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31st-octave temperaments: Difference between revisions

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By the ''31-3-comma'' is meant 617673396283947/562949953421312 = {{monzo| -49 31 }}, the amount (160.605 [[cent]]s) by which 31 just perfect fifths ([[3/2]]) exceed 18 [[octave]]s. 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.  
By the ''31-3-comma'' is meant 617673396283947/562949953421312 = {{monzo| -49 31 }}, the amount (160.605 [[cent]]s) by which 31 just perfect fifths ([[3/2]]) exceed 18 [[octave]]s. 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 ({{monzo| 72 0 -31 }}, the amount by which 31 just major thirds ([[5/4]]) fall short of 10 octaves) and the 31-7-comma ({{monzo| -87 0 0 31 }}, the amount by which 31 septimal whole tone ([[8/7]]) fall short of 6 octaves) is tempered out by {{Val list| 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.
31edo is accurate for harmonics 5 and 7, the 31-5-comma ({{monzo| 72 0 -31 }}, the amount by which 31 just major thirds ([[5/4]]) fall short of 10 octaves) and the 31-7-comma ({{monzo| -87 0 0 31 }}, the amount by which 31 septimal whole tones ([[8/7]]) fall short of 6 octaves) is tempered out by {{Val list| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527 }} EDOs. Tempering out these commas leads to the birds temperament.


== Birds ==
== Birds ==
Line 31: Line 31:
POTE generator: ~385/384 = 4.9377
POTE generator: ~385/384 = 4.9377


Vals: {{Val list| 31, 186e, 217, 248, 961cd }}
Optimal GPV sequence: {{Val list| 31, 186e, 217, 248, 961cd }}


Badness: 0.039921
Badness: 0.039921
Line 44: Line 44:
POTE generator: ~385/384 = 5.1703
POTE generator: ~385/384 = 5.1703


Vals: {{Val list| 31, 186e, 217, 248, 465 }}
Optimal GPV sequence: {{Val list| 31, 186e, 217, 248, 465 }}


Badness: 0.035680
Badness: 0.035680
Line 57: Line 57:
POTE generator: ~385/384 = 5.2248
POTE generator: ~385/384 = 5.2248


Vals: {{Val list| 31, 186e, 217, 248, 465 }}
Optimal GPV sequence: {{Val list| 31, 186e, 217, 248, 465 }}


Badness: 0.025890
Badness: 0.025890
Line 70: Line 70:
POTE generator: ~385/384 = 5.3169
POTE generator: ~385/384 = 5.3169


Vals: {{Val list| 31, 186e, 217, 248h, 465h }}
Optimal GPV sequence: {{Val list| 31, 186e, 217, 248h, 465h }}


Badness: 0.021271
Badness: 0.021271
Line 98: Line 98:
POTE generator: ~66/65 = 9.171
POTE generator: ~66/65 = 9.171


Vals: {{Val list| 31, 93f, 124bf }}
Optimal GPV sequence: {{Val list| 31, 93f, 124bf }}


Badness: 0.037885
Badness: 0.037885
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POTE generator: ~196/195 = 10.120
POTE generator: ~196/195 = 10.120


Vals: {{Val list| 31, 62e, 93, 124b, 341b }}
Optimal GPV sequence: {{Val list| 31, 62e, 93, 124b, 341b }}


Badness: 0.048582
Badness: 0.048582
Line 139: Line 139:
POTE generator: ~121/119 = 15.785
POTE generator: ~121/119 = 15.785


Vals: {{Val list| 31, 62, 93e, 155befg }}
Optimal GPV sequence: {{Val list| 31, 62, 93e, 155befg }}


Badness: 0.023421
Badness: 0.023421
Line 152: Line 152:
POTE generator: ~77/76 = 16.206
POTE generator: ~77/76 = 16.206


Vals: {{Val list| 31, 62, 155befg }}
Optimal GPV sequence: {{Val list| 31, 62, 155befg }}


Badness: 0.019963
Badness: 0.019963
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