# Augmented family

(Redirected from Niner)

The 5-limit parent comma for the augmented family is 128/125, the diesis. Its monzo is [7 0 -3, and flipping that yields ⟨⟨3 0 -7]] for the wedgie. Hence the period is 1/3 octave, and this is what is used for 5/4, the classical major third. The generator can be taken as a fifth or a semitone, and 12edo, with its excellent fifth, is an obvious tuning for 5-limit augmented, though a sharper fifth might be preferred to go with the sharp third.

## Augmented

Subgroup: 2.3.5

Comma list: 128/125

Mapping[3 0 7], 0 1 0]]

mapping generators: ~5/4, ~3
• CTE: ~5/4 = 1\3, ~3/2 = 701.9550
• POTE: ~5/4 = 1\3, ~3/2 = 706.638

### Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.

## August

Subgroup: 2.3.5.7

Comma list: 36/35, 128/125

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

Wedgie⟨⟨3 0 6 -7 1 14]]

• CTE: ~5/4 = 1\3, ~3/2 = 692.1237
• POTE: ~5/4 = 1\3, ~3/2 = 696.011

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 56/55

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

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 687.6853
• POTE: ~5/4 = 1\3, ~3/2 = 692.514

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 56/55

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

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 685.0836
• POTE: ~5/4 = 1\3, ~3/2 = 688.783

#### Augustus

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 45/44, 56/55

Mapping: [3 0 7 -1 1 11], 0 1 0 2 2 0]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 687.6853
• POTE: ~5/4 = 1\3, ~3/2 = 685.356

## Augene

Subgroup: 2.3.5.7

Comma list: 64/63, 126/125

Mapping[3 0 7 18], 0 1 0 -2]]

Wedgie⟨⟨3 0 -6 -7 -18 -14]]

• CTE: ~5/4 = 1\3, ~3/2 = 709.5948
• POTE: ~5/4 = 1\3, ~3/2 = 709.257

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 100/99

Mapping: [3 0 7 18 20], 0 1 0 -2 -2]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 713.5701
• POTE: ~5/4 = 1\3, ~3/2 = 711.177

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 64/63, 66/65

Mapping: [3 0 7 18 20 16], 0 1 0 -2 -2 -1]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 716.1234
• POTE: ~5/4 = 1\3, ~3/2 = 712.013

#### Ogene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 91/90, 100/99

Mapping: [3 0 7 18 20 -8], 0 1 0 -2 -2 4]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 711.9020
• POTE: ~5/4 = 1\3, ~3/2 = 712.609

#### Agene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 100/99

Mapping: [3 0 7 18 20 35], 0 1 0 -2 -2 -5]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 712.5722
• POTE: ~5/4 = 1\3, ~3/2 = 709.677

### Eugene

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 77/75

Mapping: [3 0 7 18 -4], 0 1 0 -2 3]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 713.0025
• POTE: ~5/4 = 1\3, ~3/2 = 714.150

## Inflated

Subgroup: 2.3.5.7

Comma list: 28/27, 128/125

Mapping[3 0 7 -6], 0 1 0 3]]

Wedgie⟨⟨3 0 9 -7 6 21]]

• CTE: ~5/4 = 1\3, ~3/2 = 717.5172
• POTE: ~5/4 = 1\3, ~3/2 = 722.719

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 55/54, 128/125

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

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 717.5172
• POTE: ~5/4 = 1\3, ~3/2 = 722.663

## Deflated

Subgroup: 2.3.5.7

Comma list: 21/20, 128/125

Mapping[3 0 7 13], 0 1 0 -1]]

Wedgie⟨⟨3 0 -3 -7 -13 -7]]

• CTE: ~5/4 = 1\3, ~3/2 = 684.8473
• POTE: ~5/4 = 1\3, ~3/2 = 681.629

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 33/32, 128/125

Mapping: [3 0 7 13 15], 0 1 0 -1 -1]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~3/2 = 679.881
• POTE: ~5/4 = 1\3, ~3/2 = 680.042

