Cloudy comma: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Ratio = 16807/16384 | | Ratio = 16807/16384 | ||
| Name = cloudy comma | | Name = cloudy comma | ||
| Color name = | | Color name = Lz<sup>5</sup>3, Laquinzo comma | ||
| Comma = yes | |||
| | |||
}} | }} | ||
'''16807/16384''', the '''cloudy comma''', is a 7-limit medium comma. It is the amount by which five [[8/7]]s fall short of one octave and exceeds [[128/125]] by [[ | '''16807/16384''', the '''cloudy comma''', is a 7-limit medium comma. It is the amount by which five [[8/7]]s 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 leads a number of regular 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) === | ||
[[Category: | Subgroup: 2.3.5.7 | ||
[[Category: | |||
[[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:Commas named for their regular temperament properties]] | |||
Latest revision as of 07:59, 3 November 2024
| Interval information |
reduced harmonic
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