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

• Sad Like Winter Leaves by Igliashon Jones
• Galticeran Sonatina by Joel Taylor

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

Music

• The Triforce of Courage (24edo) by Igliashon Jones
• 2-2-1-2-2-1-2-2-1 mode of 15 edo play by Chris Vaisvil

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

Triforce(9) with L:s = phi

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

Music

• Herman Miller's page about augmented temperament