31st-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|31}}
{{Technical data page}}
{{Infobox fractional-octave|31}}
This page collects rank-2 temperaments with a period that is 1/31 of an octave.
This page collects rank-2 temperaments with a period that is 1/31 of an octave.


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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 the following ETs: {{Optimal ET sequence| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527 }}. 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 the following ETs: {{Optimal ET sequence| 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527 }}. Tempering out these commas leads to the birds temperament.
== 31-commatic ==
Subgroup: 2.3.5
Comma list: {{monzo| -49 31 }}
{{Mapping|legend=1| -31 -49 0 | 0 0 1 }}
: mapping generators: ~531441/524288 = 1\31, ~5
[[Optimal tuning]] ([[CTE]]): ~5/4 = 386.314
[[Support]]ing [[ET]]s: {{EDOs|31, 62, 93}}
== 31-5-commatic ==
Subgroup: 2.3.5
Comma list: {{monzo| 72 0 -31 }}
{{Mapping|legend=1| 31 31 72 | 0 1 0 }}
[[Optimal tuning]] ([[CWE]]): ~128/125 = 1\31, ~3/2 = 702.133
[[Support]]ing [[ET]]s: 31, 217, 186, 248, 155, 465, 403, 279, 124, 93c, 62c, 682, 310, 620
== 31-17/13-commatic ==
A circle of 31 [[17/13]]'s closes at the octave with an error of only 2.74 cents.
Subgroup: 2.13.17
Comma list: {{Monzo|12 0 0 0 0 31 -31}}
{{Mapping|31 0 12|0 1 1|legend=2}}
: sval mapping generators: ~2.13.17 {{monzo|-5 -13 13}} = 1\31, ~13
[[Optimal tuning]] ([[CTE]]): ~13/8 = 840.488


== Birds ==
== Birds ==
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[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
[[Mapping]]: [{{val| 31 49 72 87 }}, {{val| 0 1 0 0 }}]
{{Multival|legend=1| 31 0 0 -72 -87 0 }}


[[POTE generator]]: ~1029/1024 = 5.1551
[[POTE generator]]: ~1029/1024 = 5.1551
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Badness: 0.021271
Badness: 0.021271
== 217 & 1178 ==
The 217 & 1178 temperament combines two multiples of 31, which are large equal divisions consistent in the 21-odd-limit. 1395edo, also consistent in 21-odd-limit, is also a tuning.
Subgroup: 2.3.5.7
Comma list: 4375/4374, {{monzo|-153 42 7 25}}
{{Mapping|legend=1| 31 2 -38 197 | 0 3 7 -7 }}
: mapping generators: ~562711519881/549755813888 = 1\31, ~67108864/47258883 = 608.167
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 744c, 961, 1178, 1395, 1612, 2573}}
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 820125/819896, {{monzo|-37 12 -1  6  1}}
{{Mapping|legend=1| 31 2 -38 197 -97 | 0 3 7 -7 13 }}
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 4375/4374, 225000/224939, 18753525/18743296
{{Mapping|legend=1| 31 2 -38 197 -97 99 | 0 3 7 -7 13 1 }}
: mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167
[[Optimal tuning]] ([[CTE]]): ~14553/10240 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 4225/4224, 4375/4374, 14400/14399, 14875/14872, 56595/56576
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 | 0 3 7 -7 13 1 1 }}
: mapping generators: ~45/44 = 1\31, ~1989/1400 = 608.167
[[Optimal tuning]] ([[CTE]]): ~1989/1400 = 608.167
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 4200/4199, 4225/4224, 4375/4374, 5929/5928, 5985/5984, 14875/14872
{{Mapping|legend=1| 31 2 -38 197 -97 99 111 6 | 0 3 7 -7 13 1 1 8 }}
: mapping generators: ~112651/110160 = 1\31, ~665/468 = 608.166
[[Optimal tuning]] ([[CTE]]): ~665/468 = 608.166
[[Support]]ing [[ET]]s: {{EDOs|217, 961e, 1178, 1395, 1612, 2573}}
; Music
* ''[https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening]'' by [[Eliora]] (2023) - 217 & 1178 and enneadecal in 1178edo tuning


== Prajapati ==
== 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.
The Hindu god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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== Gallium ==
== Gallium ==
The name of gallium temperament comes from the 31st element.
The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31et, while adding 13, 17, and 19 on an independent generator chain, and this considerably improves the qualities of 13-limit and beyond.  


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 81/80, 99/98, 121/120, 126/125
[[Comma list]]: 81/80, 99/98, 121/120, 126/125


[[Mapping]]: [{{val| 31 49 72 87 107 115 }}, {{val| 0 0 0 0 0 -1 }}]
{{Mapping|legend=1| 31 49 72 87 107 115 | 0 0 0 0 0 -1 }}


[[POTE generator]]: ~16807/16640 = 15.541
[[Optimal tuning]] ([[CTE]]): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)


{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155bef }}
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Comma list: 81/80, 99/98, 121/120, 126/125, 273/272
Comma list: 81/80, 99/98, 121/120, 126/125, 273/272


Mapping: [{{val| 31 49 72 87 107 115 127 }}, {{val| 0 0 0 0 0 -1 -1 }}]
Mapping: {{mapping| 31 49 72 87 107 115 127 | 0 0 0 0 0 -1 -1 }}


POTE generator: ~121/119 = 15.785
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)


{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
{{Optimal ET sequence|legend=1| 31, 62, 93e, 155befg }}
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Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272
Comma list: 81/80, 99/98, 121/120, 126/125, 153/152, 273/272


