Cloudy comma: Difference between revisions

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- temp data (addressed in misc. 7-limit temps), thereby removing the 11-limit extension as there's no justification for it
 
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== 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]]

Latest revision as of 11:38, 25 June 2026

Interval information
Ratio 16807/16384
Factorization 2-14 × 75
Monzo [-14 0 0 5⟩
Size in cents 44.12953¢
Name cloudy comma
Color name Lz53, Laquinzo comma
FJS name [math]\displaystyle{ \text{d3}^{7,7,7,7,7} }[/math]
Special properties reduced,
reduced harmonic
Tenney norm (log2 nd) 28.0368
Weil norm (log2 max(n, d)) 28.0735
Wilson norm (sopfr(nd)) 63
Comma size medium
Open this interval in xen-calc

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. In the full 7-limit, this leads to the rank-3 cloudy temperament. For edos 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.

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