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 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.

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 hemis 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quarter-tones instead of semitones.

August

Commas: 36/35, 128/125

POTE generator: 696.011

Map: [<3 0 7 -1|, <0 1 0 2|]

Wedgie: <<3 0 6 -7 1 14||

EDOs: 9, 12, 45cd, 57cd, 69cd

Badness: 0.0265

11-limit

Commas: 36/35, 45/44, 56/55

POTE generator: ~3/2 = 692.514

Map: [<3 0 7 -1 1|, <0 1 0 2 2|]

EDOs: 9, 12, 21, 33e, 45cde

Badness: 0.0202

13-limit

Commas: 27/26, 36/35, 45/44, 56/55

POTE generator: ~3/2 = 688.783

Map: [<3 0 7 -1 1 -3|, <0 1 0 2 2 3|]

EDOs: 9, 12, 21, 33ef, 54bcef

Badness: 0.0184

Augustus

Commas: 26/25, 36/35, 45/44, 56/55

POTE generator: ~3/2 = 685.356

Map: [<3 0 7 -1 1 11|, <0 1 0 2 2 0|]

EDOs: 9, 12, 21f

Badness: 0.0222

Augene

Commas: 64/63, 126/125

POTE generator: 709.257

Map: [<3 0 7 18|, <0 1 0 -2|]

Wedgie: <<3 0 -6 -7 -18 -14||

EDOs: 12, 27, 39d, 66cd

Badness: 0.0248

11-limit

Commas: 56/55, 64/63, 100/99

POTE generator: ~3/2 = 711.177

Map: [<3 0 7 18 20|, <0 1 0 -2 -2|]

EDOs: 12, 15, 27e

Badness: 0.0196

13-limit

Commas: 40/39, 56/55, 64/63, 66/65

POTE generator: ~3/2 = 712.013

Map: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|]

EDOs: 12f, 15, 27ef

Badness: 0.0208

Ogene

Commas: 56/55, 64/63, 91/90, 100/99

POTE generator: ~3/2 = 712.609

Map: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|]

EDOs: 12, 15, 27e, 69bcef

Badness: 0.0229

Agene

Commas: 56/55, 64/63, 78/77, 100/99

POTE generator: ~3/2 = 709.677

Map: [<3 0 7 18 20 35|, <0 1 0 -2 -2 -5|]

EDOs: 12f, 27e

Badness: 0.0231

Eugene

Commas: 55/54, 64/63, 77/75

POTE generator: ~3/2 = 714.150

Map: [<3 0 7 18 -4|, <0 1 0 -2 3|]

EDOs: 15, 27, 42

Badness: 0.0356

• Sad Like Winter Leaves by Igliashon Jones

• Galticeran Sonatina by Joel Taylor

Inflated

Commas: 28/27, 128/125

POTE generator: ~3/2 = 722.719

Map: [<3 0 7 -6|, <0 1 0 3|]

Wedgie: <<3 0 9 -7 6 21||

EDOs: 15, 48bc, 63bc, 78bc

Badness: 0.0547

11-limit

Commas: 28/27, 55/54, 128/125

POTE generator: ~3/2 = 722.663

Map: [<3 0 7 -6|, <0 1 0 3|]

EDOs: 15, 48bce, 63bce, 78bce

Badness: 0.0312

Deflated

Commas: 21/20, 128/125

POTE generator: ~3/2 = 681.629

Map: [<3 0 7 13|, <0 1 0 -1|]

Wedgie: <<3 0 -3 -7 -13 -7||

EDOs: 3, 9, 18bd, 21d, 30bd

Badness: 0.0591

Hexe

Commas: 50/49, 128/125

POTE generator: 710.963

Map: [<6 0 14 17|, <0 1 0 0|]

Wedgie: <<6 0 0 -14 -17 0||

EDOs: 6, 12, 30d, [[[42edo|42d]], 54cd

Badness: 0.0577

11-limit

Commas: 50/49, 56/55, 125/121

POTE generator: ~3/2 = 714.304

Map: [<6 0 14 17 21|, <0 1 0 0 0|]

EDOs: 6, 12, 30de

Badness: 0.0384

13-limit

Commas: 50/49, 56/55, 66/65, 105/104

POTE generator: ~3/2 = 710.005

Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]

EDOs: 12f

Badness: 0.0359

Hemiaug

Commas: 128/125, 245/243

POTE generator: ~28/27 = 45.118

Map: [<3 1 7 -1|, <0 2 0 5|]

Wedgie: <<6 0 15 -14 7 35||

EDOs: 24, 27, 105bc

Badness: 0.0705

11-limit

Commas: 56/55, 128/125, 245/243

POTE generator: ~28/27 = 45.788

Map: [<3 1 7 -1 1|, <0 2 0 5 5|]