Mapping: [{{val| 31 49 72 87 107 115 127 132 }}, {{val| 0 0 0 0 0 -1 -1 -1 }}]
Mapping: {{mapping| 31 49 72 87 107 115 127 132 | 0 0 0 0 0 -1 -1 -1 }}


POTE generator: ~77/76 = 16.206
Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)


{{Optimal ET sequence|legend=1| 31, 62, 155befg }}
{{Optimal ET sequence|legend=1| 31, 62, 155befg }}
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Badness: 0.019963
Badness: 0.019963


[[Category:31edo]]
{{Navbox fractional-octave}}
[[Category:Temperament collections]]
[[Category:Rank 2]]

Latest revision as of 04:54, 12 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

← 30th-octave • 31st-octave • 32nd-octave →

This page collects rank-2 temperaments with a period that is 1/31 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 the following ETs: 31, 62, 93, 124, 155, 186, 217, 248, 279, 310, 341, 372, 403, 434, 465, 496, and 527. Tempering out these commas leads to the birds temperament.

31-commatic

Subgroup: 2.3.5

Comma list: [-49 31⟩

Mapping: [⟨-31 -49 0], ⟨0 0 1]]

mapping generators: ~531441/524288 = 1\31, ~5

Optimal tuning (CTE): ~5/4 = 386.314

Supporting ETs: 31, 62, 93

31-5-commatic

Subgroup: 2.3.5

Comma list: [72 0 -31⟩

Mapping: [⟨31 31 72], ⟨0 1 0]]

Optimal tuning (CWE): ~128/125 = 1\31, ~3/2 = 702.133

Supporting ETs: 31, 217, 186, 248, 155, 465, 403, 279, 124, 93c, 62c, 682, 310, 620

31-17/13-commatic

A circle of 31 17/13's closes at the octave with an error of only 2.74 cents.

Subgroup: 2.13.17

Comma list: [12 0 0 0 0 31 -31⟩

Subgroup-val mapping: [⟨31 0 12], ⟨0 1 1]]

sval mapping generators: ~2.13.17 [-5 -13 13⟩ = 1\31, ~13

Optimal tuning (CTE): ~13/8 = 840.488

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]]

POTE generator: ~1029/1024 = 5.1551

Optimal ET sequence: 31, 124, 155, 186, 217, 248, 465

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 ET sequence: 31, 186e, 217, 248, 961cd

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 ET sequence: 31, 186e, 217, 248, 465

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 ET sequence: 31, 186e, 217, 248, 465

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 ET sequence: 31, 186e, 217, 248h, 465h

Badness: 0.021271

217 & 1178

The 217 & 1178 temperament combines two multiples of 31, which are large equal divisions consistent in the 21-odd-limit. 1395edo, also consistent in 21-odd-limit, is also a tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-153 42 7 25⟩

Mapping: [⟨31 2 -38 197], ⟨0 3 7 -7]]

mapping generators: ~562711519881/549755813888 = 1\31, ~67108864/47258883 = 608.167

Optimal tuning (CTE): ~14553/10240 = 608.167

Supporting ETs: 217, 744c, 961, 1178, 1395, 1612, 2573

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 820125/819896, [-37 12 -1 6 1⟩

Mapping: [⟨31 2 -38 197 -97], ⟨0 3 7 -7 13]]

mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167

Optimal tuning (CTE): ~14553/10240 = 608.167

Supporting ETs: 217, 961e, 1178, 1395, 1612, 2573

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 225000/224939, 18753525/18743296

Mapping: [⟨31 2 -38 197 -97 99], ⟨0 3 7 -7 13 1]]

mapping generators: ~45/44 = 1\31, ~14553/10240 = 608.167

Optimal tuning (CTE): ~14553/10240 = 608.167

Supporting ETs: 217, 961e, 1178, 1395, 1612, 2573

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 4225/4224, 4375/4374, 14400/14399, 14875/14872, 56595/56576

Mapping: [⟨31 2 -38 197 -97 99 111], ⟨0 3 7 -7 13 1 1]]

mapping generators: ~45/44 = 1\31, ~1989/1400 = 608.167

Optimal tuning (CTE): ~1989/1400 = 608.167

Supporting ETs: 217, 961e, 1178, 1395, 1612, 2573

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 4200/4199, 4225/4224, 4375/4374, 5929/5928, 5985/5984, 14875/14872

Mapping: [⟨31 2 -38 197 -97 99 111 6], ⟨0 3 7 -7 13 1 1 8]]

mapping generators: ~112651/110160 = 1\31, ~665/468 = 608.166

Optimal tuning (CTE): ~665/468 = 608.166

Supporting ETs: 217, 961e, 1178, 1395, 1612, 2573

Music

Prajapati

The Hindu god Pradjapati is said to have created the universe by speaking aloud the odd numbers from 1 to 31.

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

Optimal ET sequence: 31, 93, 124b, 155b, 186b

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 ET sequence: 31, 93f, 124bf

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 ET sequence: 31, 62e, 93, 124b, 341b

Badness: 0.048582

Gallium

The name of gallium temperament comes from the 31st element. Gallium preserves the 11-limit mapping of 31et, while adding 13, 17, and 19 on an independent generator chain, and this considerably improves the qualities of 13-limit and beyond.

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]]

Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.5276 (~144/143 = 11.0853)

Optimal ET sequence: 31, 62, 93e, 155bef

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]]

Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.4879 (~144/143 = 11.1250)

Optimal ET sequence: 31, 62, 93e, 155befg

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]]

Optimal tuning (CTE): ~45/44 = 1\31, ~13/8 = 840.1820 (~144/143 = 11.4309)

Optimal ET sequence: 31, 62, 155befg

Badness: 0.019963

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