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31st-octave temperaments: Difference between revisions

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All temperaments on this page have a period that is 1/31th of an octave.
{{Technical data page}}
{{Infobox fractional-octave|31}}
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.
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 tone ([[8/7]]) fall short of 6 octaves) is tempered out by {{Val list| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527edo }}. 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.


== Prajapati ==
== 31-commatic ==
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.
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 ==
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."
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 3136/3125, 823543/819200
 
[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
 
[[POTE generator]]: ~1029/1024 = 5.1551
 
{{Optimal ET sequence|legend=1| 31, 124, 155, 186, 217, 248, 465 }}
 
[[Badness]]: 0.099928


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 81/80, 126/125, 1029/1024
Comma list: 441/440, 3136/3125, 41503/41472


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


[[POTE generator]]: ~176/175 = 6.519
POTE generator: ~385/384 = 4.9377


{{Val list|legend=1| 31, 93, 124b, 155b, 186b }}
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 961cd }}


[[Badness]]: 0.042959
Badness: 0.039921


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 126/125, 105/104, 512/507
Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689
 
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 1 0 0 2 -2 }}]
 
POTE generator: ~385/384 = 5.1703
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.035680
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125
 
Mapping: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 1 0 0 2 -2 -2 }}]
 
POTE generator: ~385/384 = 5.2248
 
{{Optimal ET sequence|legend=1| 31, 186e, 217, 248, 465 }}
 
Badness: 0.025890
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444
 
Mapping: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 1 0 0 2 -2 -2 -2 }}]
 
POTE generator: ~385/384 = 5.3169
 
{{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


Mapping: [{{val|31 49 72 87 107 115}}, {{val|0 0 0 0 1 0}}]
Comma list: 4375/4374, {{monzo|-153 42 7 25}}


POTE generator: ~66/65 = 9.171
{{Mapping|legend=1| 31 2 -38 197 | 0 3 7 -7 }}


Vals: {{Val list| 31, 93f, 124bf }}
: mapping generators: ~562711519881/549755813888 = 1\31, ~67108864/47258883 = 608.167


Badness: 0.037885
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167


=== Kumhar ===
[[Support]]ing [[ET]]s: {{EDOs|217, 744c, 961, 1178, 1395, 1612, 2573}}
Subgroup: 2.3.5.7.11.13


Comma list: 81/80, 126/125, 144/143, 1029/1024
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val|31 49 72 87 107 115}}, {{val|0 0 0 0 1 -1}}]
Comma list: 4375/4374, 820125/819896, {{monzo|-37 12 -1 6  1}}


POTE generator: ~196/195 = 10.120
{{Mapping|legend=1| 31 2 -38 197 -97 | 0 3 7 -7 13 }}


Vals: {{Val list| 31, 62e, 93, 124b, 341b }}
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167


Badness: 0.048582
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167


== Gallium ==
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
The name of gallium temperament comes from Gallium, the 31st element.


=== 13-limit ===
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: 4225/4224, 4375/4374, 225000/224939, 18753525/18743296


[[Mapping]]: [{{val|31 49 72 87 107 115}}, {{val|0 0 0 0 0 -1}}]
{{Mapping|legend=1| 31 2 -38 197 -97 99 | 0 3 7 -7 13 1 }}


[[POTE generator]]: ~16807/16640 = 15.541
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167


{{Val list|legend=1| 31, 62, 93e, 155bef }}
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167


[[Badness]]: 0.025484
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 81/80, 99/98, 121/120, 126/125, 273/272
Comma list: 4225/4224, 4375/4374, 14400/14399, 14875/14872, 56595/56576


Mapping: [{{val|31 49 72 87 107 115 127}}, {{val|0 0 0 0 0 -1 -1}}]
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 | 0 3 7 -7 13 1 1 }}


POTE generator: ~121/119 = 15.785
: mapping generators: ~45/44 = 1\31, ~1989/1400 = 608.167


Vals: {{Val list| 31, 62, 93e, 155befg }}
[[Optimal tuning]] ([[CTE]]): ~1989/1400 = 608.167


Badness: 0.023421
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272
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: [{{val|31 49 72 87 107 115 127 132}}, {{val|0 0 0 0 0 -1 -1 -1}}]
: mapping generators: ~112651/110160 = 1\31, ~665/468 = 608.166


POTE generator: ~77/76 = 16.206
[[Optimal tuning]] ([[CTE]]): ~665/468 = 608.166


Vals: {{Val list| 31, 62, 155befg }}
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}


Badness: 0.019963
; Music


== Birds ==
* ''[https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening]'' by [[Eliora]] (2023) - 217 & 1178 and enneadecal in 1178edo tuning
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."


Subgroup: 2.3.5.7
== Prajapati ==
The Hindu god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31.


[[Comma list]]: 3136/3125, 823543/819200
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val|31 49 72 87}}, {{val|0 1 0 0}}]
[[Comma list]]: 81/80, 126/125, 1029/1024


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


[[POTE generator]]: ~1029/1024 = 5.1551
[[POTE generator]]: ~176/175 = 6.519


{{Val list|legend=1| 31, 124, 155, 186, 217, 248, 465 }}
{{Optimal ET sequence|legend=1| 31, 93, 124b, 155b, 186b }}


[[Badness]]: 0.099928
[[Badness]]: 0.042959


=== 11-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 441/440, 3136/3125, 41503/41472
Comma list: 81/80, 126/125, 105/104, 512/507


Mapping: [{{val|31 49 72 87 107}}, {{val|0 1 0 0 2}}]
Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 0 }}]


POTE generator: ~385/384 = 4.9377
POTE generator: ~66/65 = 9.171


Vals: {{Val list| 31, 186e, 217, 248, 961cd }}
{{Optimal ET sequence|legend=1| 31, 93f, 124bf }}


Badness: 0.039921
Badness: 0.037885


=== 13-limit ===
=== Kumhar ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


POTE generator: ~385/384 = 5.1703
[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)


Vals: {{Val list| 31, 186e, 217, 248, 465 }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}


Badness: 0.035680
[[Badness]]: 0.025484


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125
Comma list: 81/80, 99/98, 121/120, 126/125, 273/272


Mapping: [{{val|31 49 72 87 107 115 127}}, {{val|0 1 0 0 2 -2 -2}}]
Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}


POTE generator: ~385/384 = 5.2248
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)


Vals: {{Val list| 31, 186e, 217, 248, 465 }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}


Badness: 0.025890
Badness: 0.023421


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444
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 1 0 0 2 -2 -2 -2}}]
Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}


POTE generator: ~385/384 = 5.3169
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)


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


Badness: 0.021271
Badness: 0.019963


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