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. 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

Comma list: 4802000/4782969

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 tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220

error map: ⟨0.0000 +0.3625 -0.1667 -0.3249]

• CWE: ~2 = 1200.0000, ~3/2 = 702.3455, ~81/70 = 254.6237

error map: ⟨0.0000 +0.3904 -0.1175 -0.2640]

Minimax tuning:

• 7-odd-limit: 3 +c/14, 5 and 7 just

eigenmonzo (unchanged-interval) basis: 2.5.7

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

eigenmonzo (unchanged-interval) basis: 2.7/5

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 tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215

error map: ⟨0.0000 +0.2565 -0.3768 -0.5383 +0.2980]

• CWE: ~2 = 1200.0000, ~3/2 = 702.1829, ~81/70 = 254.6186

error map: ⟨0.0000 +0.2279 -0.4221 -0.6043 +0.1069]

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 tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
• CWE: ~2 = 1200.0000, ~3/2 = 702.1889, ~81/70 = 254.6222

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 tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
• CWE: ~2 = 1200.0000, ~3/2 = 702.2055, ~51/44 = 254.6066

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 tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
• CWE: ~2 = 1200.0000, ~3/2 = 702.2117, ~22/19 = 254.5983

Optimal ET sequence: 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh

Badness (Smith): 1.00 × 10^-3

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155

Mapping: [⟨1 0 0 -1 -7 -13 -5 -6 4], ⟨0 1 2 2 7 10 6 7 1], ⟨0 0 -4 3 -3 4 -2 -4 -5]]

Optimal tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222
• CWE: ~2 = 1200.0000, ~3/2 = 702.2359, ~22/19 = 254.6223

Optimal ET sequence: 94, 193f, 212gh, 217, 311

Badness (Smith): 0.948 × 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 tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378

error map: ⟨0.0000 +0.8543 +1.9537 -0.1940 +6.0769]

• CWE: ~2 = 1200.0000, ~3/2 = 703.5249, ~64/55 = 254.5492

error map: ⟨0.0000 +1.5699 +2.5393 +1.8714 +5.2799]

Optimal ET sequence: 75e, 80, 99e, 179e, 457bcddeeee

Badness (Smith): 4.52 × 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 tunings:

• CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
• CWE: ~2 = 1200.0000, ~3/2 = 703.8323, ~64/55 = 254.5887

Optimal ET sequence: 75e, 80, 99ef, 179ef

Badness (Smith): 4.78 × 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 tunings:

• CTE: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191

error map: ⟨0.0000 +0.4712 +0.0625 -0.1163 -1.0846]

• CWE: ~99/70 = 600.0000, ~3/2 = 702.4048, ~81/70 = 254.6179

error map: ⟨0.0000 +0.4498 +0.0245 -0.1627 -1.1262]

Optimal ET sequence: 80, 94, 118, 198, 212, 292, 330e, 410

Badness (Smith): 2.20 × 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 tunings:

• CTE: ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
• CWE: ~99/70 = 600.0000, ~3/2 = 702.4945, ~81/70 = 254.6511

Optimal ET sequence: 80f, 94, 118f, 198, 410

Badness (Smith): 2.97 × 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 tunings:

• CTE: ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
• CTE: ~99/70 = 600.0000, ~3/2 = 702.7916, ~81/70 = 254.6704

Optimal ET sequence: 80, 94, 118, 174d, 198, 490f

Badness (Smith): 2.70 × 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 tunings:

• CTE: ~55/39 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
• CWE: ~55/39 = 600.0000, ~3/2 = 702.8092, ~15/13 = 254.3396

Optimal ET sequence: 80, 104c, 118f, 198f, 420cff

Badness (Smith): 3.51 × 10^-3