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.
 Temperaments discussed elsewhere include:
-* ''[[subneutral]]'', {2401/2400, 274877906944/274658203125} → [[Breedsmic temperaments #Subneutral]]
+* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]
+* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
 == Luna ==
-Subgroup: 2.3.5
+{{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
-[[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 }}
@@ Line 18: / Line 24: @@
 ; Music
-* "Moongazing" from ''Lessor 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]]
+* "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 ==
-{{main| Hemithirds }}
-{{see also| Hemimean clan #Hemithirds }}
-Subgroup: 2.3.5.7
-[[Comma list]]: 1029/1024, 3136/3125
-[[Mapping]]: [{{val| 1 4 2 2 }}, {{val| 0 -15 2 5 }}]
-{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}
-[[POTE generator]]: ~28/25 = 193.244
-[[Minimax tuning]]:
-* [[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 }}]
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6
-* [[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 }}]
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 9/7
-{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
-[[Badness]]: 0.044284
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 385/384, 441/440, 3136/3125
-Mapping: [{{val| 1 4 2 2 7 }}, {{val| 0 -15 2 5 -22 }}]
-POTE generator: ~28/25 = 193.227
-Minimax tuning:
-* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
-: [{{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 (unchanged-intervals): 2, 11/7
-{{Optimal ET sequence|legend=1| 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: [{{val| 1 4 2 2 7 0 }}, {{val| 0 -15 2 5 -22 23 }}]
-POTE generator: ~28/25 = 193.166
-{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
-Badness: 0.021738
 == Lunatic ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~262144/234375 = 193.196
-[[POTE generator]]: ~262144/234375 = 193.196
 {{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646 }}
@@ Line 97: / Line 44: @@
 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 }}
@@ Line 106: / Line 53: @@
 == Hemiluna ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 96.591
-[[POTE generator]]: ~200/189 = 96.591
 {{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}
@@ Line 127: / Line 72: @@
 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 }}
@@ Line 140: / Line 85: @@
 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 }}
@@ Line 149: / Line 94: @@
 == Semiluna ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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
-{{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
 {{Optimal ET sequence|legend=1| 56d, 118, 292, 410 }}
@@ Line 170: / Line 113: @@
 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 }}
@@ Line 179: / Line 122: @@
 [[Category:Temperament families]]
-[[Category:Luna family]]
+[[Category:Pages with mostly numerical content]]
-[[Category:Hemithirds]]
+[[Category:Luna family| ]] <!-- main article -->
+[[Category:Luna| ]] <!-- key article -->
 [[Category:Rank 2]]