Canou family
Canou is a rank-3 temperament that tempers out the canousma, 4802000/4782969 = [4 -14 3 4⟩, a 7-limit comma measuring about 6.9 cents.
The 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 80edo. Others such as 94edo, 99edo and 118edo are more accurate; 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: 4802000/4782969
Map: [<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
EDOs: 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b
Badness: 0.001122
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], 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.
Commas: 9801/9800, 14641/14580
Map: [<2 0 0 -2 1|, <0 1 2 2 2|, <0 0 4 -3 1|]
POTE generators: ~3/2 = 702.3850, ~81/70 = 254.6168 or ~11/9 = 345.3832
EDOs: 80, 94, 118, 198, 212, 292, 330e, 410
Badness: 0.002197
13-limit
This adds 352/351, the minthma, to the comma list. It is a natural extension to the 13-limit.
Commas: 352/351, 9801/9800, 14641/14580
Map: [<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