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 ==
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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
Line 111: Line 111:
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

Revision as of 04:45, 22 February 2022

All temperaments on this page have a period that is 1/31th 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 Template:Val list EDOs. 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

Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

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. People with an interest in 31 may want to try this method themselves.

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

Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.048582

Gallium

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

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

Template:Val list

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 GPV sequence: Template:Val list

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 GPV sequence: Template:Val list

Badness: 0.019963

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