## Hexe

Subgroup: 2.3.5.7

Comma list: 50/49, 128/125

Mapping[6 0 14 17], 0 1 0 0]]

mapping generators: ~28/25, ~3

Wedgie⟨⟨6 0 0 -14 -17 0]]

• CTE: ~28/25 = 1\6, ~3/2 = 701.9550
• POTE: ~28/25 = 1\6, ~3/2 = 710.963

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 125/121

Mapping: [6 0 14 17 21], 0 1 0 0 0]]

Optimal tunings:

• CTE: ~28/25 = 1\6, ~3/2 = 701.9550
• POTE: ~28/25 = 1\6, ~3/2 = 714.304

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 56/55, 66/65, 105/104

Mapping: [6 0 14 17 21 13], 0 1 0 0 0 1]]

Optimal tunings:

• CTE: ~28/25 = 1\6, ~3/2 = 692.4327
• POTE: ~28/25 = 1\6, ~3/2 = 710.005

## Triforce

Lattice of triforce

Subgroup: 2.3.5.7

Comma list: 49/48, 128/125

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

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

Wedgie⟨⟨6 0 3 -14 -12 7]]

• CTE: ~5/4 = 1\3, ~7/4 = 952.2948
• POTE: ~5/4 = 1\3, ~7/4 = 952.951

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 77/75

Mapping: [3 0 7 6 8], 0 2 0 1 1]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~7/4 = 952.2495
• POTE: ~5/4 = 1\3, ~7/4 = 952.932

Music

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 66/65, 77/75

Mapping: [3 0 7 6 8 4], 0 2 0 1 1 3]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~7/4 = 950.8049
• POTE: ~5/4 = 1\3, ~7/4 = 951.687

Scales
• triphi, triforce[9] with L:s = phi

#### Semitriforce

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 77/75, 507/500

Mapping: [6 0 14 12 16 27], 0 2 0 1 1 -1]]

mapping generators: ~44/39, ~7/4

Optimal tunings:

• CTE: ~44/39 = 1\6, ~7/4 = 952.5307
• POTE: ~44/39 = 1\6, ~7/4 = 953.358

## Hemiaug

Subgroup: 2.3.5.7

Comma list: 128/125, 245/243

Mapping[3 1 7 -1], 0 2 0 5]]

mapping generators: ~5/4, ~14/9

Wedgie⟨⟨6 0 15 -14 7 35]]

• CTE: ~5/4 = 1\3, ~14/9 = 752.8335 (~36/35 = 47.1665)
• POTE: ~5/4 = 1\3, ~14/9 = 754.882 (~36/35 = 45.118)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 243/242

Mapping: [3 1 7 -1 1], 0 2 0 5 5]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~14/9 = 752.0512 (~36/35 = 47.9488)
• POTE: ~5/4 = 1\3, ~14/9 = 754.212 (~36/35 = 45.788)

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 128/125, 245/243

Mapping: [3 1 7 -1 1 13], 0 2 0 5 5 -1]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~14/9 = 752.1284 (~36/35 = 47.8716)
• POTE: ~5/4 = 1\3, ~14/9 = 753.750 (~36/35 = 46.250)

## Hemiug

Subgroup: 2.3.5.7

Comma list: 128/125, 1323/1250

Mapping[3 1 7 14], 0 2 0 -3]]

mapping generators: ~5/4, ~32/21

Wedgie⟨⟨6 0 -9 -14 -31 -21]]

• CTE: ~5/4 = 1\3, ~32/21 = 747.9484 (~21/20 = 52.0516)
• POTE: ~5/4 = 1\3, ~32/21 = 747.907 (~21/20 = 52.093)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 1323/1250

Mapping: [3 1 7 14 16], 0 2 0 -3 -3]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~32/21 = 748.2962 (~33/32 = 51.7038)
• POTE: ~5/4 = 1\3, ~32/21 = 748.345 (~33/32 = 51.655)

