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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''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[luna comma]], 274877906944/274658203125, which has a monzo of {{monzo| 38 -2 -15 }}.  


Temperaments discussed elsewhere include:
== Luna ==
* ''[[subneutral]]'', {2401/2400, 274877906944/274658203125} → [[Breedsmic temperaments #Subneutral]]
{{See also| Luna and hemithirds }}
 
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 }}.


== Luna ==
[[Subgroup]]: 2.3.5
Subgroup: 2.3.5


[[Comma list]]: 274877906944/274658203125
[[Comma list]]: 274877906944/274658203125


[[Mapping]]: [{{val| 1 4 2 }}, {{val| 0 -15 2 }}]
{{Mapping|legend=1| 1 -11 4 | 0 15 -2 }}
: mapping generators: ~2, ~234375/131072
 
[[Optimal tuning]]s:
* [[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]]: ~262144/234375 = 193.201
{{Optimal ET sequence|legend=1| 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559, 15031cc, 16590cc, …, 21267bccc }}


{{Val list|legend=1| 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559 }}
[[Badness]] (Sintel): 0.483


[[Badness]]: 0.020576
; 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]]


== Hemithirds ==
=== Overview to extensions ===
{{main| Hemithirds }}
[[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.
{{see also| Hemimean clan #Hemithirds }}


Subgroup: 2.3.5.7
Weak extensions of luna include hemiluna (87 & 323), semiluna (118 & 292), and subneutral (31 & 441).  


[[Comma list]]: 1029/1024, 3136/3125
Temperaments discussed elsewhere include:
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]


[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
The rest are considered below.


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
== Lunatic ==
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~28/25 = 193.244
[[Comma list]]: 4375/4374, 274877906944/274658203125


[[Minimax tuning]]:
{{Mapping|legend=1| 1 -11 4 -92 | 0 15 -2 113 }}
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 11/5 -1/10 0 1/10 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 82/35 -4/35 0 2/35 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6


{{Val list|legend=1| 31, 87, 118 }}
[[Optimal tuning]]s:
* [[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 }}


[[Badness]]: 0.044284
{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646, 8671bc, 10317bcd, 11963bbcd }}
 
[[Badness]] (Sintel): 0.961


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 385/384, 441/440, 3136/3125
Comma list: 4375/4374, 12005/11979, 172032/171875


Mapping: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
Mapping: {{mapping| 1 -11 4 -92 -114 | 0 15 -2 113 140 }}


POTE generator: ~28/25 = 193.227
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}})


Minimax tuning:
{{Optimal ET sequence|legend=0| 118, 323e, 441, 559, 1000e }}
* 11-odd-limit
: [{{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


Vals: {{Val list| 31, 87, 118 }}
Badness (Sintel): 1.94


Badness: 0.019003
== Hemiluna ==
[[Subgroup]]: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 48828125/48771072, 67108864/66976875
Subgroup: 2.3.5.7.11.13


Comma list: 196/195, 352/351, 385/384, 625/624
{{Mapping|legend=1| 1 -26 6 92 | 0 30 -4 -97 }}
: mapping generators: ~2, ~189/100


Mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
[[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 }}


POTE generator: ~28/25 = 193.166
{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}


Vals: {{Val list| 31, 56, 87, 118, 205d }}
[[Badness]] (Sintel): 4.54


Badness: 0.021738
== Semiluna ==
[[Subgroup]]: 2.3.5.7


== Lunatic ==
[[Comma list]]: 4802000/4782969, 95703125/95551488
Subgroup: 2.3.5.7
 
[[Comma list]]: 4375/4374, 274877906944/274658203125
 
[[Mapping]]: [{{val| 1 4 2 21 }}, {{val| 0 -15 2 -113 }}]


{{Multival|legend=1| 15 -2 113 -38 137 268 }}
{{Mapping|legend=1| 2 -7 6 -31 | 0 15 -2 54 }}
: mapping generators: ~10125/7168, ~4375/3456


[[POTE generator]]: ~262144/234375 = 193.196
[[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 }}


{{Val list|legend=1| 87d, 118, 323, 441, 1205, 1646 }}
{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}


[[Badness]]: 0.037966
[[Badness]] (Sintel): 4.86


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 12005/11979, 172032/171875
Comma list: 5632/5625, 9801/9800, 14641/14580
 
Mapping: [{{val| 1 4 2 21 14 }}, {{val| 0 -15 2 -113 -140 }}]
 
POTE generator: ~352/315 = 193.200
 
Vals: {{Val list| 118, 323e, 441, 559, 1000e }}


Badness: 0.058588
Mapping: {{mapping| 2 -7 6 -31 -8 | 0 15 -2 54 22 }}


== Semiluna ==
Optimal tunings:
{{see also| Canousmic temperaments #Semiluna }}
* 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}})


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 56d, 118, 292, 410 }}


[[Comma list]]: 4802000/4782969, 95703125/95551488
Badness (Sintel): 2.24


[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]
== 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.


{{Multival|legend=1| 30 -4 108 -76 87 262 }}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~2187/1960 = 193.1725
[[Comma list]]: 2401/2400, 274877906944/274658203125
 
{{Val list|legend=1| 56d, 118, 292, 410 }}
 
[[Badness]]: 0.192167
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 5632/5625, 9801/9800, 14641/14580


Mapping: [{{val| 2 8 4 23 14 }}, {{val| 0 -15 2 -54 -22 }}]
{{Mapping|legend=1| 1 -41 8 -5 | 0 60 -8 11 }}
: mapping generators: ~2, ~46875/28672


POTE generator: ~121/108 = 193.1732
[[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 }}


Vals: {{Val list| 56d, 118, 292, 410 }}
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}


Badness: 0.067783
[[Badness]] (Sintel): 1.16


[[Category:Regular temperament theory]]
[[Category:Luna family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Luna]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Hemithirds]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Luna_family"