Cloudy comma: Difference between revisions

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'''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 [[2100875/2097152]]. 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]].
'''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 [[2100875/2097152]]. 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]].


==Cloudy temperaments==
== Temperaments ==
Tempering out the cloudy comma leads a number of regular temperaments including [[Marvel temperaments|decic]], [[Starling temperaments|coblack]], [[Archytas clan|blacksmith]], [[Meantone family|cloudtone]], [[Pental family|pental]] (pentacloud), and [[Qintosec family|sengasec]]. Cloudy rank three 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.
Tempering out the cloudy comma leads a number of regular temperaments including [[Marvel temperaments|decic]], [[Starling temperaments|coblack]], [[Archytas clan|blacksmith]], [[Meantone family|cloudtone]], [[Pental family|pental]] (pentacloud), and [[Qintosec family|sengasec]]. Cloudy rank three 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.


'''<font style="font-size: 1.2em">7-limit cloudy (10&amp;15&amp;50)</font>'''<br>
=== Cloudy (10&amp;15&amp;50) ===
Comma: 16807/16384<br><br>
 
POTE generators: ~8/7 = 240.000, ~3/2 = 699.467, ~5/4 = 382.669<br><br>
Subgroup: 2.3.5.7
Map: [&lt;5 0 0 14|, &lt;0 1 0 0|, &lt;0 0 1 0|]<br><br>
 
EDOs: {{Vals| 5, 10, 15, 25, 35, 45, 50, 55, 60, 65 }}<br><br>
[[Comma list]]: 16807/16384
'''<font style="font-size: 1.2em">11-limit cloudy (10&amp;15&amp;50)</font>'''<br>
 
Commas: 385/384, 3087/3025<br><br>
[[Mapping]]: [{{val| 5 0 0 14 }}, {{val| 0 1 0 0 }}, {{val| 0 0 1 0 }}]
POTE generators: ~8/7 = 240.000, ~3/2 = 699.779, ~5/4 = 386.575<br><br>
 
Map: [&lt;5 0 0 14 21|, &lt;0 1 0 0 1|, &lt;0 0 1 0 -1|]<br><br>
Mapping generators: ~8/7, ~3, ~5
EDOs: {{Vals| 5, 10, 15, 35, 40, 50, 55, 60, 65 }}<br><br>
 
'''<font style="font-size: 1.2em">13-limit cloudy (10&amp;15&amp;50)</font>'''<br>
[[POTE generator]]s: ~3/2 = 699.467, ~5/4 = 382.669
Commas: 105/104, 144/143, 3087/3025<br><br>
 
POTE generators: ~8/7 = 240.000, ~3/2 = 698.726, ~5/4 = 384.760<br><br>
{{Val list|legend=1| 5, 10, 15, 25, 35, 45, 50, 55, 60, 65 }}
Map: [&lt;5 0 0 14 21 -1|, &lt;0 1 0 0 1 1|, &lt;0 0 1 0 -1 1|]<br><br>
 
EDOs: {{Vals| 5, 10, 15, 40, 50, 60, 65 }}
=== 11-limit cloudy (10&amp;15&amp;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
 
{{Val list|legend=1| 5, 10, 15, 35, 40, 50, 55, 60, 65 }}
 
=== 13-limit cloudy (10&amp;15&amp;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
 
{{Val list|legend=1| 5, 10, 15, 40, 50, 60, 65 }}


[[Category:7-limit]]
[[Category:7-limit]]
[[Category:Medium comma]]
[[Category:Medium comma]]
[[Category:Interval ratio]]
[[Category:Interval]]

Revision as of 05:26, 4 April 2021

Interval information
Ratio 16807/16384
Factorization 2-14 × 75
Monzo [-14 0 0 5⟩
Size in cents 44.12953¢
Name cloudy comma
FJS name [math]\displaystyle{ \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
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 2100875/2097152. 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 including decic, coblack, blacksmith, cloudtone, pental (pentacloud), and sengasec. Cloudy rank three 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

Template:Val list

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

Template:Val list

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

Template:Val list

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