Canou family: Difference between revisions

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

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]]. 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

[[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 }}

[[POTE generator]]s: ~3/2 = 702.3728, ~81/70 = 254.6253

: [[Eigenmonzo]]s: 2, 5, 7

: [[Eigenmonzo]]s: 2, 7/5

: 3/2 length = 0.8110, 81/70 length = 0.5135

: Angle (3/2, 81/70) = 73.88 deg

 

[[Badness]]: 1.122 × 10^-3

=== Extensions ===

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.  

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

Subgroup: 2.3.5.7.11

[[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, 1051d }}

[[Badness]]: 2.042 × 10^-3

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

Vals: {{Val list| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}

Badness: 2.555 × 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. 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.

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

POTE generators: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832

Mapping: [{{val| 2 0 0 -2 1 }}, {{val| 0 1 2 2 2 }}, {{val| 0 0 4 -3 1 }}]

{{Val list|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}

Badness: 2.197 × 10^-3

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.  

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

POTE generators: ~3/2 = 702.8788, ~81/70 = 254.6664 or ~11/9 = 345.3336

Mapping: [{{val| 2 0 0 -2 1 11 }}, {{val| 0 1 2 2 2 -1 }}, {{val| 0 0 4 -3 1 1 }}]

{{Val list|legend=1| 80, 94, 118, 174d, 198, 490f }}

Badness: 2.701 × 10^-3

=== Gentsemicanou ===

This 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.  

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

POTE generators: ~3/2 = 702.7876, ~15/13 = 254.3411 or ~11/9 = 345.6789

Mapping: [{{val| 2 0 0 -2 1 0 }}, {{val| 0 1 2 2 2 3 }}, {{val| 0 0 4 -3 1 5 }}]

{{Val list|legend=1| 80, 104c, 118f, 198f, 420cff }}

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

POTE generators: ~3/2 = 703.7418, ~64/55 = 254.6133

Mapping: [{{val| 1 0 0 -1 6 }}, {{val| 0 1 2 2 -2 }}, {{val| 0 0 4 -3 -3 }}]

{{Val list|legend=1| 75e, 80, 99e, 179e }}

Badness: 4.523 × 10^-3

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

POTE generators: ~3/2 = 703.8423, ~15/13 = 254.3605

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

Badness: 3.470 × 10^-3

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.  

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

POTE generators: ~3/2 = 703.8695, ~64/55 = 254.6321

Mapping: [{{val| 1 0 0 -1 6 11 }}, {{val| 0 1 2 2 -2 -5 }}, {{val| 0 0 4 -3 -3 -3 }}]

{{Val list|legend=1| 75e, 80, 99ef, 179ef }}

Badness: 4.781 × 10^-3

[[Category:Regular temperament theory]]

[[Category:Temperament family]]