Cloudy comma

Interval information

Ratio	16807/16384
Factorization	2-14 × 75
Monzo	[-14 0 0 5⟩
Size in cents	44.129532¢
Name	cloudy comma
Color name	Lz53, Laquinzo comma
FJS name	[math]\text{d3}^{7,7,7,7,7}[/math]
Special properties	reduced, reduced harmonic
Tenney height (log2 nd)	28.0368
Weil height (log2 max(n, d))	28.0735
Wilson height (sopfr(nd))	63
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.50743 bits
Comma size	medium
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16807/16384, the cloudy comma, is a 7-limit medium comma. It is the amount by which five 8/7s fall short of one octave and exceeds 128/125 by the rainy comma. 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.

Temperaments

Tempering out the cloudy comma splits the octave into 5 equal parts and maps the harmonic 7 to 4\5. It leads to a number of regular temperaments in the cloudy clan.

Cloudy rank-3 temperament can be described as the 10&15&50 temperament, which tempers out 385/384 and 3087/3025 in the 11-limit; 105/104, 144/143, and 4459/4400 in the 13-limit.

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