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
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[[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
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[[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 ===
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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 ==

Revision as of 03:32, 23 June 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, 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.

Temperaments discussed elsewhere include:

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 4 2 -5], ⟨0 -30 4 97]]

mapping generators: ~2, ~200/189

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~200/189 = 96.591 ¢

Optimal ET sequence: 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd

Badness (Smith): 0.179304

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 14641/14580, 131072/130977

Mapping: [⟨1 4 2 -5 7], ⟨0 -30 4 97 -44]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~200/189 = 96.587 ¢

Optimal ET sequence: 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: [⟨1 4 2 -5 7 7], ⟨0 -30 4 97 -44 -41]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~143/135 = 96.586 ¢

Optimal ET sequence: 87, 236, 323, 410

Badness (Smith): 0.046151

Semiluna

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 95703125/95551488

Mapping: [⟨2 8 4 23], ⟨0 -15 2 -54]]

mapping generators: ~10125/7168, ~2187/1960

Optimal tuning (POTE): ~99/70 = 600.0000 ¢, ~2187/1960 = 193.1725 ¢

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

Badness (Smith): 0.192167

11-limit

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800, 14641/14580

Mapping: [⟨2 8 4 23 14], ⟨0 -15 2 -54 -22]]

Optimal tuning (POTE): ~99/70 = 600.0000 ¢, ~121/108 = 193.1732 ¢

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

Badness (Smith): 0.067783

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