Augmented family

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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

POTE generator: ~3/2 = 706.638

Optimal GPV sequence3, 12, 27, 39, 51c, 90cc

Badness: 0.022315

Seven-limit 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]]

POTE generator: ~3/2 = 696.011

Optimal GPV sequence9, 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]]

POTE generator: ~3/2 = 692.514

Optimal GPV 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]]

POTE generator: ~3/2 = 688.783

Optimal GPV 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]]

POTE generator: ~3/2 = 685.356

Optimal GPV 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]]

POTE generator: ~3/2 = 709.257

Optimal GPV sequence12, 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]]

POTE generator: ~3/2 = 711.177

Optimal GPV 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]]

POTE generator: ~3/2 = 712.013

Optimal GPV 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]]

POTE generator: ~3/2 = 712.609

Optimal GPV 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]]

POTE generator: ~3/2 = 709.677

Optimal GPV 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]]

POTE generator: ~3/2 = 714.150

Optimal GPV sequence: 12e, 15, 27, 42

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

Optimal GPV sequence3d, 15, 48bc, 63bbc, 78bbc

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

Optimal GPV sequence: 3de, 15, 48bce, 63bbcee, 78bbcee

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

Optimal GPV sequence3, 9, 21d, 30bdd

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

Optimal GPV sequence: 3, 9, 21de, 30bdde

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]]

POTE generator: ~3/2 = 710.963

Optimal GPV sequence6, 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]]

POTE generator: ~3/2 = 714.304

Optimal GPV sequence: 6, 12, 30de

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

Optimal GPV sequence: 6f, 12f

Badness: 0.035946

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]]

POTE generator: ~14/9 = 754.882 (or ~36/35 = 45.118)

Optimal GPV sequence24, 27, 105bcc

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: ~14/9 = 754.212 (or ~36/35 = 45.788)

Optimal GPV sequence: 24, 27e, 51ce, 78cee

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: ~14/9 = 753.750 (or ~36/35 = 46.250)

Optimal GPV sequence: 24, 27e, 51ce, 78cee

Badness: 0.030159

Triforce

triforce9.jpg
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]]

POTE generator: ~7/4 = 952.951

Optimal GPV sequence6, 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]]

POTE generator: ~7/4 = 952.932

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

Badness: 0.026152

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]]

POTE generator: ~7/4 = 951.687

Optimal GPV sequence: 6, 9, 15, 24, 63cdf, 87cddff

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

POTE generator: ~7/4 = 953.358

Optimal GPV sequence: 6, 24, 54cd, 78cd, 102ccddf

Badness: 0.059160

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]]

POTE generator: ~32/21 = 747.907 (or ~21/20 = 52.093)

Optimal GPV sequence21, 24, 45c, 69cd

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: ~32/21 = 748.345 (or ~33/32 = 51.655)

Optimal GPV sequence: 21, 24, 69cd, 93ccdd

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: ~32/21 = 748.452 (or ~21/20 = 51.548)

Optimal GPV sequence: 21, 24, 69cdf, 93ccddf

Badness: 0.042289

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

POTE generator: ~48/35 = 501.980

Optimal GPV sequence9bd, 12

Badness: 0.138279

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]]

POTE generator: ~28/25 = 176.646

Optimal GPV sequence6, 21, 27, 75c, 102ccd, 129bccd

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: ~28/25 = 176.981

Optimal GPV 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]]

POTE generator: ~28/25 = 176.551

Optimal GPV 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]]

POTE generator: ~45/28 = 855.485 (or ~36/35 = 55.485)

Optimal GPV sequence18, 24, 42, 66c, 108bcc

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: ~18/11 = 855.220 (or ~36/35 = 55.220)

Optimal GPV 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]]

POTE generator: ~3/2 = 707.167

Optimal GPV sequence9, 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]]

POTE generator: ~3/2 = 706.726

Optimal GPV sequence: 9, 27e, 36, 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]]

POTE generator: ~3/2 = 706.889

Optimal GPV sequence: 9, 27e, 36, 63cee

Badness: 0.024143

Music