val| 0 0 -4 3 }}] | ||
{{Multival|legend=1|rank=3| 4 -3 -14 -4 }} | {{Multival|legend=1|rank=3| 4 -3 -14 -4 }} | ||
[[POTE generator]]s: ~3/2 = 702.3728, ~81/70 = 254.6253 | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
| Line 30: | Line 32: | ||
: Angle (3/2, 81/70) = 73.88 deg | : Angle (3/2, 81/70) = 73.88 deg | ||
{{Val list|legend=1| 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b }} | {{Val list|legend=1| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }} | ||
[[Badness]]: 1.122 × 10<sup>-3</sup> | [[Badness]]: 1.122 × 10<sup>-3</sup> | ||
[[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 | ||
= Synca = | |||
Synca, for symbiotic canou, adds the [[symbiotic comma]] to the comma list. | |||
Subgroup: 2.3.5.7.11 | |||
[[Comma list]]: 19712/19683, 42875/42768 | |||
[[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 | |||
{{Val list|legend=1| 94, 99e, 118, 193, 212, 311, 740 }} | |||
[[Badness]]: 2.042 × 10<sup>-3</sup> | |||
[[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 | |||
= Semicanou = | = Semicanou = | ||
| Line 42: | Line 61: | ||
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. | 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. | ||
Still 80edo can be used as | Still 80edo, 94edo, and 118edo can be used as tunings. Other options include [[104edo]] in 104c val. | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 9801/9800, 14641/14580 | Comma list: 9801/9800, 14641/14580 | ||
| Line 55: | Line 76: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | |||
This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit. | This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit. | ||
| Line 73: | Line 96: | ||
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. | 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 | Comma list: 351/350, 364/363, 11011/10935 | ||
| Line 86: | Line 111: | ||
= Canta = | = 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. | 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. | ||
Subgroup: 2.3.5.7 | |||
Comma list: 896/891, 472392/471625 | Comma list: 896/891, 472392/471625 | ||
| Line 99: | Line 126: | ||
== 13-limit == | == 13-limit == | ||
This adds [[351/350]], the ratwolfsma, 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. Again 80edo makes the optimal. | This adds [[351/350]], the ratwolfsma, 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. Again 80edo makes the optimal. | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 351/350, 832/825, 13013/12960 | Comma list: 351/350, 832/825, 13013/12960 | ||
| Line 112: | Line 141: | ||
== Gentcanta == | == Gentcanta == | ||
This adds [[352/351]], the minthma, as well as [[364/363]], the gentle comma, to the comma list. It is a natural extension of canta, as 896/891 factors neatly into (352/351)×(364/363). Again 80edo makes the optimal. | This adds [[352/351]], the minthma, as well as [[364/363]], the gentle comma, to the comma list. It is a natural extension of canta, as 896/891 factors neatly into (352/351)×(364/363). Again 80edo makes the optimal. | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 352/351, 364/363, 472392/471625 | Comma list: 352/351, 364/363, 472392/471625 | ||
| Line 123: | Line 154: | ||
Badness: 4.781 × 10<sup>-3</sup> | Badness: 4.781 × 10<sup>-3</sup> | ||
[[Category: | [[Category:Regular temperament theory]] | ||
[[Category:Temperament family]] | [[Category:Temperament family]] | ||
[[Category:Canou]] | [[Category:Canou]] | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||