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 105/104, 507/500

Mapping: [3 1 7 14 16 13], 0 2 0 -3 -3 -1]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~32/21 = 748.5255 (~33/32 = 51.4745)
• POTE: ~5/4 = 1\3, ~32/21 = 748.452 (~33/32 = 51.548)

## Oodako

Subgroup: 2.3.5.7

Comma list: 128/125, 2401/2400

Mapping[3 3 7 8], 0 4 0 1]]

mapping generators: ~5/4, ~28/25

Wedgie⟨⟨12 0 3 -28 -29 7]]

• CTE: ~5/4 = 1\3, ~28/25 = 175.3586
• POTE: ~5/4 = 1\3, ~28/25 = 176.646

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 2401/2400

Mapping: [3 3 7 8 10], 0 4 0 1 1]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~11/10 = 175.0533
• POTE: ~5/4 = 1\3, ~11/10 = 176.981

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 128/125, 507/500

Mapping: [3 3 7 8 10 12], 0 4 0 1 1 -2]]

Optimal tunings:

• CTE: ~5/4 = 1\3, ~11/10 = 175.2524
• POTE: ~5/4 = 1\3, ~11/10 = 176.551

## Hemisemiaug

Subgroup: 2.3.5.7

Comma list: 128/125, 12005/11664

Mapping[6 1 14 4], 0 2 0 3]]

mapping generators: ~54/49, ~45/28

Wedgie⟨⟨12 0 18 -28 -5 42]]

• CTE: ~54/49 = 1\6, ~45/28 = 853.1901 (~36/35 = 53.1901)
• POTE: ~54/49 = 1\6, ~45/28 = 855.485 (~36/35 = 55.485)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 3773/3645

Mapping: [6 1 14 4 8], 0 2 0 3 3]]

Optimal tunings:

• CTE: ~54/49 = 1\6, ~18/11 = 852.5968 (~36/35 = 52.5968)
• POTE: ~54/49 = 1\6, ~18/11 = 855.220 (~36/35 = 55.220)

## Niner

Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.

Subgroup: 2.3.5.7

Comma list: 128/125, 686/675

Mapping[9 0 21 11], 0 1 0 1]]

mapping generators: ~49/45, ~3

Wedgie⟨⟨9 0 9 -21 -11 21]]

• CTE: ~49/45 = 1\9, ~3/2 = 702.0044
• POTE: ~49/45 = 1\9, ~3/2 = 707.167

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 540/539

Mapping: [9 0 21 11 17], 0 1 0 1 1]]

Optimal tunings:

• CTE: ~12/11 = 1\9, ~3/2 = 699.6216
• POTE: ~12/11 = 1\9, ~3/2 = 706.726

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90, 128/125

Mapping: [9 0 21 11 17 19], 0 1 0 1 1 1]]

Optimal tunings:

• CTE: ~14/13 = 1\9, ~3/2 = 700.4330
• POTE: ~14/13 = 1\9, ~3/2 = 706.889

## Trug

Subgroup: 2.3.5.7

Comma list: 128/125, 360/343

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

mapping generators: ~5/4, ~48/35
• CTE: ~5/4 = 1\3, ~48/35 = 498.6367
• POTE: ~5/4 = 1\3, ~48/35 = 501.980

## Ternary

Subgroup: 2.3.5.7

Comma list: 10/9, 16/15

Mapping[3 5 7 0], 0 0 0 1]]

mapping generators: ~5/4, ~7

Wedgie⟨⟨0 0 3 0 5 7]]

• POTE: ~5/4 = 1\3, ~7/4 = 1034.013

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 10/9, 16/15, 22/21

Mapping: [3 5 7 0 2], 0 0 0 1 1]]

Optimal tunings:

• POTE: ~5/4 = 1\3, ~7/4 = 1033.153