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 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

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 701.9550

Optimal ET sequence: 3, 12, 27, 39, 51c, 90cc

Badness: 0.022315

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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 692.1237

Optimal ET sequence: 9, 12, 45cd, 57cd, 69cdd

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 687.6853

Optimal ET sequence: 9, 12, 21, 33e, 45cde

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 685.0836

Optimal ET sequence: 9, 12f, 21, 33ef, 54bceeff

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 687.6853

Optimal ET sequence: 9, 12, 21f

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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 709.5948

Optimal ET sequence: 12, 27, 39d, 66cd

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 713.5701

Optimal ET sequence: 12, 15, 27e

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 716.1234

Optimal ET sequence: 12f, 15, 27eff

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 711.9020

Optimal ET sequence: 12, 15, 27e, 69bceef

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 712.5722

Optimal ET sequence: 12f, 27e

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 713.0025

Optimal ET sequence: 12e, 15, 27, 42

Badness: 0.035554

Music

• Sad Like Winter Leaves^{[dead link]} play^{[dead link]} by Igliashon Jones
• Galticeran Sonatina^{[dead link]} play^{[dead link]} 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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 717.5172

Optimal ET sequence: 3d, 12d, 15

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 717.5172

Optimal ET sequence: 3de, 12de, 15

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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 684.8473

Optimal ET sequence: 3, 9

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~3/2 = 679.881

Optimal ET sequence: 3, 9

Badness: 0.037183

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 ]]

Optimal tuning (CTE): ~28/25 = 1\6, ~3/2 = 701.9550

Optimal ET sequence: 6, 12, 30d, 42dd, 54cdd

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]]

Optimal tuning (CTE): ~28/25 = 1\6, ~3/2 = 701.9550

Optimal ET sequence: 6, 12, 30dee, 42ddeee

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]]

Optimal tuning (CTE): ~28/25 = 1\6, ~3/2 = 692.4327

Optimal ET sequence: 6f, 12f

Badness: 0.035946

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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~7/4 = 952.2948

Optimal ET sequence: 6, 9, 15, 24, 39, 63cd, 102ccdd

Badness: 0.054988

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 tuning (CTE): ~5/4 = 1\3, ~7/4 = 952.2495

Optimal ET sequence: 6, 9, 15, 24, 39

Badness: 0.026152

Music

• The Triforce of Courage (24edo) by Igliashon Jones (2018)
• 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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~7/4 = 950.8049

Optimal ET sequence: 6, 9, 15, 24

Badness: 0.020248

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 tuning (CTE): ~44/39 = 1\6, ~7/4 = 952.5307

Optimal ET sequence: 6, 18bd, 24

Badness: 0.059160

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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~14/9 = 752.8335 (~36/35 = 47.1665)

Optimal ET sequence: 24, 27

Badness: 0.070463

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 tuning (CTE): ~5/4 = 1\3, ~14/9 = 752.0512 (~36/35 = 47.9488)

Optimal ET sequence: 24, 27e, 51ce

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~14/9 = 752.1284 (or ~36/35 = 47.8716)

Optimal ET sequence: 24, 27e, 51ce

Badness: 0.030159

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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~32/21 = 747.9484 (~21/20 = 52.0516)

Optimal ET sequence: 21, 24, 45c

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~32/21 = 748.2962 (~33/32 = 51.7038)

Optimal ET sequence: 21, 24

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~32/21 = 748.5255 (~33/32 = 51.4745)

Optimal ET sequence: 21, 24

Badness: 0.042289

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 ]]

Optimal tuning (CTE): ~5/4 = 1\3, ~28/25 = 175.3586

Optimal ET sequence: 6, 21, 27, 75c, 102ccd, 129bccd

Badness: 0.113209

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 tuning (CTE): ~5/4 = 1\3, ~11/10 = 175.0533

Optimal ET sequence: 6, 21, 27e

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]]

Optimal tuning (CTE): ~5/4 = 1\3, ~11/10 = 175.2524

Optimal ET sequence: 6, 21, 27e

Badness: 0.042323

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 ]]

Optimal tuning (CTE): ~54/49 = 1\6, ~45/28 = 853.1901 (~36/35 = 53.1901)

Optimal ET sequence: 18, 24, 42

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]]

Optimal tuning (CTE): ~54/49 = 1\6, ~18/11 = 852.5968 (~36/35 = 52.5968)

Optimal ET sequence: 24, 42e, 66ce, 108bccee

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]]

Mapping generators: ~49/45, ~3

Wedgie: ⟨⟨ 9 0 9 -21 -11 21 ]]

Optimal tuning (CTE): ~49/45 = 1\9, ~3/2 = 702.0044

Optimal ET sequence: 9, 18, 27, 63c, 90cc

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]]

Optimal tuning (CTE): ~12/11 = 1\9, ~3/2 = 699.6216

Optimal ET sequence: 9, 18e, 27e, 63cee

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]]

Optimal tuning (CTE): ~14/13 = 1\9, ~3/2 = 700.4330

Optimal ET sequence: 9, 18e, 27e

Badness: 0.024143

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

Optimal tuning (CTE): ~5/4 = 1\3, ~48/35 = 498.6367

Optimal ET sequence: 9bd, 12

Badness: 0.138279

External links

• Herman Miller's page about augmented temperament