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.


= Luna =
Temperaments discussed elsewhere include:
Subgroup: 2.3.5
* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
 
== Luna ==
{{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


{{Val list|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 }}


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


[[Comma list]]: 1029/1024, 3136/3125
[[Comma list]]: 4375/4374, 274877906944/274658203125


[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
{{Mapping|legend=1| 1 4 2 21 | 0 -15 2 -113 }}


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~262144/234375 = 193.196


[[POTE generator]]: ~28/25 = 193.244
{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646 }}


[[Minimax tuning]]:
[[Badness]]: 0.037966
* [[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 }}


[[Badness]]: 0.044284
=== 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 4 2 21 14 | 0 -15 2 -113 -140 }}


POTE generator: ~28/25 = 193.227
Optimal tuning (POTE): ~2 = 1\1, ~352/315 = 193.200


Minimax tuning:
{{Optimal ET sequence|legend=1| 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


Vals: {{Val list| 31, 87, 118 }}
Badness: 0.058588


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 4 2 -5 | 0 -30 4 97 }}


Mapping: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
: mapping generators: ~2, ~200/189


POTE generator: ~28/25 = 193.166
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 96.591


Vals: {{Val list| 31, 56, 87, 118, 205d }}
{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}


Badness: 0.021738
[[Badness]]: 0.179304


= Lunatic =
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 4375/4374, 274877906944/274658203125
Comma list: 5632/5625, 14641/14580, 131072/130977


[[Mapping]]: [{{val| 1 4 2 21 }}, {{val| 0 -15 2 -113 }}]
Mapping: {{mapping| 1 4 2 -5 7 | 0 -30 4 97 -44 }}


{{Multival|legend=1| 15 -2 113 -38 137 268 }}
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 96.587


[[POTE generator]]: ~262144/234375 = 193.196
{{Optimal ET sequence|legend=1| 87, 236, 323, 410 }}


{{Val list|legend=1| 87d, 118, 323, 441, 1205, 1646 }}
Badness: 0.100077


[[Badness]]: 0.037966
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580
Subgroup: 2.3.5.7.11


Comma list: 4375/4374, 12005/11979, 172032/171875
Mapping: {{mapping| 1 4 2 -5 7 7 | 0 -30 4 97 -44 -41 }}
 
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 }}
Optimal tuning (POTE): ~2 = 1\1, ~143/135 = 96.586


Badness: 0.058588
{{Optimal ET sequence|legend=1| 87, 236, 323, 410 }}


= Semiluna =
Badness: 0.046151
{{see also| Canousmic temperaments #Semiluna }}


Subgroup: 2.3.5.7
== Semiluna ==
[[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 }}


{{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


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


[[Badness]]: 0.192167
[[Badness]]: 0.192167


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


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


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


Badness: 0.067783
Badness: 0.067783


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Luna]]
[[Category:Luna family| ]] <!-- main article -->
[[Category:Hemithirds]]
[[Category:Luna| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 04:59, 17 July 2025

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 tempers out the luna comma, 274877906944/274658203125, which has a monzo of [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:

Luna

See also: Luna and 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

Mapping: [⟨1 4 2], ⟨0 -15 2]]

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

Optimal ET sequence: 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559

Badness: 0.020576

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

Lunatic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 274877906944/274658203125

Mapping: [⟨1 4 2 21], ⟨0 -15 2 -113]]

Optimal tuning (POTE): ~2 = 1\1, ~262144/234375 = 193.196

Optimal ET sequence: 87d, 118, 323, 441, 1205, 1646

Badness: 0.037966

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 4 2 21 14], ⟨0 -15 2 -113 -140]]

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

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

Badness: 0.058588

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 = 1\1, ~200/189 = 96.591

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

Badness: 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 = 1\1, ~200/189 = 96.587

Optimal ET sequence: 87, 236, 323, 410

Badness: 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 = 1\1, ~143/135 = 96.586

Optimal ET sequence: 87, 236, 323, 410

Badness: 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 = 1\2, ~2187/1960 = 193.1725

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

Badness: 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 = 1\2, ~121/108 = 193.1732

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

Badness: 0.067783

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