Meantone family
The 5-limit parent comma of the meantone family is the syntonic comma, 81/80. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
Meantone
- Main article: Meantone
Subgroup: 2.3.5
Comma list: 81/80
Mapping: [⟨1 0 -4], ⟨0 1 4]]
- mapping generators: ~2, ~3
Wedgie: ⟨⟨1 4 4]]
- 5-odd-limit: ~3/2 = [0 0 1/4⟩ (1/4-comma)
- 5-odd-limit diamond monotone: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
- 5-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b
Badness: 0.007381
Overview to extensions
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- Septimal meantone adds [-13 10 0 -1⟩, finding the ~7/4 at the augmented sixth,
- Flattone adds [-17 9 0 1⟩, finding the ~7/4 at the diminished seventh,
- Dominant adds [6 -2 0 -1⟩, finding the ~7/4 at the minor seventh,
- Sharptone adds [2 -3 0 1⟩, finding the ~7/4 at the major sixth,
Those all have a fifth as generator.
- Injera adds [-7 8 0 -2⟩ with a half-octave period.
- Mohajira adds [-23 11 0 2⟩ and splits the fifth in two.
- Godzilla adds [-4 -1 0 2⟩ with an ~8/7 generator, two of which give the fourth.
- Mothra adds [-10 1 0 3⟩ with an ~8/7 generator, three of which give the fifth.
- Liese adds [-9 11 0 -3⟩ with a ~10/7 generator, three of which give the twelfth.
- Squares adds [-3 9 0 -4⟩ with a ~9/7 generator, four of which give the eleventh.
- Jerome adds [3 7 0 -5⟩ and slices the fifth in five.
Temperaments discussed elsewhere include
- Plutus → Very low accuracy temperaments
- Godzilla → Slendro clan
- Mothra → Gamelismic clan
- Mohaha → Rastmic clan
- Dequarter → No-sevens subgroup temperaments
The rest are considered below.
Septimal meantone
- Main article: Meantone
In septimal meantone, nine fifths get to the interval class for 7, so that 7/4 is an augmented sixth (C-A♯), 7/6 is an augmented second (C-D♯), 7/5 is an augmented fourth (C-F♯), and 21/16 is an augmented third (C-E♯). Septimal meantone tempers out the common 7-limit commas 126/125, 225/224, and 3136/3125 and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
Subgroup: 2.3.5.7
Comma list: 81/80, 126/125
Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]
Wedgie: ⟨⟨1 4 10 4 13 12]]
- 7- and 9-odd-limit: ~3/2 = [0 0 1/4⟩ (1/4-comma)
- projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
- eigenmonzo (unchanged-interval) basis: 2.5
- 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
- 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
- 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, 503.4257 cents. The recurrence converges quickly.
Optimal ET sequence: 12, 19, 31, 81, 112b, 143b
Badness: 0.013707
Undecimal meantone (huygens)
- "Huygens" redirects here. For the Dutch mathematician, physicist and astronomer, see Wikipedia: Christiaan Huygens.
- See also: Meantone vs meanpop
Undecimal meantone maps the 11/8 to the double augmented third (C-E𝄪), and tridecimal meantone maps the 13/8 to the double augmented fifth (C-G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is the double augmented unison; 12/11 is a double diminished third; and 14/13 is a minor second.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 126/125
Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.1676
- POTE: ~2 = 1\1, ~3/2 = 696.967
Minimax tuning:
- 11-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
- projection map: [[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]
- eigenmonzo (unchanged-interval) basis: 2.11/9
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
- 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Algebraic generator: Traverse, the positive real root of x4 + 2x - 13, or 696.9529 cents.
Optimal ET sequence: 12, 19e, 31, 105, 136b
Badness: 0.017027
- Music
Tridecimal meantone
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 99/98, 105/104
Mapping: [⟨1 0 -4 -13 -25 -20], ⟨0 1 4 10 18 15]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.8552
- POTE: ~2 = 1\1, ~3/2 = 696.642
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
- eigenmonzo (unchanged-interval) basis: 2.11/9
Optimal ET sequence: 12f, 19e, 31
Badness: 0.018048
Meantonic
Dubbed meantonic here, this extension maps the 17/16 to the octave-reduced triple augmented seventh (C-B𝄪♯), and 19/16 to the quadruple augmented unison (C-C𝄪𝄪). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.
