31 comma temperaments

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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 31et is such a strong 7-limit system, 11- and 13-limit temperaments based on the cycle of 31 fifths actually make sense.

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.

Commas: 81/80, 126/125, 1029/1024

POTE generator: ~176/175 = 6.519

Map: [[<31 49 72 87 107|, <0 0 0 0 1|]

EDOs: 31, 93, 124b, 155b, 186b

Badness: 0.0430

13-limit

Commas: 81/80, 126/125, 105/104, 512/507

POTE generator: ~66/65 = 9.171

Map: [[<31 49 72 87 107 115|, <0 0 0 0 1 0|]

EDOs: 31, 93f, 124bf

Badness: 0.0379

Kumhar

Commas: 81/80, 126/125, 1029/1024, 144/143

POTE generator: ~196/195 = 10.120

Map: [[<31 49 72 87 107 115|, <0 0 0 0 1 -1|]

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

Badness: 0.0486

Gallium

The name of gallium temperament comes from Gallium, the 31st element.

Commas: 81/80, 126/125, 99/98, 121/120

POTE generator: ~16807/16640 = 15.541

Map: [<31 49 72 87 107 0|, <0 0 0 0 0 1|]

EDOs: 31, 62, 93e, 155bef

Badness: 0.0255

Birds

The birds temperament tempers out the 31-5 comma, |72 0 -31>, which is the amount (24.275 cents) by which 31 just (5/4) major thirds 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.

Commas: 3136/3125, 823543/819200

POTE generator: ~1029/1024 = 5.155

Map: [<31 49 72 87|, <0 1 0 0|]

Wedgie: <<31 0 0 -72 -87 0||

Edos: 31, 124, 155, 186, 217, 248, 465

Badness: 0.1000

11-limit

Commas: 441/440, 3136/3125, 41503/41472

POTE generator: ~385/384 = 4.938

Map: [<31 49 72 87 107|, <0 1 0 0 2|]

EDOs: 31, 186e, 217, 248, 961cd

Badness: 0.0399

13-limit

Commas: 441/440, 3136/3125, 1001/1000, 13720/13689

POTE generator: ~385/384 = 5.170

Map: [<31 49 72 87 107 115|, <0 1 0 0 2 -2|]

EDOs: 31, 186e, 217, 248, 465

Badness: 0.0357