Augmented family
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 a major third. The generator can be taken as a fifth or a semitone, and 12 EDO, 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: 128/125
Mapping: [⟨3 0 7], ⟨0 1 0]]
POTE generator: ~3/2 = 706.638
Badness: 0.022315
Seven-limit children
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 ]]
POTE generator: ~3/2 = 696.011
Badness: 0.026459
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]]
POTE generator: ~3/2 = 692.514
Vals: Template:Val list
Badness: 0.020191
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]]
POTE generator: ~3/2 = 688.783
Vals: Template:Val list
Badness: 0.018448
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]]
POTE generator: ~3/2 = 685.356
Vals: Template:Val list
Badness: 0.022236
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 ]]
POTE generator: ~3/2 = 709.257
Badness: 0.024816
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]]
POTE generator: ~3/2 = 711.177
Vals: Template:Val list
Badness: 0.019613
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]]
POTE generator: ~3/2 = 712.013
Vals: Template:Val list
Badness: 0.020785
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]]
POTE generator: ~3/2 = 712.609
Vals: Template:Val list
Badness: 0.022890
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]]
POTE generator: ~3/2 = 709.677
Vals: Template:Val list
Badness: 0.023113
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]]
POTE generator: ~3/2 = 714.150
Vals: Template:Val list
Badness: 0.035554
- Music
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 ]]
POTE generator: ~3/2 = 722.719
Badness: 0.054729
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]]
POTE generator: ~3/2 = 722.663
Vals: Template:Val list
Badness: 0.031171
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 ]]
POTE generator: ~3/2 = 681.629
Badness: 0.059079
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]]
POTE generator: ~3/2 = 680.042
Vals: Template:Val list
Badness: 0.037183
Hexe
Subgroup: 2.3.5.7
Comma list: 50/49, 128/125
Mapping: [⟨6 0 14 17], ⟨0 1 0 0]]
Wedgie: ⟨⟨ 6 0 0 -14 -17 0 ]]
POTE generator: ~3/2 = 710.963
Badness: 0.057730
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]]
POTE generator: ~3/2 = 714.304
Vals: Template:Val list
Badness: 0.038412
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]]
POTE generator: ~3/2 = 710.005
Vals: Template:Val list
Badness: 0.035946
Hemiaug
Subgroup: 2.3.5.7
Comma list: 128/125, 245/243
Mapping: [⟨3 1 7 -1], ⟨0 2 0 5]]
Wedgie: ⟨⟨ 6 0 15 -14 7 35 ]]
POTE generator: ~28/27 = 45.118
Badness: 0.070463
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 128/125, 245/243
Mapping: [⟨3 1 7 -1 1], ⟨0 2 0 5 5]]
POTE generator: ~28/27 = 45.788
Vals: Template:Val list
Badness: 0.038232
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]]
POTE generator: ~28/27 = 46.250
Vals: Template:Val list
Badness: 0.030159
Triforce
Subgroup: 2.3.5.7
Comma list: 49/48, 128/125
Mapping: [⟨3 0 7 6], ⟨0 2 0 1]]
Wedgie: ⟨⟨ 6 0 3 -14 -12 7 ]]
POTE generator: ~7/6 = 247.049
Badness: 0.054988
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 128/125
Mapping: [⟨3 0 7 6 8], ⟨0 2 0 1 1]]
POTE generator: ~7/6 = 247.068
Vals: Template:Val list
Badness: 0.026152
- Musical example
- The Triforce of Courage (tuned in 15edo) by Igliashon Jones
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]]
POTE generator: ~7/6 = 248.313
Vals: Template:Val list
Badness: 0.020248
- Scales
Semitriforce
Subgroup: 2.3.5.7.13
Comma list: 49/48, 56/55, 77/75, 507/500
Mapping: [⟨6 0 14 12 16 27], ⟨0 2 0 1 1 -1]]
POTE generator: ~7/6 = 246.642
Vals: Template:Val list
Badness: 0.059160
Hemiug
Subgroup: 2.3.5.7
Comma list: 128/125, 1323/1250
Mapping: [⟨3 1 7 14], ⟨0 2 0 -3]]
Wedgie: ⟨⟨ 6 0 -9 -14 -31 -21 ]]
POTE generator: ~21/20 = 52.093
Badness: 0.137764
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]]
POTE generator: ~21/20 = 51.655
Vals: Template:Val list
Badness: 0.068062
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]]
POTE generator: ~21/20 = 51.548
Vals: Template:Val list
Badness: 0.042289
Trug
Subgroup: 2.3.5.7
Comma list: 128/125, 360/343
Mapping: [⟨3 1 7 6], ⟨0 3 0 2]]
POTE generator: ~48/35 = 501.980
Badness: 0.138279
Oodako
Subgroup: 2.3.5.7
Comma list: 128/125, 2401/2400
Mapping: [⟨3 3 7 8], ⟨0 4 0 1]]
Wedgie: ⟨⟨ 12 0 3 -28 -29 7 ]]
POTE generator: ~8/7 = 223.354
Badness: 0.113209
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 128/125, 3773/3750
Mapping: [⟨3 3 7 8 10], ⟨0 4 0 1 1]]
POTE generator: ~8/7 = 223.019
Vals: Template:Val list
Badness: 0.059192
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]]
POTE generator: ~8/7 = 223.449
Vals: Template:Val list
Badness: 0.042323
Hemisemiaug
Subgroup: 2.3.5.7
Comma list: 128/125, 12005/11664
Mapping: [⟨6 1 14 4], ⟨0 2 0 3]]
Wedgie: ⟨⟨ 12 0 18 -28 -5 42 ]]
POTE generator: ~15/14 = 144.515
Badness: 0.210984
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]]
POTE generator: ~15/14 = 144.780
Vals: Template:Val list
Badness: 0.080738
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]]
Wedgie: ⟨⟨ 9 0 9 -21 -11 21 ]]
POTE generator: ~3/2 = 707.167
Badness: 0.067157
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]]
POTE generator: ~3/2 = 706.726
Vals: Template:Val list
Badness: 0.034861
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]]
POTE generator: ~3/2 = 706.889
Vals: Template:Val list
Badness: 0.024143