31st-octave temperaments
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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
31-17/13-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⟩
Sval 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]]
Wedgie: ⟨⟨31 0 0 -72 -87 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