31st-octave temperaments

Template:Fractional-octave navigation This page collects rank-2 temperaments with a period that is 1/31 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 tones (8/7) fall short of 6 octaves) is tempered out by the following ETs: 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

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

Optimal ET sequence: 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: [⟨31 49 72 87 107], ⟨0 1 0 0 2]]

POTE generator: ~385/384 = 4.9377

Optimal ET sequence: 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

Optimal ET sequence: 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

Optimal ET sequence: 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

Optimal ET sequence: 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.

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

Optimal ET sequence: 31, 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

Optimal ET sequence: 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

Optimal ET sequence: 31, 62e, 93, 124b, 341b

Badness: 0.048582

Gallium

The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31edo, while keeping 13, 17, 19 as independent generators. In what way is this useful is unexplained.

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

Optimal ET sequence: 31, 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

Optimal ET sequence: 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

Optimal ET sequence: 31, 62, 155befg

Badness: 0.019963