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31st-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|31}}
{{Technical data page}}
{{Infobox fractional-octave|31}}
This page collects rank-2 temperaments with a period that is 1/31 of an octave.
This page collects rank-2 temperaments with a period that is 1/31 of an octave.


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[[Support]]ing [[ET]]s: {{EDOs|31, 62, 93}}
[[Support]]ing [[ET]]s: {{EDOs|31, 62, 93}}
== 31-5-commatic ==
Subgroup: 2.3.5
Comma list: {{monzo| 72 0 -31 }}
{{Mapping|legend=1| 31 31 72 | 0 1 0 }}
[[Optimal tuning]] ([[CWE]]): ~128/125 = 1\31, ~3/2 = 702.133
[[Support]]ing [[ET]]s: 31, 217, 186, 248, 155, 465, 403, 279, 124, 93c, 62c, 682, 310, 620
== 31-17/13-commatic ==
A circle of 31 [[17/13]]'s closes at the octave with an error of only 2.74 cents.
Subgroup: 2.13.17
Comma list: {{Monzo|12 0 0 0 0 31 -31}}
{{Mapping|31 0 12|0 1 1|legend=2}}
: sval mapping generators: ~2.13.17 {{monzo|-5 -13 13}} = 1\31, ~13
[[Optimal tuning]] ([[CTE]]): ~13/8 = 840.488


== Birds ==
== Birds ==
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[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
{{Multival|legend=1| 31 0 0 -72 -87 0 }}


[[POTE generator]]: ~1029/1024 = 5.1551
[[POTE generator]]: ~1029/1024 = 5.1551
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Badness: 0.021271
Badness: 0.021271
== Euphrosynean ==
''Euphrosynean'' (217 & 1178) temperament combines two multiples of 31, which are large equal divisions consistent in the 21-odd-limit. Named after the minor planet [[wikipedia:31 Euphrosyne|31 Euphrosyne]]. 1395edo, also consistent in 21-odd-limit, is also a tuning.
Subgroup: 2.3.5.7
Comma list: 4375/4374, {{monzo|-153 42 7 25}}
{{Mapping|legend=1| 31 2 -38 197 | 0 3 7 -7 }}
: mapping generators: ~562711519881/549755813888 = 1\31, ~67108864/47258883 = 608.167
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 744c, 961, 1178, 1395, 1612, 2573}}
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 820125/819896, {{monzo|-37 12 -1  6  1}}
{{Mapping|legend=1| 31 2 -38 197 -97 | 0 3 7 -7 13 }}
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 4375/4374, 225000/224939, 18753525/18743296
{{Mapping|legend=1| 31 2 -38 197 -97 99 | 0 3 7 -7 13 1 }}
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 4225/4224, 4375/4374, 14400/14399, 14875/14872, 56595/56576
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 | 0 3 7 -7 13 1 1 }}
: mapping generators: ~45/44 = 1\31, ~1989/1400 = 608.167
[[Optimal tuning]] ([[CTE]]): ~1989/1400 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 4200/4199, 4225/4224, 4375/4374, 5929/5928, 5985/5984, 14875/14872
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 6 | 0 3 7 -7 13 1 1 8 }}
: mapping generators: ~112651/110160 = 1\31, ~665/468 = 608.166
[[Optimal tuning]] ([[CTE]]): ~665/468 = 608.166
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
; Music
* ''[https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening]'' by [[Eliora]] (2023) - 217 & 1178 and enneadecal in 1178edo tuning


== Prajapati ==
== Prajapati ==
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Comma list: 81/80, 99/98, 121/120, 126/125, 273/272
Comma list: 81/80, 99/98, 121/120, 126/125, 273/272


Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}
{{Mapping|legend=0| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}


Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)
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Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272
Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272


Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}
{{Mapping|legend=0| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}


Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)
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Badness: 0.019963
Badness: 0.019963
{{Navbox fractional-octave}}
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