# Kleismic rank three family

(Redirected from Enlil)

The kleismic rank-3 family of temperaments tempers out the kleisma, 15625/15552. If nothing else is tempered out in the 7-limit, we have a planar temperament. With an 11-limit comma we get an 11-limit temperament, and so forth.

## Rank-3 kleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552

Mapping[1 0 1 0], 0 6 5 0], 0 0 0 1]]

mapping generators: ~2, ~6/5, ~7

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.0586, ~7/4 = 968.8259

[[1 0 0 0, [0 1 0 0, [1 5/6 0 0, [0 0 0 1]
eigenmonzo (unchanged-interval) basis: 2.3.7

### 2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 325/324, 625/624

Sval mapping: [1 0 1 0 0], 0 6 5 0 14], 0 0 0 1 0]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1110, ~7/4 = 968.8259

## Enlil

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237

Mapping[1 0 1 0 6], 0 6 5 0 1], 0 0 0 1 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.0667, ~7/4 = 967.6041

[[1 0 0 0 0, [0 1 0 0 0, [1 5/6 0 0 0, [3 1/12 0 1/2 -1/2, [3 1/12 0 -1/2 1/2]
eigenmonzo (unchanged-interval) basis: 2.3.11/7

### 13-limit (aka sumatra)

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624

Mapping: [1 0 1 0 6 0], 0 6 5 0 1 14], 0 0 0 1 -1 0]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1156, ~7/4 = 967.6235

Minimax tuning:

• 13- and 15-odd-limit
[[1 0 0 0 0 0, [4/3 0 0 -2/9 -2/9 4/9, [19/9 0 0 -5/27 -5/27 10/27, [28/9 0 0 13/27 -14/27 1/27, [28/9 0 0 -14/27 13/27 1/27, [28/9 0 0 -14/27 -14/27 28/27]
eigenmonzo (unchanged-interval) basis: 2.11/7.13/7

Scales: enlil19_13

## Isimud

Subgroup: 2.3.5.7.11

Comma list: 4125/4096, 8019/8000

Mapping[1 0 1 0 9], 0 6 5 0 -21], 0 0 0 1 0]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.7584, ~7/4 = 968.8259

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1287/1280

Mapping: [1 0 1 0 9 0], 0 6 5 0 -21 14], 0 0 0 1 0 0]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8446, ~7/4 = 968.8259

## Enki

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4356

Mapping[1 0 1 0 1], 0 6 5 0 4], 0 0 0 2 1]]

mapping generators: ~2, ~6/5, ~66/25

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.0396, ~33/25 = 484.2358

[[1 0 0 0 0, [-18/13 0 6/13 -6/13 12/13, [-2/13 0 5/13 -5/13 10/13, [-2/13 0 -8/13 8/13 10/13, [0 0 0 0 1]
eigenmonzo (unchanged-interval) basis: 2.7/5.11

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624

Mapping: [1 0 1 0 1 0], 0 6 5 0 4 14], 0 0 0 2 1 0]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.1387, ~33/25 = 483.3828

Minimax tuning:

• 13- and 15-odd-limit
[[1 0 0 0 0 0, [2/3 0 -2/3 0 0 2/3, [14/9 0 -5/9 0 0 5/9, [0 0 0 1 0 0, [13/9 0 -4/9 1/2 0 4/9, [14/9 0 -14/9 0 0 14/9]
eigenmonzo (unchanged-interval) basis: 2.7.13/5