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Luna family: Difference between revisions

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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, 274877906944/274658203125} → [[Breedsmic temperaments #Subneutral]]
* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]


== Luna ==
== Luna ==
Subgroup: 2.3.5
{{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


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


[[Mapping]]: [{{val| 1 4 2 }}, {{val| 0 -15 2 }}]
{{Mapping|legend=1| 1 4 2 | 0 -15 2 }}


[[POTE generator]]: ~262144/234375 = 193.201
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~262144/234375 = 193.201


{{Optimal ET sequence|legend=1| 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559 }}
{{Optimal ET sequence|legend=1| 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559 }}
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[[Badness]]: 0.020576
[[Badness]]: 0.020576


== Hemithirds ==
; Music
{{main| Hemithirds }}
* "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]]
{{see also| Hemimean clan #Hemithirds }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 1029/1024, 3136/3125
 
[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
 
{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
 
[[POTE generator]]: ~28/25 = 193.244
 
[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: [{{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]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: [{{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, 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: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
 
POTE generator: ~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: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
 
POTE generator: ~28/25 = 193.166
 
{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
 
Badness: 0.021738


== Lunatic ==
== Lunatic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 274877906944/274658203125
[[Comma list]]: 4375/4374, 274877906944/274658203125


[[Mapping]]: [{{val| 1 4 2 21 }}, {{val| 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
 
[[POTE generator]]: ~262144/234375 = 193.196


{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646 }}
{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646 }}
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Comma list: 4375/4374, 12005/11979, 172032/171875
Comma list: 4375/4374, 12005/11979, 172032/171875


Mapping: [{{val| 1 4 2 21 14 }}, {{val| 0 -15 2 -113 -140 }}]
Mapping: {{mapping| 1 4 2 21 14 | 0 -15 2 -113 -140 }}


POTE generator: ~352/315 = 193.200
Optimal tuning (POTE): ~2 = 1\1, ~352/315 = 193.200


{{Optimal ET sequence|legend=1| 118, 323e, 441, 559, 1000e }}
{{Optimal ET sequence|legend=1| 118, 323e, 441, 559, 1000e }}
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== Hemiluna ==
== Hemiluna ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 48828125/48771072, 67108864/66976875
[[Comma list]]: 48828125/48771072, 67108864/66976875


[[Mapping]]: [{{val| 1 4 2 -5 }}, {{val| 0 -30 4 97 }}]
{{Mapping|legend=1| 1 4 2 -5 | 0 -30 4 97 }}


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
 
[[POTE generator]]: ~200/189 = 96.591


{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}
{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}
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Comma list: 5632/5625, 14641/14580, 131072/130977
Comma list: 5632/5625, 14641/14580, 131072/130977


Mapping: [{{val| 1 4 2 -5 7 }}, {{val| 0 -30 4 97 -44 }}]
Mapping: {{mapping| 1 4 2 -5 7 | 0 -30 4 97 -44 }}


POTE generator: ~200/189 = 96.587
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 96.587


{{Optimal ET sequence|legend=1| 87, 236, 323, 410 }}
{{Optimal ET sequence|legend=1| 87, 236, 323, 410 }}
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Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580
Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580


Mapping: [{{val| 1 4 2 -5 7 7 }}, {{val| 0 -30 4 97 -44 -41 }}]
Mapping: {{mapping| 1 4 2 -5 7 7 | 0 -30 4 97 -44 -41 }}


POTE generator: ~143/135 = 96.586
Optimal tuning (POTE): ~2 = 1\1, ~143/135 = 96.586


{{Optimal ET sequence|legend=1| 87, 236, 323, 410 }}
{{Optimal ET sequence|legend=1| 87, 236, 323, 410 }}
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== Semiluna ==
== Semiluna ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4802000/4782969, 95703125/95551488
[[Comma list]]: 4802000/4782969, 95703125/95551488


[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]
{{Mapping|legend=1| 2 8 4 23 | 0 -15 2 -54 }}
 
Mapping generators: ~10125/7168, ~2187/1960


{{Multival|legend=1| 30 -4 108 -76 87 262 }}
: mapping generators: ~10125/7168, ~2187/1960


[[POTE generator]]: ~2187/1960 = 193.1725
[[Optimal tuning]] ([[POTE]]): ~99/70 = 1\2, ~2187/1960 = 193.1725


{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}
{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}
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Comma list: 5632/5625, 9801/9800, 14641/14580
Comma list: 5632/5625, 9801/9800, 14641/14580


Mapping: [{{val| 2 8 4 23 14 }}, {{val| 0 -15 2 -54 -22 }}]
Mapping: {{mapping| 2 8 4 23 14 | 0 -15 2 -54 -22 }}


POTE generator: ~121/108 = 193.1732
Optimal tuning (POTE): ~99/70 = 1\2, ~121/108 = 193.1732


{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}
{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Luna family]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemithirds]]
[[Category:Luna family| ]] <!-- main article -->
[[Category:Luna| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Luna_family"