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The Luna family tempers out the Luna comma; 274877906944/274658203125, which has a monzo of |38 -2 -15>
{{Technical data page}}
The '''luna family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[luna comma]], 274877906944/274658203125, which has a monzo of {{monzo| 38 -2 -15 }}.


=Luna=
== Luna ==
Comma: 274877906944/274658203125
{{See also| Luna and hemithirds }}


POTE generator: ~262144/234375 = 193.201
Luna divides [[16/3]] into fifteen parts, two of which make [[5/4]]. It has a [[ploidacot]] signature of 12-sheared 15-cot (or beta-seph). It also divides [[9/8]] in five and [[3/2]] in three. It is part of the [[syntonic–31 equivalence continuum]] with {{nowrap| ''n'' {{=}} 15/2 }}.  


Map: [<1 4 2|, <0 -15 2|]
[[Subgroup]]: 2.3.5


EDOs: 13, 19, 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559
[[Comma list]]: 274877906944/274658203125


Badness: 0.0206
{{Mapping|legend=1| 1 -11 4 | 0 15 -2 }}
: mapping generators: ~2, ~234375/131072


=[[Gamelismic_clan#Hemithirds|Hemithirds]]=
[[Optimal tuning]]s:
Commas: 1029/1024, 3136/3125
* [[WE]]: ~2 = 1199.9803{{c}}, ~234375/131072 = 1006.7827{{c}} (~262144/234375 = 193.1976{{c}})
: [[error map]]: {{val| -0.020 +0.003 +0.042 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~234375/131072 = 1006.7987{{c}} (~262144/234375 = 193.2013{{c}})
: error map: {{val| 0.000 +0.025 +0.089 }}


POTE generator: ~28/25 = 193.244
{{Optimal ET sequence|legend=1| 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559, 15031cc, 16590cc, …, 21267bccc }}


Map: [<1 4 2 2|, <0 -15 2 5|]
[[Badness]] (Sintel): 0.483


Wedgie: <<15 -2 -5 -38 -50 -6||
; Music
* "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]]


EDOs: 13, 19, 25, 31, 87, 118
=== Overview to extensions ===
[[Hemimean clan #Hemithirds|Hemithirds]] (25 & 31), 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.


Badness: 0.0443
Weak extensions of luna include hemiluna (87 & 323), semiluna (118 & 292), and subneutral (31 & 441).  


==11-limit==
Temperaments discussed elsewhere include:
Commas: 385/384, 441/440, 3136/3125
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]


POTE generator: ~28/25 = 193.227
The rest are considered below.  


Map: [<1 4 2 2 7|, <0 -15 2 5 -22|]
== Lunatic ==
[[Subgroup]]: 2.3.5.7


EDOs: 13, 19, 25, 31, 87, 118
[[Comma list]]: 4375/4374, 274877906944/274658203125


Bad: 0.0190
{{Mapping|legend=1| 1 -11 4 -92 | 0 15 -2 113 }}


=Lunatic=
[[Optimal tuning]]s:
Commas: 4375/4374, 274877906944/274658203125
* [[WE]]: ~2 = 1199.9687{{c}}, ~234375/131072 = 1006.7781{{c}} (~262144/234375 = 193.1906{{c}})
: [[error map]]: {{val| -0.031 +0.060 +0.005 -0.025 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~234375/131072 = 1006.8041{{c}} (~262144/234375 = 193.1959{{c}})
: error map: {{val| 0.000 +0.106 +0.078 +0.035 }}


POTE generator: ~262144/234375 = 193.196
{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646, 8671bc, 10317bcd, 11963bbcd }}


Map: [<1 4 2 2|, <0 -15 2 -113|]
[[Badness]] (Sintel): 0.961


Wedgie: <<15 -2 113 -38 137 268||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 13, 19, 25, 31, 87, 118, 323, 441, 1205, 1646
Comma list: 4375/4374, 12005/11979, 172032/171875


Badness: 0.0380
Mapping: {{mapping| 1 -11 4 -92 -114 | 0 15 -2 113 140 }}
[[Category:hemithirds]]
 
[[Category:luna]]
Optimal tunings:
* WE: ~2 = 1200.000{{c}}, ~315/176 = 1006.7823{{c}} (~352/315 = 193.1967{{c}})
* CWE: ~2 = 1200.000{{c}}, ~315/176 = 1006.7998{{c}} (~352/315 = 193.2002{{c}})
 
{{Optimal ET sequence|legend=0| 118, 323e, 441, 559, 1000e }}
 
Badness (Sintel): 1.94
 
== Hemiluna ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 48828125/48771072, 67108864/66976875
 
{{Mapping|legend=1| 1 -26 6 92 | 0 30 -4 -97 }}
: mapping generators: ~2, ~189/100
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9023{{c}}, ~189/100 = 1103.3197{{c}} (~200/189 = 96.5827{{c}})
: [[error map]]: {{val| -0.098 +0.175 -0.178 +0.180 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/100 = 1103.4097{{c}} (~200/189 = 96.5903{{c}})
: error map: {{val| 0.000 +0.336 +0.047 +0.432 }}
 
{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}
 
[[Badness]] (Sintel): 4.54
 
== Semiluna ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4802000/4782969, 95703125/95551488
 
{{Mapping|legend=1| 2 -7 6 -31 | 0 15 -2 54 }}
: mapping generators: ~10125/7168, ~4375/3456
 
[[Optimal tuning]]s:
* [[WE]]: ~10125/7168 = 600.0000{{c}}, ~4375/3456 = 406.8061{{c}} (~2187/1960 = 193.1624{{c}})
: [[error map]]: {{val| -0.063 +0.358 -0.115 -0.318 }}
* [[CWE]]: ~10125/7168 = 600.0000{{c}}, ~4375/3456 = 406.8263{{c}} (~2187/1960 = 193.1737{{c}})
: error map: {{val| 0.000 +0.440 +0.034 -0.203 }}
 
{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}
 
[[Badness]] (Sintel): 4.86
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 5632/5625, 9801/9800, 14641/14580
 
Mapping: {{mapping| 2 -7 6 -31 -8 | 0 15 -2 54 22 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0093{{c}}, ~486/385 = 406.8331{{c}} (~121/108 = 193.1762{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~486/385 = 406.8272{{c}} (~121/108 = 193.1728{{c}})
 
{{Optimal ET sequence|legend=0| 56d, 118, 292, 410 }}
 
Badness (Sintel): 2.24
 
== Subneutral ==
Subneutral tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 31 & 441 }} temperament. Despite the name, the generator is only about 3 cents flat of the exact neutral third.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 274877906944/274658203125
 
{{Mapping|legend=1| 1 -41 8 -5 | 0 60 -8 11 }}
: mapping generators: ~2, ~46875/28672
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9998{{c}}, ~46875/28672 = 851.6994{{c}} (~57344/46875 = 348.3005{{c}})
: [[error map]]: {{val| -0.000 +0.013 +0.090 -0.132 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~46875/28672 = 851.6995{{c}} (~57344/46875 = 348.3005{{c}})
: error map: {{val| 0.000 +0.014 +0.090 -0.132 }}
 
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
 
[[Badness]] (Sintel): 1.16
 
[[Category:Luna family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
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