Augmented family

The 5-limit parent comma for the augmented family is 128/125, the diesis. 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 701.955 (~16/15 = 98.045)

error map: ⟨0.000 0.000 +13.686]

• POTE: ~5/4 = 400.000, ~3/2 = 706.638 (~16/15 = 93.362)

error map: ⟨0.000 +4.683 +13.686]

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

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 692.124 (~16/15 = 107.876)

error map: ⟨0.000 -9.831 +13.686 +15.422]

• POTE: ~5/4 = 400.000, ~3/2 = 696.011 (~16/15 = 103.989)

error map: ⟨0.000 -5.944 +13.686 +23.195]

Optimal ET sequence: 9, 12, 45cd

Badness (Smith): 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]]

Wedgie: ⟨⟨ 3 0 6 6 -7 1 -1 14 14 -4 ]]

Optimal tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 687.685 (~16/15 = 112.315)
• POTE: ~5/4 = 400.000, ~3/2 = 692.514 (~16/15 = 107.486)

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

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 685.084 (~16/15 = 114.916)
• POTE: ~5/4 = 400.000, ~3/2 = 688.783 (~16/15 = 111.217)

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

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 687.685 (~16/15 = 112.315)
• POTE: ~5/4 = 400.000, ~3/2 = 685.356 (~16/15 = 114.644)

Optimal ET sequence: 9, 12, 21f

Badness (Smith): 0.022236

Augene

Main article: 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 709.595 (~21/20 = 90.405)

error map: ⟨0.000 +7.640 +13.686 +11.984]

• POTE: ~5/4 = 400.000, ~3/2 = 709.257 (~21/20 = 90.743)

error map: ⟨0.000 +7.302 +13.686 +12.660]

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

Badness (Smith): 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]]

Wedgie: ⟨⟨ 3 0 -6 -6 -7 -18 -20 -14 -14 4 ]]

Optimal tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 713.570 (~21/20 = 86.430)
• POTE: ~5/4 = 400.000, ~3/2 = 711.177 (~21/20 = 88.823)

Optimal ET sequence: 12, 15, 27e

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 716.123 (~21/20 = 83.877)
• POTE: ~5/4 = 400.000, ~3/2 = 712.013 (~21/20 = 87.987)

Optimal ET sequence: 12f, 15, 27eff

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 711.902 (~21/20 = 88.098)
• POTE: ~5/4 = 400.000, ~3/2 = 712.609 (~21/20 = 87.391)

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

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 712.572 (~21/20 = 87.428)
• POTE: ~5/4 = 400.000, ~3/2 = 709.677 (~21/20 = 90.323)

Optimal ET sequence: 12f, 27e

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 713.002 (~21/20 = 86.998)
• POTE: ~5/4 = 400.000, ~3/2 = 714.150 (~21/20 = 85.850)

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

Badness (Smith): 0.035554

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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 717.517 (~25/24 = 82.483)

error map: ⟨0.000 +15.562 +13.686 -16.274]

• POTE: ~5/4 = 400.000, ~3/2 = 722.719 (~25/24 = 77.281)

error map: ⟨0.000 +20.764 +13.686 -0.668]

Optimal ET sequence: 3d, 12d, 15

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 717.382 (~25/24 = 82.618)
• POTE: ~5/4 = 400.000, ~3/2 = 722.663 (~25/24 = 77.337)

Optimal ET sequence: 3de, 12de, 15

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 684.847 (~16/15 = 115.153)

error map: ⟨0.000 -17.108 +13.686 -53.673]

• POTE: ~5/4 = 400.000, ~3/2 = 681.629 (~16/15 = 118.371)

error map: ⟨0.000 -20.326 +13.686 -50.455]

Optimal ET sequence: 3, 6b, 9

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~3/2 = 679.881 (~16/15 = 120.119)
• POTE: ~5/4 = 400.000, ~3/2 = 680.042 (~16/15 = 119.958)

Optimal ET sequence: 3, 9

Badness (Smith): 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 tunings:

• CTE: ~28/25 = 200.000, ~3/2 = 701.955 (~16/15 = 98.045)

error map: ⟨0.000 0.000 +13.686 +31.174]

• POTE: ~28/25 = 200.000, ~3/2 = 710.963 (~16/15 = 89.037)

error map: ⟨0.000 +9.008 +13.686 +31.174]

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

Badness (Smith): 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 tunings:

• CTE: ~28/25 = 200.000, ~3/2 = 701.955 (~16/15 = 98.045)
• POTE: ~28/25 = 200.000, ~3/2 = 714.304 (~16/15 = 85.696)

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

Badness (Smith): 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 tunings:

• CTE: ~28/25 = 200.000, ~3/2 = 692.433 (~21/20 = 92.433)
• POTE: ~28/25 = 200.000, ~3/2 = 710.005 (~16/15 = 89.995)

Optimal ET sequence: 6f, 12f

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~7/4 = 952.295 (~35/32 = 152.295)

error map: ⟨0.000 +2.635 +13.686 -16.531]

• POTE: ~5/4 = 400.000, ~7/4 = 952.951 (~35/32 = 152.951)

error map: ⟨0.000 +3.947 +13.686 -15.875]

Optimal ET sequence: 6, 9, 15, 24, 39

Badness (Smith): 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]]

Wedgie: ⟨⟨ 6 0 3 3 -14 -12 -16 7 7 -2 ]]

Optimal tunings:

• CTE: ~5/4 = 400.000, ~7/4 = 952.250 (~12/11 = 152.250)
• POTE: ~5/4 = 400.000, ~7/4 = 952.932 (~12/11 = 152.932)

