Augmented family
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The augmented family of temperaments tempers out the diesis a.k.a. augmented comma, 128/125, the amount by which three 5/4 major thirds fall short of an octave, and so identifies the major third with the 1/3-octave. Hence it has the same 400-cent 5/4-approximations as 12edo.
Augmented
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. Its ploidacot is triploid monocot.
Subgroup: 2.3.5
Comma list: 128/125
Mapping: [⟨3 0 7], ⟨0 1 0]]
- mapping generators: ~5/4, ~3
- WE: ~5/4 = 399.0128 ¢, ~3/2 = 704.8937 ¢ (~16/15 = 93.1320 ¢)
- error map: ⟨-2.962 -0.023 +6.776]
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 705.0691 ¢ (~16/15 = 94.9309 ¢)
- error map: ⟨0.000 +3.114 +13.686]
Optimal ET sequence: 3, 9, 12, 27, 39, 51c, 90cc
Badness (Sintel): 0.523
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.
Augene
Subgroup: 2.3.5.7
Comma list: 64/63, 126/125
Mapping: [⟨3 0 7 18], ⟨0 1 0 -2]]
- WE: ~5/4 = 398.7461 ¢, ~3/2 = 707.0335 ¢ (~21/20 = 90.4587 ¢)
- error map: ⟨-3.762 +1.317 +4.909 +2.060]
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 709.3249 ¢ (~21/20 = 90.6751 ¢)
- error map: ⟨0.000 +7.370 +13.686 +12.524]
Optimal ET sequence: 12, 27, 39d, 66cd
Badness (Sintel): 0.628
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 tunings:
- WE: ~5/4 = 398.4962 ¢, ~3/2 = 708.5030 ¢ (~21/20 = 88.4895 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 711.6031 ¢ (~21/20 = 88.3969 ¢)
Optimal ET sequence: 12, 15, 27e
Badness (Sintel): 0.648
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:
- WE: ~5/4 = 398.0488 ¢, ~3/2 = 708.5402 ¢ (~21/20 = 87.5574 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 712.6704 ¢ (~21/20 = 87.3296 ¢)
Optimal ET sequence: 12f, 15, 27eff
Badness (Sintel): 0.859
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:
- WE: ~5/4 = 398.6473 ¢, ~3/2 = 710.1987 ¢ (~21/20 = 87.0959 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 712.5057 ¢ (~21/20 = 87.4943 ¢)
Optimal ET sequence: 12, 15, 27e, 69bceef
Badness (Sintel): 0.946
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:
- WE: ~5/4 = 398.5229 ¢, ~3/2 = 707.0562 ¢ (~21/20 = 89.9897 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 710.1903 ¢ (~21/20 = 89.8097 ¢)
Optimal ET sequence: 12f, 27e, 66cdeeef
Badness (Sintel): 0.955
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:
- WE: ~5/4 = 399.1743 ¢, ~3/2 = 712.6763 ¢ (~21/20 = 85.6723 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 713.9414 ¢ (~21/20 = 86.0586 ¢)
Optimal ET sequence: 12e, 15, 27, 42
Badness (Sintel): 1.18
August
Subgroup: 2.3.5.7
Comma list: 36/35, 128/125
Mapping: [⟨3 0 7 -1], ⟨0 1 0 2]]
- WE: ~5/4 = 399.1036 ¢, ~3/2 = 694.4509 ¢ (~16/15 = 103.7564 ¢)
- error map: ⟨-2.689 -10.193 +7.412 +15.594]
