# Biyatismic clan

(Redirected from Biyatismic temperaments)

The biyatismic clan of rank-3 temperaments tempers out the biyatisma, 121/120 = [-3 -1 -1 0 2.

Temperaments discussed elsewhere are:

Considered below are zeus, artemis, oxpecker, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see Rank-4 temperament #Biyatismic (121/120).

## Protomere

Subgroup: 2.3.5.11

Comma list: 121/120

Sval mapping[1 0 1 2], 0 1 1 1], 0 0 -2 -1]]

Mapping generators: ~2, ~3, ~11/10

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466

## Zeus

Main article: Zeus

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175

Mapping[1 0 1 4 2], 0 1 1 -1 1], 0 0 -2 3 1]]

Mapping to lattice: [0 1 -1 2 0], 0 1 1 -1 1]]

Lattice basis:

11/10, 11/8
Angle (11/10, 11/8) = 87.464 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881

[[1 0 0 0 0, [11/9 10/9 -1/3 -2/9 0, [22/9 2/9 1/3 -4/9 0, [22/9 2/9 -2/3 5/9 0, [10/3 2/3 0 -1/3 0]
eigenmonzo (unchanged-interval) basis: 2.9/5.9/7

Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5

Zeus11[22] hobbit transversal

33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
11/6, 15/8, 64/33, 2

Zeus11[24] hobbit transversal

33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
11/6, 15/8, 64/33, 2

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350

Mapping: [1 0 1 4 2 7], 0 1 1 -1 1 -2], 0 0 -2 3 -1 -1]]

Mapping to lattice: [0 1 -1 2 0 -3], 0 1 1 -1 1 -2]]

Lattice basis:

11/10 length = 0.7898, 11/8 length = 1.002
Angle (11/10, 11/8) = 106.7439 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582

Minimax tuning:

• 13-odd-limit
[[1 0 0 0 0 0, [11/9 10/9 -1/3 -2/9 0 0, [22/9 2/9 1/3 -4/9 0 0, [22/9 2/9 -2/3 5/9 0 0, [10/3 2/3 0 -1/3 0 0, [14/3 -8/3 1 1/3 0 0]
eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
• 15-odd-limit
[[1 0 0 0 0 0, [0 1 0 0 0 0, [11/5 1/5 2/5 -2/5 0 0, [11/5 1/5 -3/5 3/5 0 0, [13/5 3/5 1/5 -1/5 0 0, [38/5 -12/5 1/5 -1/5 0 0]
eigenmonzo (unchanged-interval) basis: 2.3.7/5

Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5

Zeus13[22] hobbit transversal

260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2

### Tinia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 176/175

Mapping: [1 0 1 4 2 2], 0 1 1 -1 1 1], 0 0 -2 3 -1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666

## Artemis

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224

Mapping[1 0 1 -3 2], 0 1 1 4 1], 0 0 -2 -4 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.8719, ~11/10 = 158.3232

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195

Mapping: [1 0 1 -3 2 -5], 0 1 1 4 1 6], 0 0 -2 -4 -1 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7090, ~11/10 = 158.7117

### Diana

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273

Mapping: [1 0 1 -3 2 7], 0 1 1 4 1 -2], 0 0 -2 -4 -1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.9789, ~11/10 = 159.0048

## Oxpecker

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125

Mapping[1 0 1 2 2], 0 1 1 1 1], 0 0 -2 -6 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.8882, ~11/10 = 155.7756

### Woodpecker

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 126/125

Mapping: [1 0 1 2 2 2], 0 1 1 1 1 1], 0 0 -2 -6 -1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5946, ~11/10 = 154.8652

## Aphrodite

Aphrodite tempers out the squalentine comma, 64827/64000, in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)3 with 8/7.

### 7-limit (squalentine)

Subgroup: 2.3.5.7

Comma list: 64827/64000

Mapping[1 0 1 3], 0 1 1 0], 0 0 -4 -3]]

Mapping generators: ~2, ~3, ~21/20

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.2144, ~21/20 = 78.5694

Projection pairs: 5 320000/64827 7 64000/9261 to 2.3.7/5

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440

Mapping[1 0 1 3 2], 0 1 1 0 1], 0 0 -4 -3 -2]]

Mapping generators: ~2, ~3, ~22/21

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.3200, ~21/20 = 78.6421

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 351/350, 441/440

Mapping: [1 0 1 3 2 6], 0 1 1 0 1 -1], 0 0 -4 -3 -2 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.1158, ~21/20 = 78.5211

#### Eros

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 352/351

Mapping: [1 0 1 3 2 7], 0 1 1 0 1 -2], 0 0 -4 -3 -2 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.5014, ~21/20 = 78.6143

#### Inanna

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 275/273

Mapping: [1 0 1 3 2 1], 0 1 1 0 1 2], 0 0 -4 -3 -2 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.7754, ~21/20 = 79.6096