31st-octave temperaments

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All temperaments on this page have a period that is 1/31th of an octave.

By the 31-3-comma is meant 617673396283947/562949953421312 = [-49 31, the amount (160.605 cents) by which 31 just perfect fifths (3/2) 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. This approach leads to the prajapati temperament and the gallium temperament.

31edo is accurate for harmonics 5 and 7, the 31-5-comma ([72 0 -31, the amount by which 31 just 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 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, [72 0 -31, and the 31-7 comma, ([-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: [31 49 72 87], 0 1 0 0]]

Wedgie⟨⟨31 0 0 -72 -87 0]]

POTE generator: ~1029/1024 = 5.1551

Vals31, 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: [31 49 72 87 107], 0 1 0 0 2]]

POTE generator: ~385/384 = 4.9377

Vals: 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: [31 49 72 87 107 115], 0 1 0 0 2 -2]]

POTE generator: ~385/384 = 5.1703

Vals: 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: [31 49 72 87 107 115 127], 0 1 0 0 2 -2 -2]]

POTE generator: ~385/384 = 5.2248

Vals: 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: [31 49 72 87 107 115 127 132], 0 1 0 0 2 -2 -2 -2]]

POTE generator: ~385/384 = 5.3169

Vals: 31, 186e, 217, 248h, 465h

Badness: 0.021271

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

Vals31, 93, 124b, 155b, 186b

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: 31, 93f, 124bf

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: 31, 62e, 93, 124b, 341b

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

Vals31, 62, 93e, 155bef

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: 31, 62, 93e, 155befg

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: 31, 62, 155befg

Badness: 0.019963