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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.


By the ''31-3-comma'' is meant 617673396283947/562949953421312 = {{monzo| -49 31 }}, the amount (160.605 [[cent]]s) by which 31 just perfect fifths ([[3/2]]) exceed 18 [[octave]]s. This may not seem like much of a comma, but since [[31edo]] is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense. This approach leads to the prajapati temperament and the gallium temperament.  
By the ''31-3-comma'' is meant 617673396283947/562949953421312 = {{monzo| -49 31 }}, the amount (160.605 [[cent]]s) by which 31 just perfect fifths ([[3/2]]) exceed 18 [[octave]]s. This may not seem like much of a comma, but since [[31edo]] is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense. This approach leads to the prajapati temperament and the gallium 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: {{val list| 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 ==
Line 13: Line 52:


[[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


{{Val list|legend=1| 31, 124, 155, 186, 217, 248, 465 }}
{{Optimal ET sequence|legend=1| 31, 124, 155, 186, 217, 248, 465 }}


[[Badness]]: 0.099928
[[Badness]]: 0.099928
Line 31: Line 68:
POTE generator: ~385/384 = 4.9377
POTE generator: ~385/384 = 4.9377


Optimal GPV sequence: {{Val list| 31, 186e, 217, 248, 961cd }}
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 961cd }}


Badness: 0.039921
Badness: 0.039921
Line 44: Line 81:
POTE generator: ~385/384 = 5.1703
POTE generator: ~385/384 = 5.1703


Optimal GPV sequence: {{Val list| 31, 186e, 217, 248, 465 }}
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}


Badness: 0.035680
Badness: 0.035680
Line 57: Line 94:
POTE generator: ~385/384 = 5.2248
POTE generator: ~385/384 = 5.2248


Optimal GPV sequence: {{Val list| 31, 186e, 217, 248, 465 }}
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}


Badness: 0.025890
Badness: 0.025890
Line 70: Line 107:
POTE generator: ~385/384 = 5.3169
POTE generator: ~385/384 = 5.3169


Optimal GPV sequence: {{Val list| 31, 186e, 217, 248h, 465h }}
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248h, 465h }}


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
Line 85: Line 195:
[[POTE generator]]: ~176/175 = 6.519
[[POTE generator]]: ~176/175 = 6.519


{{Val list|legend=1| 31, 93, 124b, 155b, 186b }}
{{Optimal ET sequence|legend=1| 31, 93, 124b, 155b, 186b }}


[[Badness]]: 0.042959
[[Badness]]: 0.042959
Line 98: Line 208:
POTE generator: ~66/65 = 9.171
POTE generator: ~66/65 = 9.171


Optimal GPV sequence: {{Val list| 31, 93f, 124bf }}
{{Optimal ET sequence|legend=1| 31, 93f, 124bf }}


Badness: 0.037885
Badness: 0.037885
Line 111: Line 221:
POTE generator: ~196/195 = 10.120
POTE generator: ~196/195 = 10.120


Optimal GPV sequence: {{Val list| 31, 62e, 93, 124b, 341b }}
{{Optimal ET sequence|legend=1| 31, 62e, 93, 124b, 341b }}


Badness: 0.048582
Badness: 0.048582


== 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)


{{Val list|legend=1| 31, 62, 93e, 155bef }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}


[[Badness]]: 0.025484
[[Badness]]: 0.025484
Line 135: Line 245:
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 GPV sequence: {{Val list| 31, 62, 93e, 155befg }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}


Badness: 0.023421
Badness: 0.023421
Line 148: Line 258:
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 GPV sequence: {{Val list| 31, 62, 155befg }}
{{Optimal ET sequence|legend=1| 31, 62, 155befg }}


Badness: 0.019963
Badness: 0.019963


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