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 694.6812 ¢ (~16/15 = 105.3188 ¢)
- error map: ⟨0.000 -7.274 +13.686 +20.537]
Optimal ET sequence: 9, 12, 45cd
Badness (Sintel): 0.670
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 tunings:
- WE: ~5/4 = 398.9225 ¢, ~3/2 = 690.6486 ¢ (~16/15 = 107.1966 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 690.8519 ¢ (~16/15 = 109.1481 ¢)
Optimal ET sequence: 9, 12, 21, 33e
Badness (Sintel): 0.668
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:
- WE: ~5/4 = 399.0956 ¢, ~3/2 = 687.2261 ¢ (~16/15 = 110.9651 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 687.5057 ¢ (~16/15 = 112.4943 ¢)
Optimal ET sequence: 9, 12f, 21, 33ef
Badness (Sintel): 0.762
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:
- WE: ~5/4 = 400.4230 ¢, ~3/2 = 686.0809 ¢ (~16/15 = 114.7650 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 685.8446 ¢ (~16/15 = 114.1554 ¢)
Badness (Sintel): 0.919
Inflated
Subgroup: 2.3.5.7
Comma list: 28/27, 128/125
Mapping: [⟨3 0 7 -6], ⟨0 1 0 3]]
- WE: ~5/4 = 398.4023 ¢, ~3/2 = 719.8327 ¢ (~25/24 = 76.9719 ¢)
- error map: ⟨-3.762 +1.317 +4.909 +2.060]
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 721.0196 ¢ (~25/24 = 78.9804 ¢)
- error map: ⟨0.000 +19.065 +13.686 -5.767]
Optimal ET sequence: 3d, 12d, 15
Badness (Sintel): 1.39
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:
- WE: ~5/4 = 398.4016 ¢, ~3/2 = 719.7758 ¢ (~25/24 = 77.0275 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 720.9386 ¢ (~25/24 = 79.0614 ¢)
Optimal ET sequence: 3de, 12de, 15
Badness (Sintel): 1.03
Deflated
Subgroup: 2.3.5.7
Comma list: 21/20, 128/125
Mapping: [⟨3 0 7 13], ⟨0 1 0 -1]]
- WE: ~5/4 = 401.9566 ¢, ~3/2 = 684.9634 ¢ (~16/15 = 118.9497 ¢)
- error map: ⟨+5.870 -11.122 +27.382 -34.224]
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 682.2587 ¢ (~16/15 = 117.7413 ¢)
- error map: ⟨0.000 -19.696 +13.686 -51.085]
Optimal ET sequence: 3, 6b, 9
Badness (Sintel): 1.50
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:
- WE: ~5/4 = 402.1799 ¢, ~3/2 = 683.7477 ¢ (~16/15 = 120.6120 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~3/2 = 680.0162 ¢ (~16/15 = 119.9838 ¢)
Badness (Sintel): 1.23
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
- WE: ~28/25 = 199.0488 ¢, ~3/2 = 707.5815 ¢ (~25/24 = 88.6137 ¢)
- error map: ⟨+5.870 -11.122 +27.382 -34.224]
- CWE: ~28/25 = 200.0000 ¢, ~3/2 = 708.6907 ¢ (~25/24 = 91.3093 ¢)
- error map: ⟨0.000 +6.735 +13.686 +31.174]
Optimal ET sequence: 6, 12, 30d, 42dd, 54cdd
Badness (Sintel): 1.46
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:
- WE: ~28/25 = 198.6942 ¢, ~3/2 = 709.6404 ¢ (~25/24 = 85.1362 ¢)
- CWE: ~28/25 = 200.0000 ¢, ~3/2 = 711.8043 ¢ (~25/24 = 88.1957 ¢)
Optimal ET sequence: 6, 12, 30dee, 42ddeee
Badness (Sintel): 1.27
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:
