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31st-octave temperaments: Difference between revisions

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__FORCETOC__
{{Technical data page}}
=31-3-comma=
{{Infobox fractional-octave|31}}
By the ''31-3-comma'' is meant 617673396283947/562949953421312 = <-49 31|, the amount (160.605 cents) by which 31 just (3/2) fifths exceed 18 octaves. This may not seem like much of a comma, but since 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.
This page collects rank-2 temperaments with a period that is 1/31 of an octave.


=Prajapati=
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.  
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.


Commas: 81/80, 126/125, 1029/1024
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.


POTE generator: ~176/175 = 6.519
== 31-commatic ==
Subgroup: 2.3.5


Map: [<31 49 72 87 107|, <0 0 0 0 1|]
Comma list: {{monzo| -49 31 }}


EDOs: 31, 93, 124b, 155b, 186b
{{Mapping|legend=1| -31 -49 0 | 0 0 1 }}


Badness: 0.0430
: mapping generators: ~531441/524288 = 1\31, ~5


==13-limit==
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.314
Commas: 81/80, 126/125, 105/104, 512/507


POTE generator: ~66/65 = 9.171
[[Support]]ing [[ET]]s: {{EDOs|31, 62, 93}}


Map: [<31 49 72 87 107 115|, <0 0 0 0 1 0|]
== 31-5-commatic ==
Subgroup: 2.3.5


EDOs: 31, 93f, 124bf
Comma list: {{monzo| 72 0 -31 }}


Badness: 0.0379
{{Mapping|legend=1| 31 31 72 | 0 1 0 }}


==Kumhar==
[[Optimal tuning]] ([[CWE]]): ~128/125 = 1\31, ~3/2 = 702.133
Commas: 81/80, 126/125, 1029/1024, 144/143


POTE generator: ~196/195 = 10.120
[[Support]]ing [[ET]]s: 31, 217, 186, 248, 155, 465, 403, 279, 124, 93c, 62c, 682, 310, 620


Map: [<31 49 72 87 107 115|, <0 0 0 0 1 -1|]
== 31-17/13-commatic ==
A circle of 31 [[17/13]]'s closes at the octave with an error of only 2.74 cents.


EDOs: 31, 62e, 93, 124b, 341b
Subgroup: 2.13.17


Badness: 0.0486
Comma list: {{Monzo|12 0 0 0 0 31 -31}}


=Gallium=
{{Mapping|31 0 12|0 1 1|legend=2}}
The name of gallium temperament comes from Gallium, the 31st element.


Commas: 81/80, 126/125, 99/98, 121/120
: sval mapping generators: ~2.13.17 {{monzo|-5 -13 13}} = 1\31, ~13


POTE generator: ~16807/16640 = 15.541
[[Optimal tuning]] ([[CTE]]): ~13/8 = 840.488


Map: [<31 49 72 87 107 0|, <0 0 0 0 0 1|]
== Birds ==
The birds temperament tempers out the 31-5 comma, {{monzo| 72 0 -31 }}, and the 31-7 comma, ({{monzo| -87 0 0 31 }}. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it."


EDOs: 31, 62, 93e, 155bef
Subgroup: 2.3.5.7


Badness: 0.0255
[[Comma list]]: 3136/3125, 823543/819200


=Birds=
[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
The birds temperament tempers out the ''31-5 comma'', |72 0 -31>, which is the amount (24.275 cents) by which 31 just major thirds (5/4) fall short of ten octaves. The name comes from Isaiah 31:5 "As birds flying, so wil the Lord of hostes defend Ierusalem, defending also hee will deliuer it, and passing ouer, he will preserue it." It also tempers out the 31-7 comma, but sadly, combining the two commas leads to torsion.


Commas: 3136/3125, 823543/819200
[[POTE generator]]: ~1029/1024 = 5.1551


POTE generator: ~1029/1024 = 5.1551
{{Optimal ET sequence|legend=1| 31, 124, 155, 186, 217, 248, 465 }}


Map: [<31 49 72 87|, <0 1 0 0|]
[[Badness]]: 0.099928


Wedgie: <<31 0 0 -72 -87 0||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 31, 124, 155, 186, 217, 248, 465
Comma list: 441/440, 3136/3125, 41503/41472


Badness: 0.0999
Mapping: [{{val| 31 49 72 87 107 }}, {{val| 0 1 0 0 2 }}]


==11-limit==
POTE generator: ~385/384 = 4.9377
Commas: 441/440, 3136/3125, 41503/41472


POTE generator: ~385/384 = 4.9377
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 961cd }}


Map: [<31 49 72 87 107|, <0 1 0 0 2|]
Badness: 0.039921


EDOs: 31, 186e, 217, 248, 961cd
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0399
Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689


==13-limit==
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 1 0 0 2 -2 }}]
Commas: 441/440, 1001/1000, 3136/3125, 13720/13689


POTE generator: ~385/384 = 5.1703
POTE generator: ~385/384 = 5.1703


Map: [<31 49 72 87 107 115|, <0 1 0 0 2 -2|]
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.035680


EDOs: 31, 186e, 217, 248, 465
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.0357
Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125


==17-limit==
Mapping: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 1 0 0 2 -2 -2 }}]
Commas: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125


POTE generator: ~385/384 = 5.2248
POTE generator: ~385/384 = 5.2248


Map: [<31 49 72 87 107 115 127|, <0 1 0 0 2 -2 -2|]
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.025890


EDOs: 31, 186e, 217, 248, 465
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0259
Comma list: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444


==19-limit==
Mapping: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 1 0 0 2 -2 -2 -2 }}]
Commas: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444


POTE generator: ~385/384 = 5.3169
POTE generator: ~385/384 = 5.3169


Map: [<31 49 72 87 107 115 127 132|, <0 1 0 0 2 -2 -2 -2|]
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248h, 465h }}
 
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 ==
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
 
[[Comma list]]: 81/80, 126/125, 1029/1024
 
[[Mapping]]: [{{val| 31 49 72 87 107 }}, {{val| 0 0 0 0 1 }}]
 
[[POTE generator]]: ~176/175 = 6.519
 
{{Optimal ET sequence|legend=1| 31, 93, 124b, 155b, 186b }}
 
[[Badness]]: 0.042959
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 126/125, 105/104, 512/507
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 0 }}]
 
POTE generator: ~66/65 = 9.171
 
{{Optimal ET sequence|legend=1| 31, 93f, 124bf }}
 
Badness: 0.037885
 
=== Kumhar ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 126/125, 144/143, 1029/1024
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 -1 }}]
 
POTE generator: ~196/195 = 10.120
 
{{Optimal ET sequence|legend=1| 31, 62e, 93, 124b, 341b }}
 
Badness: 0.048582
 
== Gallium ==
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
 
[[Comma list]]: 81/80, 99/98, 121/120, 126/125
 
{{Mapping|legend=1| 31 49 72 87 107 115 | 0 0 0 0 0 -1 }}
 
[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)
 
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}
 
[[Badness]]: 0.025484
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)
 
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
 
Badness: 0.023421
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}
 
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)


EDOs: 31, 186e, 217, 248h, 465h
{{Optimal ET sequence|legend=1| 31, 62, 155befg }}


Badness: 0.0213
Badness: 0.019963


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