Subgroup: 2.3.5.7.11.13.17
Comma list: 66/65, 81/80, 99/98, 105/104, 121/119
Mapping: [⟨1 0 -4 -13 -25 -20 -37], ⟨0 1 4 10 18 15 26]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.6486
- POTE: ~2 = 1\1, ~3/2 = 696.377
Optimal ET sequence: 12fg, 19eg, 31, 50e
Badness: 0.019037
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 66/65, 77/76, 81/80, 99/98, 105/104, 121/119
Mapping: [⟨1 0 -4 -13 -25 -20 -37 -40], ⟨0 1 4 10 18 15 26 28]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.5551
- POTE: ~2 = 1\1, ~3/2 = 696.273
Optimal ET sequence: 12fghh, 19egh, 31, 50e
Badness: 0.017846
Meantoid
Dubbed meantoid here, this extension maps 17/16~19/18 to the augmented unison (C-C♯) and 19/16 to the augmented second (C-D♯). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverted, so genuine septendecimal and undevicesimal harmony cannot be expected.
Subgroup: 2.3.5.7.11.13.17
Comma list: 51/50, 66/65, 81/80, 85/84, 99/98
Mapping: [⟨1 0 -4 -13 -25 -20 -7], ⟨0 1 4 10 18 15 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.0360
- POTE: ~2 = 1\1, ~3/2 = 696.448
Optimal ET sequence: 12f, 19eg, 31g
Badness: 0.019433
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 51/50, 57/56, 66/65, 81/80, 85/84, 99/98
Mapping: [⟨1 0 -4 -13 -25 -20 -7 -10], ⟨0 1 4 10 18 15 7 9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.2161
- POTE: ~2 = 1\1, ~3/2 = 696.394
Optimal ET sequence: 12f, 19egh, 31gh
Badness: 0.017437
Huygens
Dubbed huygens here, this extension is perhaps the most practical, as it maps 17/16 to the minor second (C-D♭), and 19/16 to the minor third (C-E♭), suitable for a system generated by a mildly tempered fifth.
Subgroup: 2.3.5.7.11.13.17
Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
Mapping: [⟨1 0 -4 -13 -25 -20 12], ⟨0 1 4 10 18 15 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.9080
- POTE: ~2 = 1\1, ~3/2 = 697.003
Badness: 0.019982
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
Mapping: [⟨1 0 -4 -13 -25 -20 12 9], ⟨0 1 4 10 18 15 -5 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.9308
- POTE: ~2 = 1\1, ~3/2 = 697.140
Badness: 0.018047
Grosstone
Grosstone maps 13/8 to the double diminished seventh (C-B♭♭♭).
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 -25 29], ⟨0 1 4 10 18 -16]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.2582
- POTE: ~2 = 1\1, ~3/2 = 697.264
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [8/13 0 0 1/26 0 -1/26⟩
- eigenmonzo basis (unchanged-interval basis): 2.13/7
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Optimal ET sequence: 12, 31, 43, 74
Badness: 0.025899
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 -25 29 12], ⟨0 1 4 10 18 -16 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.2996
- POTE: ~2 = 1\1, ~3/2 = 697.335
Optimal ET sequence: 12, 31, 43, 74g
Badness: 0.020889
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 -25 29 12 9], ⟨0 1 4 10 18 -16 -5 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.3271
- POTE: ~2 = 1\1, ~3/2 = 697.380
Optimal ET sequence: 12, 31, 43, 74gh
Badness: 0.017611
Meridetone
Meridetone maps the 13/8 to the quadruple augmented fourth (C-F𝄪𝄪).
