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
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
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
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
Badness: 0.035946
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)
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)
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)
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
Badness: 0.138279
Ternary
Subgroup: 2.3.5.7
Comma list: 10/9, 16/15
Mapping: [⟨3 5 7 0], ⟨0 0 0 1]]
- mapping generators: ~5/4, ~7
Wedgie: ⟨⟨0 0 3 0 5 7]]
Optimal tuning (POTE): ~5/4 = 1\3, ~7/4 = 1034.013
Optimal ET sequence: 3d, 6, 9bd, 15bbd
Badness: 0.028673
11-limit
Subgroup: 2.3.5.7.11
Comma list: 10/9, 16/15, 22/21
Mapping: [⟨3 5 7 0 2], ⟨0 0 0 1 1]]
Optimal tuning (POTE): ~5/4 = 1\3, ~7/4 = 1033.153
Optimal ET sequence: 3de, 6, 15bbdee
Badness: 0.025592