Canou family

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Canou is a rank-3 temperament that tempers out the canousma, 4802000/4782969 = [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 temperament.

Comma: 4802000/4782969

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: 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.

Commas: 9801/9800, 14641/14580

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: 80, 94, 118, 198, 212, 292, 330e, 410

Badness: 2.197 × 10-3

13-limit

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: 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.

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: 80, 104c, 118f, 198f, 420cff

Badness: 3.511 × 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.

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: 75e, 80, 99e, 179e

Badness: 4.523 × 10-3

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.

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: 75ef, 80, 99e, 104c, 179e, 184c, 203ce

Badness: 3.470 × 10-3

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.

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: 75e, 80, 99ef, 179ef

Badness: 4.781 × 10-3