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

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, 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.

It 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.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969

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

Lattice basis:

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

Optimal tuning (CTE): ~3/2 = 702.3175, ~81/70 = 254.6220

Minimax tuning:

Eigenmonzo basis: 2.5.7
  • 9-odd-limit: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
Eigenmonzo basis: 2.7/5

Optimal GPV sequence19, 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

2.3.5.7.17 subgroup

Subgroup: 2.3.5.7.17

Comma list: 1225/1224, 295936/295245

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

Optimal tuning (CTE): ~3/2 = 702.3458, ~81/70 = 254.6233

Optimal GPV sequence: 94, 99, 193, 217, 292, 311, 410, 1131, 1541b

Badness: 0.775 × 10-3

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 1216/1215, 1225/1224, 1445/1444

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

Optimal tuning (CTE): ~3/2 = 702.3233, ~81/70 = 254.6279

Optimal GPV sequence: 94, 99, 118, 193, 217, 292h, 311, 410, 721

Badness: 0.548 × 10-3

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]]

Optimal tuning (CTE): ~3/2 = 702.2115, ~81/70 = 254.6215

Optimal GPV sequence94, 99e, 118, 193, 212, 311, 740, 1051d

Badness: 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): ~3/2 = 702.2075, ~81/70 = 254.6183

Optimal GPV sequence: 94, 118f, 193f, 212, 217, 311, 740, 1051d

Badness: 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): ~3/2 = 702.2296, ~51/44 = 254.6012

Optimal GPV sequence: 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg

Badness: 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): ~3/2 = 702.2355, ~22/19 = 254.5930

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

Badness: 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): ~3/2 = 702.8093, ~64/55 = 254.3378

Optimal GPV sequence75e, 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): ~3/2 = 703.6228, ~64/55 = 254.3447

Optimal GPV 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 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.

Natural extensions arise up to the 19-limit, and 410edo provides a satisfactory tuning solution to any of them.

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): ~3/2 = 702.4262, ~81/70 = 254.6191

Optimal GPV sequence80, 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): ~3/2 = 702.4802, ~81/70 = 254.6526

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

Badness: 2.974 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1089/1088, 1225/1224, 14641/14580

Mapping: [2 0 0 -2 1 -11 -10], 0 1 2 2 2 5 6], 0 0 -4 3 -1 6 -2]]

Optimal tuning (CTE): ~3/2 = 702.4415, ~81/70 = 254.6663

Optimal GPV sequence: 94, 118f, 198g, 212g, 292, 410

Badness: 2.421 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1089/1088, 1216/1215, 1225/1224, 1445/1444

Mapping: [2 0 0 -2 1 -11 -10 -12], 0 1 2 2 2 5 6 7], 0 0 -4 3 -1 6 -2 -4]]

Optimal tuning (CTE): ~3/2 = 702.4030, ~81/70 = 254.6870

Optimal GPV sequence: 94, 118f, 198gh, 212gh, 292h, 410, 622ef

Badness: 2.177 × 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): ~3/2 = 702.5374, ~81/70 = 254.6819

Optimal GPV 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): ~3/2 = 702.7417, ~15/13 = 254.3382

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

Badness: 3.511 × 10-3