31st-octave temperaments: Difference between revisions
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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 {{ | 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. | ||
== Birds == | == Birds == | ||
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[[Comma list]]: 3136/3125, 823543/819200 | [[Comma list]]: 3136/3125, 823543/819200 | ||
[[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 | ||
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Comma list: 441/440, 3136/3125, 41503/41472 | Comma list: 441/440, 3136/3125, 41503/41472 | ||
Mapping: [{{val|31 49 72 87 107}}, {{val|0 1 0 0 2}}] | Mapping: [{{val| 31 49 72 87 107 }}, {{val| 0 1 0 0 2 }}] | ||
POTE generator: ~385/384 = 4.9377 | POTE generator: ~385/384 = 4.9377 | ||
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Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689 | 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}}] | Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 1 0 0 2 -2 }}] | ||
POTE generator: ~385/384 = 5.1703 | POTE generator: ~385/384 = 5.1703 | ||
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Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125 | 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}}] | Mapping: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 1 0 0 2 -2 -2 }}] | ||
POTE generator: ~385/384 = 5.2248 | POTE generator: ~385/384 = 5.2248 | ||
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Comma list: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444 | 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}}] | 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 | POTE generator: ~385/384 = 5.3169 | ||
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[[Comma list]]: 81/80, 126/125, 1029/1024 | [[Comma list]]: 81/80, 126/125, 1029/1024 | ||
[[Mapping]]: [{{val|31 49 72 87 107}}, {{val|0 0 0 0 1}}] | [[Mapping]]: [{{val| 31 49 72 87 107 }}, {{val| 0 0 0 0 1 }}] | ||
[[POTE generator]]: ~176/175 = 6.519 | [[POTE generator]]: ~176/175 = 6.519 | ||
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Comma list: 81/80, 126/125, 105/104, 512/507 | 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}}] | 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 | ||
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Comma list: 81/80, 126/125, 144/143, 1029/1024 | 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}}] | Mapping: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 1 -1 }}] | ||
POTE generator: ~196/195 = 10.120 | POTE generator: ~196/195 = 10.120 | ||
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== Gallium == | == Gallium == | ||
The name of gallium temperament comes from | The name of gallium temperament comes from the 31st element. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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[[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]]: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 0 -1 }}] | ||
[[POTE generator]]: ~16807/16640 = 15.541 | [[POTE generator]]: ~16807/16640 = 15.541 | ||
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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: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 0 0 0 0 -1 -1 }}] | ||
POTE generator: ~121/119 = 15.785 | POTE generator: ~121/119 = 15.785 | ||
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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: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 0 0 0 0 -1 -1 -1 }}] | ||
POTE generator: ~77/76 = 16.206 | POTE generator: ~77/76 = 16.206 | ||