Luna family: Difference between revisions

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Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[subneutral]]'', {2401/2400, 274877906944/274658203125} → [[Breedsmic temperaments #Subneutral]]
* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]


== Luna ==
== Luna ==
Subgroup: 2.3.5
[[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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== Hemithirds ==
== Hemithirds ==
{{main| Hemithirds }}
{{Main| Hemithirds }}
{{see also| Hemimean clan #Hemithirds }}
{{See also| Hemimean clan #Hemithirds }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 3136/3125
[[Comma list]]: 1029/1024, 3136/3125


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


{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}


[[POTE generator]]: ~28/25 = 193.244
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
* [[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 }}]
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
* [[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 }}]
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo]]s (unchanged-intervals): 2, 9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
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Comma list: 385/384, 441/440, 3136/3125
Comma list: 385/384, 441/440, 3136/3125


Mapping: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}


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


Minimax tuning:
Minimax tuning:
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Comma list: 196/195, 352/351, 385/384, 625/624
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 }}]
Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}


POTE generator: ~28/25 = 193.166
Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166


{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
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== 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 }}
{{Multival|legend=1| 15 -2 113 -38 137 268 }}


[[POTE generator]]: ~262144/234375 = 193.196
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}
{{Multival|legend=1| 30 -4 -97 -76 -238 -214 }}


[[POTE generator]]: ~200/189 = 96.591
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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
: mapping generators: ~10125/7168, ~2187/1960


{{Multival|legend=1| 30 -4 108 -76 87 262 }}
{{Multival|legend=1| 30 -4 108 -76 87 262 }}


[[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:Luna family| ]] <!-- main article -->
[[Category:Hemithirds]]
[[Category:Luna| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 08:28, 5 September 2023

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

Temperaments discussed elsewhere include:

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 Lessor Groove (2020) Bandcamp | YouTube – atmospheric-electro, luna[25] in 1000edo by Xotla

Hemithirds

Main article: Hemithirds
See also: Hemimean clan § Hemithirds

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3136/3125

Mapping: [⟨1 4 2 2], ⟨0 -15 2 5]]

Wedgie: ⟨⟨ 15 -2 -5 -38 -50 -6 ]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.244

Minimax tuning:

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [11/5 -1/10 0 1/10⟩, [5/2 -1/4 0 1/4⟩]
eigenmonzo (unchanged-interval) basis: 2.7/3
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [82/35 -4/35 0 2/35⟩, [20/7 -2/7 0 1/7⟩]
eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 25, 31, 87, 118

Badness: 0.044284

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 3136/3125

Mapping: [⟨1 4 2 2 7], ⟨0 -15 2 5 -22]]

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

Minimax tuning:

  • 11-odd-limit: ~28/25 = [5/27 0 0 1/27 -1/27⟩
[[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]
Eigenmonzos (unchanged-intervals): 2, 11/7

Optimal ET sequence: 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: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166

Optimal ET sequence: 31, 56, 87, 118, 205d

Badness: 0.021738

Lunatic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 274877906944/274658203125

Mapping: [⟨1 4 2 21], ⟨0 -15 2 -113]]

Wedgie: ⟨⟨ 15 -2 113 -38 137 268 ]]

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

Wedgie: ⟨⟨ 30 -4 -97 -76 -238 -214 ]]

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

Wedgie: ⟨⟨ 30 -4 108 -76 87 262 ]]

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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