# Dimipent family

The dimipent family tempers out the major diesis aka diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as 12edo.

## Dimipent

Subgroup: 2.3.5

Comma list: 648/625

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

• CTE: ~6/5 = 1\4, ~3/2 = 696.9833 (~25/24 = 96.9833)
• POTE: ~6/5 = 1\4, ~3/2 = 699.507 (~25/24 = 99.507)

## Diminished

Deutsch

Subgroup: 2.3.5.7

Comma list: 36/35, 50/49

Mapping[4 0 3 5], 0 1 1 1]]

Wedgie⟨⟨4 4 4 -3 -5 -2]]

• CTE: ~6/5 = 1\4, ~3/2 = 691.9545 (~21/20 = 91.9545)
• POTE: ~6/5 = 1\4, ~3/2 = 699.523 (~21/20 = 99.523)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 56/55

Mapping: [4 0 3 5 14], 0 1 1 1 0]]

Wedgie⟨⟨4 4 4 0 -3 -5 -14 -2 -14 -14]]

Optimal tunings:

• CTE: ~6/5 = 1\4, ~3/2 = 691.9546 (~21/20 = 91.9546)
• POTE: ~6/5 = 1\4, ~3/2 = 709.109 (~15/14 = 109.109)

Optimal ET sequence: 4, 8d, 12, 32cddee, 44cddeee

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 40/39, 50/49, 66/65

Mapping: [4 0 3 5 14 15], 0 1 1 1 0 0]]

Optimal tunings:

• CTE: ~6/5 = 1\4, ~3/2 = 691.9544 (~21/20 = 91.9544)
• POTE: ~6/5 = 1\4, ~3/2 = 713.773 (~15/14 = 113.773)

Optimal ET sequence: 4, 8d, 12f, 20cdef

### Demolished

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 50/49

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

Optimal tunings:

• CTE: ~6/5 = 1\4, ~3/2 = 687.7473 (~21/20 = 87.7473)
• POTE: ~6/5 = 1\4, ~3/2 = 689.881 (~21/20 = 89.881)

Optimal ET sequence: 12, 28, 40de

### Cohedim

This temperament has been documented in Graham Breed's temperament finder as hemidim, the same name as 11-limit 4e & 24 and 13-limit 4ef & 24. For 11-limit 8bce & 12 temperament, cohedim arguably makes more sense.

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 125/121

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

mapping generators: ~6/5, ~11/7

Optimal tunings:

• CTE: ~6/5 = 1\4, ~11/7 = 793.4233 (~12/11 = 106.5767)
• POTE: ~6/5 = 1\4, ~11/7 = 798.4212 (~12/11 = 101.679)

Optimal ET sequence: 8bce, 12

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 50/49, 66/65, 125/121

Mapping: [4 1 4 6 6 7], 0 2 2 2 3 3]]

Optimal tunings:

• CTE: ~6/5 = 1\4, ~11/7 = 791.3751 (~12/11 = 108.6249)
• POTE: ~6/5 = 1\4, ~11/7 = 797.701 (~12/11 = 102.299)

Optimal ET sequence: 8bcef, 12f

## Hemidim

Subgroup: 2.3.5.7

Comma list: 49/48, 648/625

Mapping[4 0 3 8], 0 2 2 1]]

mapping generators: ~6/5, ~7/4

Wedgie⟨⟨8 8 4 -6 -16 -13]]

• CTE: ~6/5 = 1\4, ~7/4 = 948.2575 (~36/35 = 48.2575)
• POTE: ~6/5 = 1\4, ~7/4 = 947.445 (~36/35 = 47.445)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 243/242

Mapping: [4 0 3 8 -2], 0 2 2 1 5]]

Optimal tunings:

• CTE: ~6/5 = 1\4, ~7/4 = 949.8722 (~36/35 = 49.8722)
• POTE: ~6/5 = 1\4, ~7/4 = 948.342 (~36/35 = 48.342)

Optimal ET sequence: 4e, 20ce, 24

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 243/242

Mapping: [4 0 3 8 -2 -1], 0 2 2 1 5 5]]

Optimal tunings:

• CTE: ~6/5 = 1\4, ~7/4 = 949.3664 (~36/35 = 49.3664)
• POTE: ~6/5 = 1\4, ~7/4 = 947.775 (~36/35 = 47.775)

Optimal ET sequence: 4ef, 24

## Semidim

Subgroup: 2.3.5.7

Comma list: 245/243, 392/375

Mapping[8 0 6 -3], 0 1 1 2]]

mapping generators: ~15/14, ~3

Wedgie⟨⟨8 8 16 -6 3 15]]

• CTE: ~15/14 = 1\8, ~3/2 = 702.7651 (~36/35 = 47.2349)
• POTE: ~15/14 = 1\8, ~3/2 = 707.014 (~36/35 = 42.986)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 245/243

Mapping: [8 0 6 -3 15], 0 1 1 2 1]]

Optimal tunings:

• CTE: ~12/11 = 1\8, ~3/2 = 702.6622 (~36/35 = 47.3378)
• POTE: ~12/11 = 1\8, ~3/2 = 706.645 (~36/35 = 43.355)

Optimal ET sequence: 8d, 24, 32c

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 507/500

Mapping: [8 0 6 -3 15 17], 0 1 1 2 1 1]]

Optimal tunings:

• CTE: ~12/11 = 1\8, ~3/2 = 701.9520 (~36/35 = 48.0480)
• POTE: ~12/11 = 1\8, ~3/2 = 707.376 (~36/35 = 42.624)

Optimal ET sequence: 8d, 24, 32cf