# Sengic family

(Redirected from Sengic)

The sengic family of rank-3 temperaments tempers out the senga a.k.a. sengic comma, 686/675.

Temperament discussed elsewhere include sensigh (→ Sensamagic family). Considered below are demeter and krypton.

## Sengic

Sengic is naturally a 2.3.5.7.13 subgroup temperament due to the identity 686/675 = (91/90)(196/195) and 91/90 = (169/168)(196/195). This identifies the last generator as 13/12~14/13~15/14. The 7-limit parent was discovered and named in 2005, whereas the extension was noted by Keenan Pepper in 2011[1].

Subgroup: 2.3.5.7

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

mapping generators: ~2, ~3, ~15/14

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 703.7873, ~15/14 = 129.6451

Projection pairs: ~5 = 3375/686, ~7 = 675/98 to 2.3.7/5

### 2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 91/90, 169/168

Sval mapping: [1 0 2 1 2], 0 1 0 1 1], 0 0 3 2 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 704.5918, ~14/13 = 129.7585

## Demeter

Subgroup: 2.3.5.7.11

Comma list: 441/440, 686/675

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.518, ~15/14 = 130.039

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187 to 2.3.11

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 352/351

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.113, ~14/13 = 129.673

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27 to 2.3.11

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 136/135, 154/153, 169/168

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.147, ~14/13 = 129.700

Projection pairs: ~5 = 2725888/531441, ~7 = 15488/2187, ~13 = 352/27, ~17 = 340736/19683 to 2.3.11

## Krypton

Subgroup: 2.3.5.7.11

Comma list: 56/55, 540/539

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.978, ~12/11 = 132.544

Projection pairs: ~5 = 6912/1331, ~7 = 854/121 to 2.3.11

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 706.029, ~14/13 = 132.428