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The '''luna family''' tempers out the hemithirds comma aka luna comma, 274877906944/274658203125, which has a monzo of {{monzo| 38 -2 -15 }}.
{{Technical data page}}
The '''luna family''' tempers out the [[luna comma]], 274877906944/274658203125, which has a monzo of {{monzo| 38 -2 -15 }}. It divides [[16/3]] into fifteen parts, two of which make [[5/4]]. It also divides [[9/8]] in five and [[3/2]] in three.


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]
* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]


== Luna ==
== Luna ==
{{see also|Luna and hemithirds}}
[[Hemimean clan #Hemithirds|Hemithirds]], a strong extension of [[didacus]] that tempers out [[1029/1024]] alongside [[3136/3125]], might be considered the main [[7-limit]] extension to luna of practical interest, as [[7/4]] is found at only 5 generators. However, luna is an undoubted microtemperament of the [[5-limit]] in its own right, supported by systems like [[118edo]] and [[441edo]], and therefore merits consideration of other extensions to the 7-limit, which are documented on this page. Though of very high complexity, they continue the level of accuracy of 5-limit luna.
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


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; Music
; Music
* "Moongazing" from ''Lessor Groove'' (2020) [https://xotla.bandcamp.com/track/moongazing-luna-25 Bandcamp] | [https://www.youtube.com/watch?v=EWuVnLOcaRg YouTube] – atmospheric-electro, luna[25] in 1000edo by [[Xotla]]
* "Moongazing" from ''Lesser Groove'' (2020) [https://xotla.bandcamp.com/track/moongazing-luna-25 Bandcamp] | [https://www.youtube.com/watch?v=EWuVnLOcaRg YouTube] – atmospheric-electro, luna[25] in 1000edo by [[Xotla]]
 
== Hemithirds ==
{{Main| Hemithirds }}
{{See also| Hemimean clan #Hemithirds }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 3136/3125
 
{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
 
{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
 
[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
 
{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
 
[[Badness]]: 0.044284
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 3136/3125
 
Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227
 
Minimax tuning:
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 0 0 -5/9 5/9 }}, {{monzo| 64/27 0 0 2/27 -2/27 }}, {{monzo| 79/27 0 0 5/27 -5/27 }}, {{monzo| 79/27 0 0 -22/27 22/27 }}]
: Eigenmonzos (unchanged-intervals): 2, 11/7
 
{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}
 
Badness: 0.019003
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 385/384, 625/624
 
Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166
 
{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
 
Badness: 0.021738


== Lunatic ==
== Lunatic ==
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{{Mapping|legend=1| 1 4 2 21 | 0 -15 2 -113 }}
{{Mapping|legend=1| 1 4 2 21 | 0 -15 2 -113 }}
{{Multival|legend=1| 15 -2 113 -38 137 268 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~262144/234375 = 193.196
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~262144/234375 = 193.196
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: mapping generators: ~2, ~200/189
: mapping generators: ~2, ~200/189
{{Multival|legend=1| 30 -4 -97 -76 -238 -214 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 96.591
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 96.591
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: mapping generators: ~10125/7168, ~2187/1960
: mapping generators: ~10125/7168, ~2187/1960
{{Multival|legend=1| 30 -4 108 -76 87 262 }}


[[Optimal tuning]] ([[POTE]]): ~99/70 = 1\2, ~2187/1960 = 193.1725
[[Optimal tuning]] ([[POTE]]): ~99/70 = 1\2, ~2187/1960 = 193.1725
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Luna family| ]] <!-- main article -->
[[Category:Luna family| ]] <!-- main article -->
[[Category:Luna| ]] <!-- key article -->
[[Category:Luna| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Luna_family"