# 31st-octave temperaments

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

## 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

### 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

### 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

### 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

### 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

## 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

### 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

### 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

### 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

### 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

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

### 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

### 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

## 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)

### 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)