Canou family

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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

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 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 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.

Comma list: 4802000/4782969

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

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

Wedgie⟨⟨⟨4 -3 -14 -4]]]

Minimax tuning:

  • 7-odd-limit: 3 +c/14, 5 and 7 just
Eigenmonzos: 2, 5, 7
  • 9-odd-limit: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
Eigenmonzos: 2, 7/5

Lattice basis:

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

Vals75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b

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

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.

Comma list: 9801/9800, 14641/14580

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

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

Vals80, 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.

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

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

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

Vals80, 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: [2 0 0 -2 1 0], 0 1 2 2 2 3], 0 0 4 -3 1 5]]

Vals80, 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.

Comma list: 896/891, 472392/471625

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

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

Vals75e, 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.

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|]

Vals75ef, 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.

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

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

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

Vals75e, 80, 99ef, 179ef

Badness: 4.781 × 10-3