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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 }}.


= Luna =
== Luna ==
{{See also| Luna and hemithirds }}


Subgroup: 2.3.5
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 }}.
 
[[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]]: 0.0206
; 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: 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: 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]]: 0.0443
{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646, 8671bc, 10317bcd, 11963bbcd }}


== 11-limit ==
[[Badness]] (Sintel): 0.961


=== 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: 2, 11/7


{{Val list|legend=1| 31, 87, 118 }}
Badness (Sintel): 1.94


Badness: 0.0190
== Hemiluna ==
[[Subgroup]]: 2.3.5.7


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


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 1 -26 6 92 | 0 30 -4 -97 }}
: mapping generators: ~2, ~189/100


Comma list: 196/195, 352/351, 385/384, 625/624
[[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 }}


Mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}


POTE generator: ~28/25 = 193.166
[[Badness]] (Sintel): 4.54


{{Val list|legend=1| 31, 56, 87, 118, 205d }}
== Semiluna ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0217
[[Comma list]]: 4802000/4782969, 95703125/95551488


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


Subgroup: 2.3.5.7
[[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 }}


[[Comma list]]: 4375/4374, 274877906944/274658203125
{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}


[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 -113 }}]
[[Badness]] (Sintel): 4.86


{{Multival|legend=1| 15 -2 113 -38 137 268 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~262144/234375 = 193.196
Comma list: 5632/5625, 9801/9800, 14641/14580
 
{{Val list|legend=1| 87d, 118, 323, 441, 1205, 1646 }}
 
[[Badness]]: 0.0380
 
= Semiluna =
 
{{see also| Canousmic temperaments #Semiluna }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 4802000/4782969, 95703125/95551488


[[Mapping]]: [{{val| 2 8 4 23 }}, {{val| 0 -15 2 -54 }}]
Mapping: {{mapping| 2 -7 6 -31 -8 | 0 15 -2 54 22 }}


[[POTE generator]]: ~2187/1960 = 193.1725
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}})


{{Val list|legend=1| 56d, 118, 292, 410 }}
{{Optimal ET sequence|legend=0| 56d, 118, 292, 410 }}


[[Badness]]: 0.1922
Badness (Sintel): 2.24


== 11-limit ==
== 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: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 5632/5625, 9801/9800, 14641/14580
[[Comma list]]: 2401/2400, 274877906944/274658203125


POTE generator: ~121/108 = 193.1732
{{Mapping|legend=1| 1 -41 8 -5 | 0 60 -8 11 }}
: mapping generators: ~2, ~46875/28672


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


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


Badness: 0.0678
[[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]]

Latest revision as of 12:05, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The luna family of temperaments tempers out the luna comma, 274877906944/274658203125, which has a monzo of [38 -2 -15⟩.

Luna

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 n = 15/2.

Subgroup: 2.3.5

Comma list: 274877906944/274658203125

Mapping: [⟨1 -11 4], ⟨0 15 -2]]

mapping generators: ~2, ~234375/131072

Optimal tunings:

  • WE: ~2 = 1199.9803 ¢, ~234375/131072 = 1006.7827 ¢ (~262144/234375 = 193.1976 ¢)
error map: ⟨-0.020 +0.003 +0.042]
  • CWE: ~2 = 1200.0000 ¢, ~234375/131072 = 1006.7987 ¢ (~262144/234375 = 193.2013 ¢)
error map: ⟨0.000 +0.025 +0.089]

Optimal ET sequence: 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559, 15031cc, 16590cc, …, 21267bccc

Badness (Sintel): 0.483

Music
  • "Moongazing" from Lesser Groove (2020) – Bandcamp | YouTube – atmospheric-electro, Luna[25] in 1000edo by Xotla

Overview to extensions

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.

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

Temperaments discussed elsewhere include:

The rest are considered below.

Lunatic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 274877906944/274658203125

Mapping: [⟨1 -11 4 -92], ⟨0 15 -2 113]]

Optimal tunings:

  • WE: ~2 = 1199.9687 ¢, ~234375/131072 = 1006.7781 ¢ (~262144/234375 = 193.1906 ¢)
error map: ⟨-0.031 +0.060 +0.005 -0.025]
  • CWE: ~2 = 1200.0000 ¢, ~234375/131072 = 1006.8041 ¢ (~262144/234375 = 193.1959 ¢)
error map: ⟨0.000 +0.106 +0.078 +0.035]

Optimal ET sequence: 87d, 118, 323, 441, 1205, 1646, 8671bc, 10317bcd, 11963bbcd

Badness (Sintel): 0.961

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 12005/11979, 172032/171875

Mapping: [⟨1 -11 4 -92 -114], ⟨0 15 -2 113 140]]

Optimal tunings:

  • WE: ~2 = 1200.000 ¢, ~315/176 = 1006.7823 ¢ (~352/315 = 193.1967 ¢)
  • CWE: ~2 = 1200.000 ¢, ~315/176 = 1006.7998 ¢ (~352/315 = 193.2002 ¢)

Optimal ET sequence: 118, 323e, 441, 559, 1000e

Badness (Sintel): 1.94

Hemiluna

Subgroup: 2.3.5.7

Comma list: 48828125/48771072, 67108864/66976875

Mapping: [⟨1 -26 6 92], ⟨0 30 -4 -97]]

mapping generators: ~2, ~189/100

Optimal tunings:

  • WE: ~2 = 1199.9023 ¢, ~189/100 = 1103.3197 ¢ (~200/189 = 96.5827 ¢)
error map: ⟨-0.098 +0.175 -0.178 +0.180]
  • CWE: ~2 = 1200.0000 ¢, ~189/100 = 1103.4097 ¢ (~200/189 = 96.5903 ¢)
error map: ⟨0.000 +0.336 +0.047 +0.432]

Optimal ET sequence: 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: [⟨2 -7 6 -31], ⟨0 15 -2 54]]

mapping generators: ~10125/7168, ~4375/3456

Optimal tunings:

  • WE: ~10125/7168 = 600.0000 ¢, ~4375/3456 = 406.8061 ¢ (~2187/1960 = 193.1624 ¢)
error map: ⟨-0.063 +0.358 -0.115 -0.318]
  • CWE: ~10125/7168 = 600.0000 ¢, ~4375/3456 = 406.8263 ¢ (~2187/1960 = 193.1737 ¢)
error map: ⟨0.000 +0.440 +0.034 -0.203]

Optimal ET sequence: 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: [⟨2 -7 6 -31 -8], ⟨0 15 -2 54 22]]

Optimal tunings:

  • WE: ~99/70 = 600.0093 ¢, ~486/385 = 406.8331 ¢ (~121/108 = 193.1762 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~486/385 = 406.8272 ¢ (~121/108 = 193.1728 ¢)

Optimal ET sequence: 56d, 118, 292, 410

Badness (Sintel): 2.24

Subneutral

Subneutral tempers out 2401/2400, the breedsma, and may be described as the 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: [⟨1 -41 8 -5], ⟨0 60 -8 11]]

mapping generators: ~2, ~46875/28672

Optimal tunings:

  • WE: ~2 = 1199.9998 ¢, ~46875/28672 = 851.6994 ¢ (~57344/46875 = 348.3005 ¢)
error map: ⟨-0.000 +0.013 +0.090 -0.132]
  • CWE: ~2 = 1200.0000 ¢, ~46875/28672 = 851.6995 ¢ (~57344/46875 = 348.3005 ¢)
error map: ⟨0.000 +0.014 +0.090 -0.132]

Optimal ET sequence: 31, …, 348, 379, 410, 441, 1354, 1795, 2236

Badness (Sintel): 1.16

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