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
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. Three make 14/9; four make 9/5. It therefore splits the large septimal diesis, 49/48, into three equal parts, making two distinct interseptimal intervals related to the 35th harmonic.
A basic tuning option would be 99edo, although 80edo is even simpler and distinctive. More intricate tunings are provided by 311edo and 410edo, 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]]
- mapping generators: ~2, ~3, ~81/70
Lattice basis:
- 3/2 length = 0.8110, 81/70 length = 0.5135
- Angle (3/2, 81/70) = 73.88 deg
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
- 7-odd-limit: 3 +c/14, 5 and 7 just
- 9-odd-limit: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
Optimal ET sequence: 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b
Badness (Smith): 1.122 × 10-3
Complexity spectrum: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5
Undecimal canou
The fifth is in the range where a stack of four (i.e. a major third) can serve as ~19/15 and a stack of five (i.e. a major seventh) can serve as ~19/10, tempering out 1216/1215. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot 22/19, so it only makes sense to temper out their difference, 1540/1539. The implied 11-limit comma is the symbiotic comma, which suggests the wilschisma should also be tempered out in the 13-limit.
Since the syntonic comma has been split in two, it is natural to map 19/17 to the mean of 9/8 and 10/9, tempering out 1445/1444. From a commatic point of view, notice the other 11-limit comma, 42875/42768, is S34 × S352, suggesting tempering out 595/594 (S34 × S35), 1156/1155 (S34), and 1225/1224 (S35), which coincides with above. Finally, we can map 23/20 to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out 760/759. 311edo remains an excellent tuning in all the limits.
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]]
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
Optimal ET sequence: 94, 99e, 118, 193, 212, 311, 740, 1051d
Badness (Smith): 2.04 × 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]]
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
Optimal ET sequence: 94, 118f, 193f, 212, 217, 311, 740, 1051d
Badness (Smith): 2.56 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 595/594, 833/832, 1156/1155, 19712/19683
Mapping: [⟨1 0 0 -1 -7 -13 -5], ⟨0 1 2 2 7 10 6], ⟨0 0 -4 3 -3 4 -2]]
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
Optimal ET sequence: 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg
Badness (Smith): 1.49 × 10-3
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215
Mapping: [⟨], ⟨1 0 0 -1 -7 -13 -5 -6], ⟨0 1 2 2 7 10 6 7], ⟨0 0 -4 3 -3 4 -2 -4]]
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
Optimal ET sequence: 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh
Badness (Smith): 1.00 × 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. It has a natural extension to the 13-limit since 896/891 = (352/351)(364/363), named gentcanta in earlier materials.
Subgroup: 2.3.5.7.11
Comma list: 896/891, 472392/471625
Mapping: [⟨1 0 0 -1 6], ⟨0 1 2 2 -2], ⟨0 0 4 -3 -3]]
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
Optimal ET sequence: 75e, 80, 99e, 179e
Badness: 4.523 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 472392/471625
Mapping: [⟨1 0 0 -1 6 11], ⟨0 1 2 2 -2 -5], ⟨0 0 4 -3 -3 -3]]
Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
Optimal ET sequence: 75e, 80, 99ef, 179ef
Badness: 4.781 × 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 99/70~140/99. 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 about 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, and 11/10.
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
Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
Optimal ET sequence: 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]]
Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
Optimal ET sequence: 80f, 94, 118f, 198, 410
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]]
Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
Optimal ET sequence: 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]]
Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
Optimal ET sequence: 80, 104c, 118f, 198f, 420cff
Badness: 3.511 × 10-3