Cloudy comma
Ratio | 16807/16384 |
Factorization | 2^{-14} × 7^{5} |
Monzo | [-14 0 0 5⟩ |
Size in cents | 44.129532¢ |
Name | cloudy comma |
Color name | Lz^{5}3, Laquinzo comma |
FJS name | [math]\text{d3}^{7,7,7,7,7}[/math] |
Special properties | reduced, reduced harmonic |
Tenney height (log_{2} nd) | 28.0368 |
Weil height (log_{2} max(n, d)) | 28.0735 |
Wilson height (sopfr (nd)) | 63 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.50743 bits |
Comma size | medium |
open this interval in xen-calc |
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