Canou family
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
- Main article: Canou temperament
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: [⟨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
- 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
Vals: 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b
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
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
Synca, for symbiotic canou, adds the symbiotic comma and the wilschisma to the comma list.
Subgroup: 2.3.5.7.11
Comma list: 19712/19683, 42875/42768
Mapping: [⟨1 0 0 -1 -7], ⟨0 1 2 2 7], ⟨0 0 -4 3 -3]]
POTE generators: ~3/2 = 702.2549, ~81/70 = 254.6291
Vals: 94, 99e, 118, 193, 212, 311, 740, 1051d
Badness: 2.042 × 10-3
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
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 19712/19683, 42875/42768
Mapping: [⟨1 0 0 -1 -7 -13], ⟨0 1 2 2 7 10], ⟨0 0 -4 3 -3 4]]
POTE generators: ~3/2 = 702.1807, ~81/70 = 254.6239
Vals: 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
Mapping: [⟨2 0 0 -2 1], ⟨0 1 2 2 2], ⟨0 0 -4 3 -1]]
Mapping generators: ~99/70, ~3, ~81/70
POTE generators: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832
Vals: 80, 94, 118, 198, 212, 292, 330e, 410
Badness: 2.197 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 14641/14580
Mapping: [⟨2 0 0 -2 1 -11], ⟨0 1 2 2 2 5], ⟨0 0 -4 3 -1 6]]
POTE generators: ~3/2 = 702.5046, ~81/70 = 254.6501 or ~11/9 = 345.3499
Badness: 2.974 × 10-3
Semicanoumint
This extension was named semicanou in the earlier materials. It adds 352/351, the minthma, to the comma list, so that the flat ~11/9 simultaneously represents ~39/32.
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 9801/9800, 14641/14580
Mapping: [⟨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
Vals: 80, 94, 118, 174d, 198, 490f
Badness: 2.701 × 10-3
Semicanouwolf
This extension was named gentsemicanou in the earlier materials. It 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.
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 11011/10935
Mapping: [⟨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
Vals: 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.
Subgroup: 2.3.5.7
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]]
Badness: 4.523 × 10-3
Cantawolf
This extension was named canta in the earlier materials. It 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|]
Vals: 75ef, 80, 99e, 104c, 179e, 184c, 203ce
Badness: 3.470 × 10-3
Cantamint
This extension was named gentcanta in the earlier materials. It 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
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]]
Badness: 4.781 × 10-3