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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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[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.314
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.314


[[Support]]ing [[ET]]s: {{EDOs|31, 62, 93, 124}}
[[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
== 217 & 1178 ==
The 217 & 1178 temperament combines two multiples of 31, which are large equal divisions consistent in the 21-odd-limit. 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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== Gallium ==
== Gallium ==
The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31edo, while keeping 13, 17, 19 as independent generators. In what way is this useful is unexplained.
The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31et, while adding 13, 17, and 19 on an independent generator chain, and this considerably improves the qualities of 13-limit and beyond.  


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 81/80, 99/98, 121/120, 126/125
[[Comma list]]: 81/80, 99/98, 121/120, 126/125


[[Mapping]]: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 0 -1 }}]
{{Mapping|legend=1| 31 49 72 87 107 115 | 0 0 0 0 0 -1 }}


[[POTE generator]]: ~16807/16640 = 15.541
[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)


{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}
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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: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 0 0 0 0 -1 -1 }}]
Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}


POTE generator: ~121/119 = 15.785
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)


{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
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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: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 0 0 0 0 -1 -1 -1 }}]
Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}


POTE generator: ~77/76 = 16.206
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)


{{Optimal ET sequence|legend=1| 31, 62, 155befg }}
{{Optimal ET sequence|legend=1| 31, 62, 155befg }}


Badness: 0.019963
Badness: 0.019963
{{Navbox fractional-octave}}
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