EDOs: 24, 27e, 51ce, 78ce

Badness: 0.0382

13-limit

Commas: 56/55, 91/90, 128/125, 245/243

POTE generator: ~28/27 = 46.250

Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]

EDOs: 24, 27e, 51ce, 78ce

Badness: 0.0302

Triforce

Commas: 49/48, 128/125

POTE generator: ~7/6 = 247.049

Map: [<3 0 7 6|, <0 2 0 1|]

Wedgie: <<6 0 3 -14 -12 7||

EDOs: 6, 9, 15, 24, 39, 63cd, 102cd

Badness: 0.0202

triforce9.jpg

11-limit

Commas: 56/55, 77/75, 128/125

POTE generator: ~7/6 = 247.068

Map: [<3 0 7 6 8|, <0 2 0 1 1|]

EDOs: 6, 9, 15, 24, 39, 63cd, 102cd

Badness: 0.0262

Musical example

• The Triforce of Courage (tuned in 15edo) by Igliashon Jones

13-limit

Commas: 49/48, 56/55, 66/65, 77/75

POTE generator: ~7/6 = 248.313

Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]

EDOs: 6, 9, 15, 24, 63cdf, 87cdf

Badness: 0.0202

Triforce(9) with L:s = phi

Semitriforce

Commas: 49/48, 56/55, 77/75, 507/500

POTE generator: ~7/6 = 246.642

Map: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]

EDOs: 6, 24, 54cd, 78cd, 102cdf

Badness: 0.0592

Hemiug

Commas: 128/125, 1323/1250

POTE generator: ~21/20 = 52.093

Map: [<3 1 7 14|, <0 2 0 -3|]

Wedgie: <<6 0 -9 -14 -31 -21||

EDOs: 21, 24, 45c, 69cd

Badness: 0.1378

11-limit

Commas: 56/55, 128/125, 1323/1250

POTE generator: ~21/20 = 51.655

Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]

EDOs: 21, 24, 69cd, 93cd

Badness: 0.0681

13-limit

Commas: 56/55, 66/65, 105/104, 507/500

POTE generator: ~21/20 = 51.548

Map: [<3 1 7 14 16 13|, <0 2 0 -3 -3 -1|]

EDOs: 21, 24, 69cdf, 93cdf

Badness: 0.0423

Trug

Commas: 128/125, 360/343

POTE generator: ~48/35 = 501.980

Map: [<3 1 7 6|, <0 3 0 2|]

EDOs: 9bd, 12

Badness: 0.1383

Oodako

Commas: 128/125, 2401/2400

POTE generator: ~8/7 = 223.3540873874507317

Map: [<3 3 7 8|, <0 4 0 1|]

Wedgie: <<12 0 3 -28 -29 7||

EDOs: 6, 21, 27, 75c, 102cd, 129bcd

Badness: 0.1132

11-limit

Commas: 56/55, 128/125, 3773/3750

POTE generator: ~8/7 = 223.019

Map: [<3 3 7 8 10|, <0 4 0 1 1|]

EDOs: 6, 21, 27e

Badness: 0.0592

13-limit

Commas: 56/55, 78/77, 128/125, 507/500

POTE generator: ~8/7 = 223.449

Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]

EDOs: 6, 21, 27e

Badness: 0.0423

Hemisemiaug

Commas: 128/125, 12005/11664

POTE generator: ~15/14 = 144.515

Map: [<6 1 14 4|, <0 2 0 3|]

Wedgie: <<12 0 18 -28 -5 42||

Edos: 18, 24, 42, 66c, 108bc

Badness: 0.2110

11-limit

Commas: 56/55, 128/125, 3773/3645

POTE generator: ~15/14 = 144.780

Map: [<6 1 14 4 8|, <0 2 0 3 3|]

EDOs: 24, 42e, 66ce, 108bce

Badness: 0.0807

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.

Commas: 128/125, 686/675

POTE generator: ~3/2 = 707.167

Map: [<9 0 21 11|, <0 1 0 1|]

Wedgie: <<9 0 9 -21 -11 21||

EDOs: 9, 18, 27, 63c, 90c

Badness: 0.0672

11-limit

Commas: 56/55, 128/125, 540/539

POTE generator: ~3/2 = 706.726

Map: [<9 0 21 11 17|, <0 1 0 1 1|]

EDOs: 9, 27e, 36, 63ce

Badness: 0.0349

13-limit

Commas: 56/55, 78/77, 91/90, 128/125

POTE generator: ~3/2 = 706.889

Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|]

EDOs: 9, 27e, 36, 63ce

Badness: 0.0241

Music

• Herman Miller's page about augmented temperament