Cloudy comma: Difference between revisions
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'''16807/16384''', the '''cloudy comma''', or systematically the '''5-7-comma''', is a [[medium comma]] in the [[7-limit]] with a size of about 44.1 [[cent]]s. It is the amount by which five [[8/7]]'s fall short of an octave. In the [[2.5.7 subgroup]], it exceeds [[128/125]] by the [[rainy comma]]. | |||
'''16807/16384''', the '''cloudy comma''', is a 7-limit | |||
== Temperaments == | == Temperaments == | ||
Tempering out | [[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]]. | ||
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& | === Cloudy (10 & 15 & 50) === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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{{Optimal ET sequence|legend=1| 5, 10, 15, 25, 35, 45, 50, 55, 60, 65 }} | {{Optimal ET sequence|legend=1| 5, 10, 15, 25, 35, 45, 50, 55, 60, 65 }} | ||
=== 11-limit cloudy (10& | === 11-limit cloudy (10 & 15 & 50) === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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{{Optimal ET sequence|legend=1| 5, 10, 15, 35, 40, 50, 55, 60, 65 }} | {{Optimal ET sequence|legend=1| 5, 10, 15, 35, 40, 50, 55, 60, 65 }} | ||
=== 13-limit cloudy (10& | === 13-limit cloudy (10 & 15 & 50) === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 10:53, 25 June 2026
| Interval information |
reduced harmonic
16807/16384, the cloudy comma, or systematically the 5-7-comma, is a medium comma in the 7-limit with a size of about 44.1 cents. It is the amount by which five 8/7's fall short of an octave. In the 2.5.7 subgroup, it exceeds 128/125 by the rainy comma.
Temperaments
Tempering out this comma splits the octave into 5 equal parts and maps the harmonic seventh to 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.
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: [⟨5 0 0 14], ⟨0 1 0 0], ⟨0 0 1 0]]
Mapping generators: ~8/7, ~3, ~5
POTE generators: ~3/2 = 699.467, ~5/4 = 382.669
Optimal ET sequence: 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: [⟨5 0 0 14 21], ⟨0 1 0 0 1], ⟨0 0 1 0 -1]]
POTE generators: ~3/2 = 699.779, ~5/4 = 386.575
Optimal ET sequence: 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: [⟨5 0 0 14 21 -1], ⟨0 1 0 0 1 1], ⟨0 0 1 0 -1 1]]
POTE generators: ~3/2 = 698.726, ~5/4 = 384.760