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 81/80, 99/98, 126/125
Mapping: [⟨1 0 -4 -13 -25 -39], ⟨0 1 4 10 18 27]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.5155
- POTE: ~2 = 1\1, ~3/2 = 697.529
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [14/25 -2/25 0 0 0 1/25⟩
- eigenmonzo (unchanged-interval) basis: 2.13/9
Optimal ET sequence: 12f, 31f, 43
Badness: 0.026421
Meridetonic
Subgroup: 2.3.5.7.11.13.17
Comma list: 78/77, 81/80, 99/98, 126/125, 273/272
Mapping: [⟨1 0 -4 -13 -25 -39 -56], ⟨0 1 4 10 18 27 38]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.5076
- POTE: ~2 = 1\1, ~3/2 = 697.514
Optimal ET sequence: 12fg, 31fg, 43
Badness: 0.027706
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 78/77, 81/80, 99/98, 126/125, 153/152, 273/272
Mapping: [⟨1 0 -4 -13 -25 -39 -56 -59], ⟨0 1 4 10 18 27 38 40]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.4848
- POTE: ~2 = 1\1, ~3/2 = 697.481
Optimal ET sequence: 12fghh, 31fgh, 43
Badness: 0.025315
Meridetoid
Subgroup: 2.3.5.7.11.13.17
Comma list: 51/50, 78/77, 81/80, 85/84, 99/98
Mapping: [⟨1 0 -4 -13 -25 -39 -7], ⟨0 1 4 10 18 27 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.6098
- POTE: ~2 = 1\1, ~3/2 = 697.376
Optimal ET sequence: 12f, 31fg, 43g
Badness: 0.027518
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 51/50, 57/56, 78/77, 81/80, 85/84, 99/98
Mapping: [⟨1 0 -4 -13 -25 -39 -7 -10], ⟨0 1 4 10 18 27 7 9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.7012
- POTE: ~2 = 1\1, ~3/2 = 697.316
Optimal ET sequence: 12f, 19effgh, 31fgh, 43gh
Badness: 0.023613
Sauveuric
Subgroup: 2.3.5.7.11.13.17
Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
Mapping: [⟨1 0 -4 -13 -25 -39 12], ⟨0 1 4 10 18 27 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.5384
- POTE: ~2 = 1\1, ~3/2 = 697.644
Badness: 0.023881
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
Mapping: [⟨1 0 -4 -13 -25 -39 12 9], ⟨0 1 4 10 18 27 -5 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.5550
- POTE: ~2 = 1\1, ~3/2 = 697.715
Badness: 0.020540
Hemimeantone
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 169/168
Mapping: [⟨1 0 -4 -13 -25 -5], ⟨0 2 8 20 36 11]]
- mapping generators: ~2, ~26/15
Optimal tunings:
- CTE: ~2 = 1\1, ~26/15 = 948.6109
- POTE: ~2 = 1\1, ~26/15 = 948.465
Optimal ET sequence: 19e, 43, 62
Badness: 0.031433
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
Mapping: [⟨1 0 -4 -13 -25 -5 -22], ⟨0 2 8 20 36 11 33]]
Optimal tunings:
- CTE: ~2 = 1\1, ~26/15 = 948.6173
- POTE: ~2 = 1\1, ~26/15 = 948.477
Optimal ET sequence: 19eg, 43, 62
Badness: 0.023380
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
Mapping: [⟨1 0 -4 -13 -25 -5 -22 -25], ⟨0 2 8 20 36 11 33 37]]
Optimal tunings:
- CTE: ~2 = 1\1, ~19/11 = 948.6088
- POTE: ~2 = 1\1, ~19/11 = 948.473
Optimal ET sequence: 19egh, 43, 62
Badness: 0.018952
Semimeantone
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 847/845
Mapping: [⟨2 0 -8 -26 -50 -59], ⟨0 1 4 10 18 21]]
- mapping generators: ~55/39, ~3
Optimal tunings:
- CTE: ~55/39 = 1\2, ~3/2 = 697.1678
- POTE: ~55/39 = 1\2, ~3/2 = 697.005
Optimal ET sequence: 12f, 38deefff, 50eff, 62, 136b
Badness: 0.040668
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
Mapping: [⟨2 0 -8 -26 -50 -59 5], ⟨0 1 4 10 18 21 1]]
Optimal tunings:
- CTE: ~17/12 = 1\2, ~3/2 = 697.1740
- POTE: ~17/12 = 1\2, ~3/2 = 696.927
Optimal ET sequence: 12f, 50eff, 62, 136bg
Badness: 0.031491
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
Mapping: [⟨2 0 -8 -26 -50 -59 5 -1], ⟨0 1 4 10 18 21 1 3]]