- WE: ~28/25 = 198.4492 ¢, ~3/2 = 704.4994 ¢ (~25/24 = 89.2973 ¢)
- CWE: ~28/25 = 200.0000 ¢, ~3/2 = 706.6050 ¢ (~16/15 = 93.3950 ¢)
Badness (Sintel): 1.49
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
- WE: ~5/4 = 399.7480 ¢, ~7/4 = 952.3507 ¢ (~35/32 = 152.8547 ¢)
- error map: ⟨-0.756 +2.746 +11.922 -17.987]
- CWE: ~5/4 = 400.0000 ¢, ~7/4 = 952.7463 ¢ (~35/32 = 152.7463 ¢)
- error map: ⟨0.000 +3.538 +13.686 -16.080]
Optimal ET sequence: 6, 9, 15, 24, 39
Badness (Sintel): 1.39
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 tunings:
- WE: ~5/4 = 399.7654 ¢, ~7/4 = 952.3730 ¢ (~12/11 = 152.8421 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~7/4 = 952.7447 ¢ (~12/11 = 152.7447 ¢)
Optimal ET sequence: 6, 9, 15, 24, 39
Badness (Sintel): 0.865
- Music
- The Triforce of Courage (24edo)[dead link] by Igliashon Jones (2018)
- 2-2-1-2-2-1-2-2-1 mode of 15 edo play by Chris Vaisvil (2013)
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:
- WE: ~5/4 = 399.7107 ¢, ~7/4 = 950.9983 ¢ (~12/11 = 151.5768 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~7/4 = 951.4465 ¢ (~12/11 = 151.4465 ¢)
Optimal ET sequence: 6f, 9, 15, 24
Badness (Sintel): 0.837
- 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:
- WE: ~44/39 = 199.8321 ¢, ~7/4 = 952.5580 ¢ (~40/39 = 46.6024 ¢)
- CWE: ~44/39 = 200.0000 ¢, ~7/4 = 953.2005 ¢ (~40/39 = 46.7995 ¢)
Optimal ET sequence: 6, 18bd, 24
Badness (Sintel): 2.44
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
- WE: ~5/4 = 398.9278 ¢, ~14/9 = 752.8583 ¢ (~36/35 = 44.9973 ¢)
- error map: ⟨-3.217 +2.689 +6.181 -3.462]
- CWE: ~5/4 = 400.0000 ¢, ~14/9 = 754.2078 ¢ (~36/35 = 45.7922 ¢)
- error map: ⟨0.000 +6.461 +13.686 +2.213]
Badness (Sintel): 1.78
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:
- WE: ~5/4 = 398.8946 ¢, ~14/9 = 752.1272 ¢ (~36/35 = 45.6619 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~14/9 = 753.5000 ¢ (~36/35 = 46.5000 ¢)
Optimal ET sequence: 24, 27e, 51ce
Badness (Sintel): 1.26
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:
- WE: ~5/4 = 399.1053 ¢, ~14/9 = 752.0643 ¢ (~36/35 = 46.1463 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~14/9 = 753.3806 ¢ (~36/35 = 46.6194 ¢)
Optimal ET sequence: 24, 27e, 51ce
Badness (Sintel): 1.25
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
- WE: ~5/4 = 400.1805 ¢, ~32/21 = 748.2436 ¢ (~21/20 = 52.1174 ¢)
- error map: ⟨+0.542 -5.287 +14.950 -11.030]
- CWE: ~5/4 = 400.0000 ¢, ~32/21 = 747.9138 ¢ (~21/20 = 52.0862 ¢)
- error map: ⟨0.000 -6.127 +13.686 -12.567]
Optimal ET sequence: 21, 24, 45c
Badness (Sintel): 3.49
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:
- WE: ~5/4 = 400.0637 ¢, ~32/21 = 748.4638 ¢ (~33/32 = 51.6637 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~32/21 = 748.3383 ¢ (~33/32 = 51.6617 ¢)
Badness (Sintel): 2.25
