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. | ||
= 31-3-comma = | == 31-3-comma == | ||
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. | 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. | ||
= 31-5-comma and 31-7-comma = | == 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. | 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. | ||
= Prajapati = | == 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. | 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: 2.3.5.7.11 | |||
[[Comma list]]: 81/80, 126/125, 1029/1024 | [[Comma list]]: 81/80, 126/125, 1029/1024 | ||
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[[Badness]]: 0.042959 | [[Badness]]: 0.042959 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 81/80, 126/125, 105/104, 512/507 | Comma list: 81/80, 126/125, 105/104, 512/507 | ||
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Badness: 0.037885 | Badness: 0.037885 | ||
== Kumhar == | === 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}}] | Mapping: [{{val|31 49 72 87 107 115}}, {{val|0 0 0 0 1 -1}}] | ||
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Badness: 0.048582 | Badness: 0.048582 | ||
= Gallium = | == Gallium == | ||
The name of gallium temperament comes from Gallium, the 31st element. | The name of gallium temperament comes from Gallium, the 31st element. | ||
Subgroup: 2.3.5.7.11.13 | |||
[[Comma list]]: 81/80, 99/98, 121/120, 126/125 | [[Comma list]]: 81/80, 99/98, 121/120, 126/125 | ||
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[[Badness]]: 0.025484 | [[Badness]]: 0.025484 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 81/80, 99/98, 121/120, 126/125, 273/272 | Comma list: 81/80, 99/98, 121/120, 126/125, 273/272 | ||
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Badness: 0.023421 | Badness: 0.023421 | ||
== 19-limit == | === 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 | Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272 | ||
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Badness: 0.019963 | Badness: 0.019963 | ||
= Birds = | == 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. | 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. | ||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 3136/3125, 823543/819200 | [[Comma list]]: 3136/3125, 823543/819200 | ||
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[[Badness]]: 0.099928 | [[Badness]]: 0.099928 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 441/440, 3136/3125, 41503/41472 | Comma list: 441/440, 3136/3125, 41503/41472 | ||
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Badness: 0.039921 | Badness: 0.039921 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689 | Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689 | ||
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Badness: 0.035680 | Badness: 0.035680 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125 | Comma list: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125 | ||
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Badness: 0.025890 | Badness: 0.025890 | ||
== 19-limit == | === 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 | Comma list: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444 | ||