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 | ||
| Line 31: | Line 35: | ||
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 | ||
| Line 103: | Line 119: | ||
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 | ||
| Line 114: | Line 132: | ||
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 | ||
| Line 125: | Line 145: | ||
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 | ||
Revision as of 22:28, 11 June 2021
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 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
31EDO is accurate for harmonics 5 and 7, the 31-5-comma ([72 0 -31⟩, the amount by which 31 major thirds (5/4) fall short of 10 octaves) and the 31-7-comma ([-87 0 0 31⟩, the amount by which 31 septimal whole tone (8/7) fall short of 6 octaves) is tempered out by Template:Val list, and 527 EDOs. Tempering out these commas leads to the birds temperament.
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.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 1029/1024
Mapping: [⟨31 49 72 87 107], ⟨0 0 0 0 1]]
POTE generator: ~176/175 = 6.519
Badness: 0.042959
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 126/125, 105/104, 512/507
Mapping: [⟨31 49 72 87 107 115], ⟨0 0 0 0 1 0]]
POTE generator: ~66/65 = 9.171
Vals: Template:Val list
Badness: 0.037885
Kumhar
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 126/125, 144/143, 1029/1024
Mapping: [⟨31 49 72 87 107 115], ⟨0 0 0 0 1 -1]]
POTE generator: ~196/195 = 10.120
Vals: Template:Val list
Badness: 0.048582
Gallium
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
Mapping: [⟨31 49 72 87 107 115], ⟨0 0 0 0 0 -1]]
POTE generator: ~16807/16640 = 15.541
Badness: 0.025484
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 121/120, 126/125, 273/272
Mapping: [⟨31 49 72 87 107 115 127], ⟨0 0 0 0 0 -1 -1]]
POTE generator: ~121/119 = 15.785
Vals: Template:Val list
Badness: 0.023421
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
Mapping: [⟨31 49 72 87 107 115 127 132], ⟨0 0 0 0 0 -1 -1 -1]]
POTE generator: ~77/76 = 16.206
Vals: Template:Val list
Badness: 0.019963
Birds
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.
Subgroup: 2.3.5.7
Comma list: 3136/3125, 823543/819200
Mapping: [⟨31 49 72 87], ⟨0 1 0 0]]
Wedgie: ⟨⟨ 31 0 0 -72 -87 0 ]]
POTE generator: ~1029/1024 = 5.1551
Badness: 0.099928
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 41503/41472
Mapping: [⟨31 49 72 87 107], ⟨0 1 0 0 2]]
POTE generator: ~385/384 = 4.9377
Vals: Template:Val list
Badness: 0.039921
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 1001/1000, 3136/3125, 13720/13689
Mapping: [⟨31 49 72 87 107 115], ⟨0 1 0 0 2 -2]]
POTE generator: ~385/384 = 5.1703
Vals: Template:Val list
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: [⟨31 49 72 87 107 115 127], ⟨0 1 0 0 2 -2 -2]]
POTE generator: ~385/384 = 5.2248
Vals: Template:Val list
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: [⟨31 49 72 87 107 115 127 132], ⟨0 1 0 0 2 -2 -2 -2]]
POTE generator: ~385/384 = 5.3169
Vals: Template:Val list
Badness: 0.021271