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:
- WE: ~5/4 = 399.8855 ¢, ~32/21 = 748.2378 ¢ (~33/32 = 51.5332 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~32/21 = 748.4655 ¢ (~33/32 = 51.5345 ¢)
Badness (Sintel): 1.75
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
- WE: ~5/4 = 399.0296 ¢, ~28/25 = 176.2174 ¢ (~49/48 = 46.5949 ¢)
- error map: ⟨-2.911 +0.004 +6.894 -0.371]
- CWE: ~5/4 = 400.0000 ¢, ~28/25 = 176.2984 ¢ (~49/48 = 47.4031 ¢)
- error map: ⟨0.000 +3.239 +13.686 +7.473]
Optimal ET sequence: 6, 21, 27, 75c, 102ccd, 129bccd
Badness (Sintel): 2.86
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:
- WE: ~5/4 = 398.6615 ¢, ~11/10 = 176.3886 ¢ (~49/48 = 45.8843 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~11/10 = 176.5471 ¢ (~49/48 = 46.9059 ¢)
Optimal ET sequence: 6, 21, 27e
Badness (Sintel): 1.96
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:
- WE: ~5/4 = 398.8612 ¢, ~11/10 = 176.0486 ¢ (~49/48 = 46.7640 ¢)
- CWE: ~5/4 = 400.0000 ¢, ~11/10 = 176.3326 ¢ (~49/48 = 47.3348 ¢)
Optimal ET sequence: 6, 21, 27e
Badness (Sintel): 1.75
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
- WE: ~54/49 = 199.5469 ¢, ~45/28 = 853.5468 ¢ (~36/35 = 55.3594 ¢)
- error map: ⟨-2.719 +4.686 +7.342 -9.998]
- CWE: ~54/49 = 200.0000 ¢, ~45/28 = 854.7144 ¢ (~36/35 = 54.7144 ¢)
- error map: ⟨0.000 +7.474 +13.686 -4.683]
Optimal ET sequence: 18, 24, 42
Badness (Sintel): 5.34
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:
- WE: ~54/49 = 199.5188 ¢, ~18/11 = 853.1623 ¢ (~36/35 = 55.0872 ¢)
- CWE: ~54/49 = 200.0000 ¢, ~18/11 = 854.3545 ¢ (~36/35 = 54.3545 ¢)
Optimal ET sequence: 18e, 24, 42e, 66ce, 108bccee
Badness (Sintel): 2.67
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
- WE: ~49/45 = 133.0272 ¢, ~3/2 = 705.5438 ¢ (~36/35 = 40.4075 ¢)
- error map: ⟨-2.755 +0.834 +7.259 -2.737]
- CWE: ~49/45 = 133.3333 ¢, ~3/2 = 705.5157 ¢ (~36/35 = 38.8490 ¢)
- error map: ⟨0.000 +3.561 +13.686 +3.356]
Optimal ET sequence: 9, 18, 27, 63c, 90cc
Badness (Sintel): 1.70
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:
- WE: ~12/11 = 132.9553 ¢, ~3/2 = 704.7217 ¢ (~36/35 = 39.9453 ¢)
- CWE: ~12/11 = 133.3333 ¢, ~3/2 = 704.5723 ¢ (~36/35 = 37.9056 ¢)
Optimal ET sequence: 9, 18e, 27e, 63cee
Badness (Sintel): 1.15
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:
- WE: ~14/13 = 133.0143 ¢, ~3/2 = 705.1969 ¢ (~36/35 = 40.1256 ¢)
- CWE: ~14/13 = 133.3333 ¢, ~3/2 = 705.0176 ¢ (~36/35 = 38.3510 ¢)
Optimal ET sequence: 9, 18e, 27e
Badness (Sintel): 0.998
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
- WE: ~5/4 = 398.2337 ¢, ~48/35 = 499.7635 ¢ (~15/14 = 101.5299 ¢)
- error map: ⟨-2.755 +0.834 +7.259 -2.737]
- CWE: ~5/4 = 400.0000 ¢, ~48/35 = 500.9654 ¢ (~15/14 = 100.9654 ¢)
- error map: ⟨0.000 +3.561 +13.686 +3.356]
Optimal ET sequence: 3b, 9bd, 12
Badness (Sintel): 3.50