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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.
All temperaments on this page have a period that is 1/31th 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 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.
== 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}}]
{{multival|legend=1|31 0 0 -72 -87 0}}
[[POTE generator]]: ~1029/1024 = 5.1551
{{Val list|legend=1| 31, 124, 155, 186, 217, 248, 465 }}
[[Badness]]: 0.099928
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 41503/41472
Mapping: [{{val|31 49 72 87 107}}, {{val|0 1 0 0 2}}]
POTE generator: ~385/384 = 4.9377
Vals: {{Val list| 31, 186e, 217, 248, 961cd }}
Badness: 0.039921
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
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
Vals: {{Val list| 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
Vals: {{Val list| 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
Vals: {{Val list| 31, 186e, 217, 248h, 465h }}
Badness: 0.021271


== Prajapati ==
== Prajapati ==
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Badness: 0.019963
Badness: 0.019963
== 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}}]
{{multival|legend=1|31 0 0 -72 -87 0}}
[[POTE generator]]: ~1029/1024 = 5.1551
{{Val list|legend=1| 31, 124, 155, 186, 217, 248, 465 }}
[[Badness]]: 0.099928
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 41503/41472
Mapping: [{{val|31 49 72 87 107}}, {{val|0 1 0 0 2}}]
POTE generator: ~385/384 = 4.9377
Vals: {{Val list| 31, 186e, 217, 248, 961cd }}
Badness: 0.039921
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
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
Vals: {{Val list| 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
Vals: {{Val list| 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
Vals: {{Val list| 31, 186e, 217, 248h, 465h }}
Badness: 0.021271


[[Category:31edo]]
[[Category:31edo]]
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