31st-octave temperaments: Difference between revisions
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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) | 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. | ||
Commas: 3136/3125, 823543/819200 | Commas: 3136/3125, 823543/819200 | ||
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EDOs: 31, 124, 155, 186, 217, 248, 465 | EDOs: 31, 124, 155, 186, 217, 248, 465 | ||
Badness: 0. | Badness: 0.0999 | ||
==11-limit== | ==11-limit== | ||
Revision as of 07:37, 23 May 2019
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 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.
Commas: 3136/3125, 823543/819200
POTE generator: ~1029/1024 = 5.1551
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: 0.0999
11-limit
Commas: 441/440, 3136/3125, 41503/41472
POTE generator: ~385/384 = 4.9377
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, 1001/1000, 3136/3125, 13720/13689
POTE generator: ~385/384 = 5.1703
Map: [<31 49 72 87 107 115|, <0 1 0 0 2 -2|]
EDOs: 31, 186e, 217, 248, 465
Badness: 0.0357
17-limit
Commas: 441/440, 833/832, 1001/1000, 1225/1224, 3136/3125
POTE generator: ~385/384 = 5.2248
Map: [<31 49 72 87 107 115 127|, <0 1 0 0 2 -2 -2|]
EDOs: 31, 186e, 217, 248, 465
Badness: 0.0259
19-limit
Commas: 343/342, 441/440, 476/475, 833/832, 1001/1000, 1445/1444
POTE generator: ~385/384 = 5.3169
Map: [<31 49 72 87 107 115 127 132|, <0 1 0 0 2 -2 -2 -2|]
EDOs: 31, 186e, 217, 248h, 465h
Badness: 0.0213