|
|
| Line 8: |
Line 8: |
|
| |
|
| == Temperaments == | | == Temperaments == |
| [[Tempering out]] this comma splits the octave into 5 equal parts and maps the [[7/4|harmonic seventh]] to [[5edo|4\5]]. For equal divisions ''N'' up to 65, the comma is tempered out if and only if 5 divides ''N''. Examples are [[5edo]], [[10edo]], [[15edo]], [[50edo]], [[55edo]] and [[65edo]]. Rank-2 temperaments tempering it out include [[blackwood]], [[coblack]], [[cloudtone]], [[sengasec]], [[warlock]], [[pentacloud]], [[decic]], [[pentadecal]], and [[quinkee]]. | | [[Tempering out]] this comma splits the octave into 5 equal parts and maps the [[7/4|harmonic seventh]] to [[5edo|4\5]]. In the full 7-limit, this leads to the rank-3 [[cloudy]] temperament. For [[edo]]s ''N'' up to 65, the comma is tempered out if and only if 5 divides ''N''. Examples are [[5edo]], [[10edo]], [[15edo]], [[50edo]], [[55edo]] and [[65edo]]. Rank-2 temperaments tempering it out include [[blackwood]], [[coblack]], [[cloudtone]], [[sengasec]], [[warlock]], [[pentacloud]], [[decic]], [[pentadecal]], and [[quinkee]]. |
| | |
| The cloudy rank-3 temperament can be described as the 10 & 15 & 50 temperament, which can be extended to the 11-limit by tempering out 385/384 and 3087/3025, and the 13-limit by tempering out 105/104, 144/143, and 4459/4400.
| |
| | |
| === Cloudy (10 & 15 & 50) ===
| |
| Subgroup: 2.3.5.7
| |
| | |
| [[Comma list]]: 16807/16384
| |
| | |
| [[Mapping]]: [{{val| 5 0 0 14 }}, {{val| 0 1 0 0 }}, {{val| 0 0 1 0 }}]
| |
| | |
| Mapping generators: ~8/7, ~3, ~5
| |
| | |
| [[POTE generator]]s: ~3/2 = 699.467, ~5/4 = 382.669
| |
| | |
| {{Optimal ET sequence|legend=1| 5, 10, 15, 25, 35, 45, 50, 55, 60, 65 }}
| |
| | |
| === 11-limit cloudy (10 & 15 & 50) ===
| |
| Subgroup: 2.3.5.7.11
| |
| | |
| Comma list: 385/384, 3087/3025
| |
| | |
| Mapping: [{{val| 5 0 0 14 21 }}, {{val| 0 1 0 0 1 }}, {{val| 0 0 1 0 -1 }}]
| |
| | |
| POTE generators: ~3/2 = 699.779, ~5/4 = 386.575
| |
| | |
| {{Optimal ET sequence|legend=1| 5, 10, 15, 35, 40, 50, 55, 60, 65 }}
| |
| | |
| === 13-limit cloudy (10 & 15 & 50) ===
| |
| Subgroup: 2.3.5.7.11.13
| |
| | |
| Comma list: 105/104, 144/143, 3087/3025
| |
| | |
| Mapping: [{{val| 5 0 0 14 21 -1 }}, {{val| 0 1 0 0 1 1 }}, {{val| 0 0 1 0 -1 1 }}]
| |
| | |
| POTE generators: ~3/2 = 698.726, ~5/4 = 384.760
| |
| | |
| {{Optimal ET sequence|legend=1| 5, 10, 15, 40, 50, 60, 65 }}
| |
|
| |
|
| [[Category:Cloudy]] | | [[Category:Cloudy]] |
| [[Category:Commas named for their regular temperament properties]] | | [[Category:Commas named for their regular temperament properties]] |