Luna family: Difference between revisions

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


Luna divides [[16/3]] into fifteen parts, two of which make [[5/4]]. It also divides [[9/8]] in five and [[3/2]] in three.
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
[[Subgroup]]: 2.3.5
Line 11: Line 11:
[[Comma list]]: 274877906944/274658203125
[[Comma list]]: 274877906944/274658203125


{{Mapping|legend=1| 1 4 2 | 0 -15 2 }}
{{Mapping|legend=1| 1 -11 4 | 0 15 -2 }}
: mapping generators: ~2, ~262144/234375
: mapping generators: ~2, ~234375/131072


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~262144/234375 = 193.201{{c}}
[[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 }}


{{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, 15031cc, 16590cc, …, 21267bccc }}


[[Badness]] (Smith): 0.020576
[[Badness]] (Sintel): 0.483


; Music
; Music
Line 24: Line 28:


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


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


== Lunatic ==
== Lunatic ==
Line 35: Line 42:
[[Comma list]]: 4375/4374, 274877906944/274658203125
[[Comma list]]: 4375/4374, 274877906944/274658203125


{{Mapping|legend=1| 1 4 2 21 | 0 -15 2 -113 }}
{{Mapping|legend=1| 1 -11 4 -92 | 0 15 -2 113 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~262144/234375 = 193.196{{c}}
[[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 }}


{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646 }}
{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646, 8671bc, 10317bcd, 11963bbcd }}


[[Badness]] (Smith): 0.037966
[[Badness]] (Sintel): 0.961


=== 11-limit ===
=== 11-limit ===
Line 48: Line 59:
Comma list: 4375/4374, 12005/11979, 172032/171875
Comma list: 4375/4374, 12005/11979, 172032/171875


Mapping: {{mapping| 1 4 2 21 14 | 0 -15 2 -113 -140 }}
Mapping: {{mapping| 1 -11 4 -92 -114 | 0 15 -2 113 140 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~352/315 = 193.200{{c}}
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 }}
{{Optimal ET sequence|legend=0| 118, 323e, 441, 559, 1000e }}


Badness (Smith): 0.058588
Badness (Sintel): 1.94


== Hemiluna ==
== Hemiluna ==
Line 61: Line 74:
[[Comma list]]: 48828125/48771072, 67108864/66976875
[[Comma list]]: 48828125/48771072, 67108864/66976875


{{Mapping|legend=1| 1 4 2 -5 | 0 -30 4 97 }}
{{Mapping|legend=1| 1 -26 6 92 | 0 30 -4 -97 }}
: mapping generators: ~2, ~200/189
: mapping generators: ~2, ~189/100


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~200/189 = 96.591{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}


[[Badness]] (Smith): 0.179304
[[Badness]] (Sintel): 4.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 5632/5625, 14641/14580, 131072/130977
 
Mapping: {{mapping| 1 4 2 -5 7 | 0 -30 4 97 -44 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~200/189 = 96.587{{c}}
 
{{Optimal ET sequence|legend=0| 87, 236, 323, 410 }}
 
Badness (Smith): 0.100077
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580
 
Mapping: {{mapping| 1 4 2 -5 7 7 | 0 -30 4 97 -44 -41 }}
 
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~143/135 = 96.586{{c}}
 
{{Optimal ET sequence|legend=0| 87, 236, 323, 410 }}
 
Badness (Smith): 0.046151


== Semiluna ==
== Semiluna ==
Line 101: Line 92:
[[Comma list]]: 4802000/4782969, 95703125/95551488
[[Comma list]]: 4802000/4782969, 95703125/95551488


{{Mapping|legend=1| 2 8 4 23 | 0 -15 2 -54 }}
{{Mapping|legend=1| 2 -7 6 -31 | 0 15 -2 54 }}
: mapping generators: ~10125/7168, ~2187/1960
: mapping generators: ~10125/7168, ~4375/3456


[[Optimal tuning]] ([[POTE]]): ~99/70 = 600.0000{{c}}, ~2187/1960 = 193.1725{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}


[[Badness]] (Smith): 0.192167
[[Badness]] (Sintel): 4.86


=== 11-limit ===
=== 11-limit ===
Line 115: Line 110:
Comma list: 5632/5625, 9801/9800, 14641/14580
Comma list: 5632/5625, 9801/9800, 14641/14580


Mapping: {{mapping| 2 8 4 23 14 | 0 -15 2 -54 -22 }}
Mapping: {{mapping| 2 -7 6 -31 -8 | 0 15 -2 54 22 }}


Optimal tuning (POTE): ~99/70 = 600.0000{{c}}, ~121/108 = 193.1732{{c}}
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 }}
{{Optimal ET sequence|legend=0| 56d, 118, 292, 410 }}


Badness (Smith): 0.067783
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:Temperament families]]
[[Category:Luna family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[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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