Optimal ET sequence: 6, 9, 15, 24, 39

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~7/4 = 950.805 (~12/11 = 150.805)
• POTE: ~5/4 = 400.000, ~7/4 = 951.687 (~12/11 = 151.687)

Optimal ET sequence: 6, 9, 15, 24

Badness (Smith): 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 tunings:

• CTE: ~44/39 = 200.000, ~7/4 = 952.531 (~40/39 = 47.469)
• POTE: ~44/39 = 200.000, ~7/4 = 953.358 (~40/39 = 46.642)

Optimal ET sequence: 6, 18bd, 24

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~14/9 = 752.834 (~36/35 = 47.166)

error map: ⟨0.000 +3.712 +13.686 -4.658]

• POTE: ~5/4 = 400.000, ~14/9 = 754.882 (~36/35 = 45.118)

error map: ⟨0.000 +7.808 +13.686 +5.583]

Optimal ET sequence: 24, 27

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~14/9 = 752.051 (~36/35 = 47.949)
• POTE: ~5/4 = 400.000, ~14/9 = 754.212 (~36/35 = 45.788)

Optimal ET sequence: 24, 27e, 51ce

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~14/9 = 752.128 (~36/35 = 47.872)
• POTE: ~5/4 = 400.000, ~14/9 = 753.750 (~36/35 = 46.250)

Optimal ET sequence: 24, 27e, 51ce

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~32/21 = 747.948 (~21/20 = 52.052)

error map: ⟨0.000 -6.058 +13.686 -12.671]

• POTE: ~5/4 = 400.000, ~32/21 = 747.907 (~21/20 = 52.093)

error map: ⟨0.000 -6.143 +13.686 -12.544]

Optimal ET sequence: 21, 24, 45c

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~32/21 = 748.296 (~33/32 = 51.704)
• POTE: ~5/4 = 400.000, ~32/21 = 748.345 (~33/32 = 51.655)

Optimal ET sequence: 21, 24

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~32/21 = 748.525 (~33/32 = 51.475)
• POTE: ~5/4 = 400.000, ~32/21 = 748.452 (~33/32 = 51.548)

Optimal ET sequence: 21, 24

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~28/25 = 175.359

error map: ⟨0.000 -0.521 +13.686 +6.533]

• POTE: ~5/4 = 400.000, ~28/25 = 176.646

error map: ⟨0.000 +4.629 +13.686 +7.820]

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

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~11/10 = 175.053
• POTE: ~5/4 = 400.000, ~11/10 = 176.981

Optimal ET sequence: 6, 21, 27e

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~11/10 = 175.252
• POTE: ~5/4 = 400.000, ~11/10 = 176.551

Optimal ET sequence: 6, 21, 27e

Badness (Smith): 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 tunings:

• CTE: ~54/49 = 200.000, ~45/28 = 853.190 (~36/35 = 53.190)

error map: ⟨0.000 +4.425 +13.686 -9.256]

• POTE: ~54/49 = 200.000, ~45/28 = 855.485 (~36/35 = 55.485)

error map: ⟨0.000 +9.015 +13.686 -2.371]

Optimal ET sequence: 18, 24, 42

Badness (Smith): 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 tunings:

• CTE: ~54/49 = 200.000, ~18/11 = 852.597 (~36/35 = 52.597)
• POTE: ~54/49 = 200.000, ~18/11 = 855.220 (~36/35 = 55.220)

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

Badness (Smith): 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 tunings:

• CTE: ~49/45 = 133.333, ~3/2 = 702.004 (~36/35 = 35.338)

error map: ⟨0.000 +0.049 +13.686 -0.155]

• POTE: ~49/45 = 133.333, ~3/2 = 707.167 (~36/35 = 40.501)

error map: ⟨0.000 +5.212 +13.686 +5.008]

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

Badness (Smith): 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 tunings:

• CTE: ~12/11 = 133.333, ~3/2 = 699.622 (~36/35 = 32.955)
• POTE: ~12/11 = 133.333, ~3/2 = 706.726 (~36/35 = 40.059)

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

Badness (Smith): 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 tunings:

• CTE: ~14/13 = 133.333, ~3/2 = 700.433 (~36/35 = 33.766)
• POTE: ~14/13 = 133.333, ~3/2 = 706.889 (~36/35 = 40.222)

Optimal ET sequence: 9, 18e, 27e

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~48/35 = 498.637 (~15/14 = 98.637)

error map: ⟨0.000 -6.045 +13.686 +28.447]

• POTE: ~5/4 = 400.000, ~48/35 = 501.980 (~15/14 = 101.980)

error map: ⟨0.000 +3.986 +13.686 +35.134]

Optimal ET sequence: 9bd, 12

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~7/4 = 968.826 (~28/25 = 168.826)

error map: ⟨0.000 +98.045 +13.686 0.000]

• POTE: ~5/4 = 400.000, ~7/4 = 1034.013 (~15/14 = 165.987)

error map: ⟨0.000 +98.045 +13.686 +65.187]

Optimal ET sequence: 3d, 6, 9bd

Badness (Smith): 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 tunings:

• CTE: ~5/4 = 400.000, ~7/4 = 961.874 (~11/10 = 161.874)
• POTE: ~5/4 = 400.000, ~7/4 = 1033.153 (~12/11 = 166.847)

Optimal ET sequence: 3de, 6

Badness (Smith): 0.025592

External links

• Herman Miller's page about augmented temperament ^{[dead link]}