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31st-octave temperaments: Difference between revisions

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__FORCETOC__
All temperaments on this page have a period that is 1/31th of an octave.
=31-3-comma=
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.


=Prajapati=
= 31-3-comma =
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.
By the ''31-3-comma'' is meant 617673396283947/562949953421312 = {{monzo|-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 [[31edo|31EDO]] is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.


Commas: 81/80, 126/125, 1029/1024
= 31-5-comma and 31-7-comma =
31EDO is accurate for harmonics 5 and 7, the 31-5-comma ({{monzo|72 0 -31}}, the amount by which 31 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|527]] EDOs. Tempering out these commas leads to the birds temperament.


POTE generator: ~176/175 = 6.519
= 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.


Map: [<31 49 72 87 107|, <0 0 0 0 1|]
[[Comma list]]: 81/80, 126/125, 1029/1024


EDOs: 31, 93, 124b, 155b, 186b
[[Mapping]]: [{{val|31 49 72 87 107}}, {{val|0 0 0 0 1}}]


Badness: 0.0430
[[POTE generator]]: ~176/175 = 6.519


==13-limit==
{{Val list|legend=1| 31, 93, 124b, 155b, 186b }}
Commas: 81/80, 126/125, 105/104, 512/507
 
[[Badness]]: 0.042959
 
== 13-limit ==
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
POTE generator: ~66/65 = 9.171


Map: [<31 49 72 87 107 115|, <0 0 0 0 1 0|]
Vals: {{Val list| 31, 93f, 124bf }}


EDOs: 31, 93f, 124bf
Badness: 0.037885


Badness: 0.0379
== Kumhar ==
Commas: 81/80, 126/125, 1029/1024, 144/143


==Kumhar==
Mapping: [{{val|31 49 72 87 107 115}}, {{val|0 0 0 0 1 -1}}]
Commas: 81/80, 126/125, 1029/1024, 144/143


POTE generator: ~196/195 = 10.120
POTE generator: ~196/195 = 10.120


Map: [<31 49 72 87 107 115|, <0 0 0 0 1 -1|]
Vals: {{Val list| 31, 62e, 93, 124b, 341b }}
 
Badness: 0.048582
 
= Gallium =
The name of gallium temperament comes from Gallium, the 31st element.
 
[[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}}]


EDOs: 31, 62e, 93, 124b, 341b
[[POTE generator]]: ~16807/16640 = 15.541


Badness: 0.0486
{{Val list|legend=1| 31, 62, 93e, 155bef }}


=Gallium=
[[Badness]]: 0.025484
The name of gallium temperament comes from Gallium, the 31st element.
 
== 17-limit ==
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}}]
 
POTE generator: ~121/119 = 15.785
 
Vals: {{Val list| 31, 62, 93e, 155befg }}
 
Badness: 0.023421
 
== 19-limit ==
Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272


Commas: 81/80, 126/125, 99/98, 121/120
Mapping: [{{val|31 49 72 87 107 115 127 132}}, {{val|0 0 0 0 0 -1 -1 -1}}]


POTE generator: ~16807/16640 = 15.541
POTE generator: ~77/76 = 16.206


Map: [<31 49 72 87 107 0|, <0 0 0 0 0 1|]
Vals: {{Val list| 31, 62, 155befg }}


EDOs: 31, 62, 93e, 155bef
Badness: 0.019963


Badness: 0.0255
= Birds =
The birds temperament tempers out the ''31-5 comma'', {{monzo|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.


=Birds=
[[Comma list]]: 3136/3125, 823543/819200
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
[[Mapping]]: [{{val|31 49 72 87}}, {{val|0 1 0 0}}]


POTE generator: ~1029/1024 = 5.1551
{{multival|legend=1|31 0 0 -72 -87 0}}


Map: [<31 49 72 87|, <0 1 0 0|]
[[POTE generator]]: ~1029/1024 = 5.1551


Wedgie: <<31 0 0 -72 -87 0||
{{Val list|legend=1| 31, 124, 155, 186, 217, 248, 465 }}


EDOs: 31, 124, 155, 186, 217, 248, 465
[[Badness]]: 0.099928


Badness: 0.0999
== 11-limit ==
Comma list: 441/440, 3136/3125, 41503/41472


==11-limit==
Mapping: [{{val|31 49 72 87 107}}, {{val|0 1 0 0 2}}]
Commas: 441/440, 3136/3125, 41503/41472


POTE generator: ~385/384 = 4.9377
POTE generator: ~385/384 = 4.9377


Map: [<31 49 72 87 107|, <0 1 0 0 2|]
Vals: {{Val list| 31, 186e, 217, 248, 961cd }}


EDOs: 31, 186e, 217, 248, 961cd
Badness: 0.039921


Badness: 0.0399
== 13-limit ==
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|]
Vals: {{Val list| 31, 186e, 217, 248, 465 }}


EDOs: 31, 186e, 217, 248, 465
Badness: 0.035680


Badness: 0.0357
== 17-limit ==
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|]
Vals: {{Val list| 31, 186e, 217, 248, 465 }}


EDOs: 31, 186e, 217, 248, 465
Badness: 0.025890


Badness: 0.0259
== 19-limit ==
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|]
Vals: {{Val list| 31, 186e, 217, 248h, 465h }}
 
EDOs: 31, 186e, 217, 248h, 465h


Badness: 0.0213
Badness: 0.021271


[[Category:31-comma]]
[[Category:31-comma]]
[[Category:Temperament]]
[[Category:Temperament]]
[[Category:Rank 2]]
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