Canou family: Difference between revisions

'''Canou''' is a rank-3 temperament that tempers out the [[canousma]], 4802000/4782969 = {{monzo|4 -14 3 4}}, a 7-limit comma measuring about 6.9 cents.

The 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 a [[980/729]] at about 510 cents, an audibly off perfect fourth. Three of them make a [[14/9]]; four of them make a [[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.

Decent amount of harmonic resources are provided by a simple 9-note scale. [[Flora Canou]] commented:

:''— It sounds somewhat like a Phrygian scale but the abundance of small intervals of [[28/27]] makes it melodically active.''

14- and 19-note scales are also possible. See [[canou scales]] for more information.  

For tunings, a basic option would be [[80edo]]. Others such as [[94edo]], [[99edo]] and [[118edo]] are more accurate; [[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|amicable temperament]].

Map: [<1 0 0 -1|, <0 1 2 2|, <0 0 -4 3|]

Wedgie: <<<4 -3 -14 -4 |||

POTE generators: ~3/2 = 702.3728, ~81/70 = 254.6253

EDOs: {{EDOs|75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b}}

Badness: 1.122 × 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 can be used as a tuning. Other options include 94edo, 118edo, and [[104edo]] in 104c val.

Map: [<2 0 0 -2 1|, <0 1 2 2 2|, <0 0 4 -3 1|]

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

EDOs: {{EDOs|80, 94, 118, 198, 212, 292, 330e, 410}}

Badness: 2.197 × 10^-3

This adds [[352/351]], the minthma, to the comma list. It is a natural extension to the 13-limit.

Commas: 352/351, 9801/9800, 14641/14580

Map: [<2 0 0 -2 1 11|, <0 1 2 2 2 -1|, <0 0 4 -3 1 1|]

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

EDOs: {{EDOs|80, 94, 118, 174d, 198, 490f}}

Badness: 2.701 × 10^-3

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.  

Commas: 351/350, 364/363, 11011/10935

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

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

EDOs: {{EDOs|80, 104c, 118f, 198f, 420cff}}

Badness: 3.511 × 10^-3

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.  

Commas: 896/891, 472392/471625

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

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

EDOs: {{EDOs|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.  

Commas: 351/350, 832/825, 13013/12960

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

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

EDOs: {{EDOs|75ef, 80, 99e, 104c, 179e, 184c, 203ce}}

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.  

Commas: 352/351, 364/363, 472392/471625

Map: [<1 0 0 -1 6 11|, <0 1 2 2 -2 -5|, <0 0 4 -3 -3 -3|]

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

EDOs: {{EDOs|75e, 80, 99ef, 179ef}}

Badness: 4.781 × 10^-3

[[Category:Theory]]

[[Category:Temperament family]]

[[Category:Planar temperament]]

[[Category:Canou]]