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31st-octave temperaments: Difference between revisions

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{{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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31edo is accurate for harmonics 5 and 7, the 31-5-comma ({{monzo| 72 0 -31 }}, the amount by which 31 just major thirds ([[5/4]]) fall short of 10 octaves) and the 31-7-comma ({{monzo| -87 0 0 31 }}, the amount by which 31 septimal whole tones ([[8/7]]) fall short of 6 octaves) is tempered out by the following ETs: {{Optimal ET sequence| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527 }}. Tempering out these commas leads to the birds temperament.
31edo is accurate for harmonics 5 and 7, the 31-5-comma ({{monzo| 72 0 -31 }}, the amount by which 31 just major thirds ([[5/4]]) fall short of 10 octaves) and the 31-7-comma ({{monzo| -87 0 0 31 }}, the amount by which 31 septimal whole tones ([[8/7]]) fall short of 6 octaves) is tempered out by the following ETs: {{Optimal ET sequence| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527 }}. Tempering out these commas leads to the birds temperament.
== 31-commatic ==
Subgroup: 2.3.5
Comma list: {{monzo| -49 31 }}
{{Mapping|legend=1| -31 -49 0 | 0 0 1 }}
: mapping generators: ~531441/524288 = 1\31, ~5
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.314
[[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 ==
The Hindu god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31. People with an interest in 31 may want to try this method themselves.
The Hindu god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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== Gallium ==
== Gallium ==
The name of gallium temperament comes from the 31st element.
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 }}
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Badness: 0.019963
Badness: 0.019963


[[Category:31edo]]
{{Navbox fractional-octave}}
[[Category:Temperament collections]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/31st-octave_temperaments"