Cloudy comma: Difference between revisions

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| Comma = yes

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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 [[cent]]s. 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]].

'''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 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~~]].

[[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]].

~~Cloudy~~ rank-3 temperament can be described as the 10&~~amp;~~15&~~amp;~~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~~.

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&~~amp;~~15&~~amp;~~50) ===

=== Cloudy (10 & 15 & 50) ===

Subgroup: 2.3.5.7

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{{Optimal ET sequence|legend=1| 5, 10, 15, 25, 35, 45, 50, 55, 60, 65 }}

=== 11-limit cloudy (10&~~amp;~~15&~~amp;~~50) ===

=== 11-limit cloudy (10 & 15 & 50) ===

Subgroup: 2.3.5.7.11

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{{Optimal ET sequence|legend=1| 5, 10, 15, 35, 40, 50, 55, 60, 65 }}

=== 13-limit cloudy (10&~~amp;~~15&~~amp;~~50) ===

=== 13-limit cloudy (10 & 15 & 50) ===

Subgroup: 2.3.5.7.11.13

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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 [[cent]]s. 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]].
-'''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 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]].
+[[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]].
-Cloudy rank-3 temperament can be described as the 10&amp;15&amp;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.
+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&amp;15&amp;50) ===
+=== Cloudy (10 & 15 & 50) ===
 Subgroup: 2.3.5.7
@@ Line 26: / Line 25: @@
 {{Optimal ET sequence|legend=1| 5, 10, 15, 25, 35, 45, 50, 55, 60, 65 }}
-=== 11-limit cloudy (10&amp;15&amp;50) ===
+=== 11-limit cloudy (10 & 15 & 50) ===
 Subgroup: 2.3.5.7.11
@@ Line 37: / Line 36: @@
 {{Optimal ET sequence|legend=1| 5, 10, 15, 35, 40, 50, 55, 60, 65 }}
-=== 13-limit cloudy (10&amp;15&amp;50) ===
+=== 13-limit cloudy (10 & 15 & 50) ===
 Subgroup: 2.3.5.7.11.13