Xenharmonic Wiki

Luna family: Difference between revisions

← Older edit
VisualWikitext
13-limit hemithirds copied from gamelismic clan
mNo edit summary
 
(38 intermediate revisions by 9 users not shown)
Line 1: Line 1:
The Luna family tempers out the Luna comma; 274877906944/274658203125, which has a monzo of |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:
Comma: 274877906944/274658203125
* ''[[Subneutral]]'' (+2401/2400) → [[Breedsmic temperaments #Subneutral|Breedsmic temperaments]]
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]


POTE generator: ~262144/234375 = 193.201
== Luna ==
{{see also|Luna and hemithirds}}


Map: [<1 4 2|, <0 -15 2|]
[[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.


EDOs: 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559
[[Subgroup]]: 2.3.5


Badness: 0.0206
[[Comma list]]: 274877906944/274658203125


= Hemithirds =
{{Mapping|legend=1| 1 4 2 | 0 -15 2 }}
{{main|Hemithirds}}
{{see also|Gamelismic clan #Hemithirds}}


Commas: 1029/1024, 3136/3125
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~262144/234375 = 193.201


POTE generator: ~28/25 = 193.244
{{Optimal ET sequence|legend=1| 25, 31, 56, 87, 118, 323, 441, 559, 1000, 1559 }}


Map: [<1 4 2 2|, <0 -15 2 5|]
[[Badness]]: 0.020576


Wedgie: <<15 -2 -5 -38 -50 -6||
; Music
* "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]]


EDOs: {{EDOs|31, 87, 118}}
== Lunatic ==
[[Subgroup]]: 2.3.5.7


Badness: 0.0443
[[Comma list]]: 4375/4374, 274877906944/274658203125


== 11-limit ==
{{Mapping|legend=1| 1 4 2 21 | 0 -15 2 -113 }}
Commas: 385/384, 441/440, 3136/3125


POTE generator: ~28/25 = 193.227
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~262144/234375 = 193.196


Map: [<1 4 2 2 7|, <0 -15 2 5 -22|]
{{Optimal ET sequence|legend=1| 87d, 118, 323, 441, 1205, 1646 }}


EDOs: {{EDOs|31, 87, 118}}
[[Badness]]: 0.037966


Badness: 0.0190
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 13-limit ==
Comma list: 4375/4374, 12005/11979, 172032/171875
Commas: 196/195, 352/351, 1001/1000, 1029/1024


POTE generator: ~28/25 = 193.166
Mapping: {{mapping| 1 4 2 21 14 | 0 -15 2 -113 -140 }}


Map: [<1 4 2 2 7 0|, <0 -15 2 5 -22 23|]
Optimal tuning (POTE): ~2 = 1\1, ~352/315 = 193.200


EDOs: {{EDOs|31, 56, 87, 118, 205d}}
{{Optimal ET sequence|legend=1| 118, 323e, 441, 559, 1000e }}


Badness: 0.0217
Badness: 0.058588


= Lunatic =
== Hemiluna ==
Commas: 4375/4374, 274877906944/274658203125
[[Subgroup]]: 2.3.5.7


POTE generator: ~262144/234375 = 193.196
[[Comma list]]: 48828125/48771072, 67108864/66976875


Map: [<1 4 2 2|, <0 -15 2 -113|]
{{Mapping|legend=1| 1 4 2 -5 | 0 -30 4 97 }}


Wedgie: <<15 -2 113 -38 137 268||
: mapping generators: ~2, ~200/189


EDOs: 87d, 118, 323, 441, 1205, 1646
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~200/189 = 96.591


Badness: 0.0380
{{Optimal ET sequence|legend=1| 87, 236, 323, 410, 733, 1056, 1789bd, 2845bdd }}


[[Category:Theory]]
[[Badness]]: 0.179304
[[Category:Temperament family]]
 
[[Category:Luna]]
=== 11-limit ===
[[Category:Hemithirds]]
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 = 1\1, ~200/189 = 96.587
 
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 87, 236, 323, 410 }}
 
Badness: 0.046151
 
== Semiluna ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4802000/4782969, 95703125/95551488
 
{{Mapping|legend=1| 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|legend=1| 56d, 118, 292, 410 }}
 
[[Badness]]: 0.192167
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 5632/5625, 9801/9800, 14641/14580
 
Mapping: {{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|legend=1| 56d, 118, 292, 410 }}
 
Badness: 0.067783
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Luna family| ]] <!-- main article -->
[[Category:Luna| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Luna_family"