31 comma temperaments
31-3-comma[edit]
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[edit]
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[edit]
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[edit]
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[edit]
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[edit]
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[edit]
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[edit]
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