Cloudy comma

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
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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 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

Optimal ET sequence: 5, 10, 15, 40, 50, 60, 65