Optimal tunings:
- CTE: ~17/12 = 1\2, ~3/2 = 697.1871
- POTE: ~17/12 = 1\2, ~3/2 = 696.906
Optimal ET sequence: 12f, 50eff, 62
Badness: 0.024206
Meanpop
- See also: Meantone vs meanpop
Meanpop maps the 11/8 to the double diminished fifth (C-G𝄫), and tridecimal meanpop still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the double diminished third; 12/11~13/12, double augmented unison; and 14/13, minor second.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 385/384
Mapping: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]
- mapping generator: ~2, ~3
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.5311
- POTE: ~2 = 1\1, ~3/2 = 696.434
Minimax tuning:
- 11-odd-limit: ~3/2 = [0 0 1/4⟩
- [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [-3 0 5/2 0 0⟩, [11 0 -13/4 0 0⟩]
- eigenmonzo (unchanged-interval) basis: 2.5
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
- 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x3 + 6x - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Optimal ET sequence: 12e, 19, 31, 81, 112b
Badness: 0.021543
- Music
- Scott Joplin's "The Entertainer" tuned into meanpop[dead link]
- Twinkle canon – 50 edo by Claudi Meneghin
Tridecimal meanpop
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 24 -20], ⟨0 1 4 10 -13 15]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.3563
- POTE: ~2 = 1\1, ~3/2 = 696.211
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [4/7 0 0 0 -1/28 1/28⟩
- eigenmonzo (unchanged-interval) basis: 2.13/11
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
Optimal ET sequence: 19, 31, 50, 81
Badness: 0.020883
Meanpoppic
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
Mapping: [⟨1 0 -4 -13 24 -20 -37], ⟨0 1 4 10 -13 15 26]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.3508
- POTE: ~2 = 1\1, ~3/2 = 696.194
Optimal ET sequence: 19g, 31, 50, 81, 131bd
Badness: 0.019953
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
Mapping: [⟨1 0 -4 -13 24 -20 -37 -40], ⟨0 1 4 10 -13 15 26 28]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.3471
- POTE: ~2 = 1\1, ~3/2 = 696.188
Optimal ET sequence: 19gh, 31, 50, 81
Badness: 0.017791
Meanpoid
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 24 -20 12], ⟨0 1 4 10 -13 15 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.4388
- POTE: ~2 = 1\1, ~3/2 = 696.408
Badness: 0.022870
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
Mapping: [⟨1 0 -4 -13 24 -20 12 9], ⟨0 1 4 10 -13 15 -5 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.4838
- POTE: ~2 = 1\1, ~3/2 = 696.499
Optimal ET sequence: 12ef, 19, 31
Badness: 0.020488
Meanplop
Subgroup: 2.3.5.7.11.13
Comma list: 65/64, 78/77, 81/80, 91/90
Mapping: [⟨1 0 -4 -13 24 10], ⟨0 1 4 10 -13 -4]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.2827
- POTE: ~2 = 1\1, ~3/2 = 696.202
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [11/13 0 0 0 -1/13⟩
- Eigenmonzo (unchanged-interval) basis: 2.11
Optimal ET sequence: 12e, 19, 31f
Badness: 0.027666
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 52/51, 65/64, 78/77, 81/80, 91/90
Mapping: [⟨1 0 -4 -13 24 10 12], ⟨0 1 4 10 -13 -4 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.4069
- POTE: ~2 = 1\1, ~3/2 = 696.414
Badness: 0.026836
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 39/38, 52/51, 65/64, 77/76, 81/80, 91/90
Mapping: [⟨1 0 -4 -13 24 10 12 9], ⟨0 1 4 10 -13 -4 -5 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.4731
- POTE: ~2 = 1\1, ~3/2 = 696.497
Badness: 0.023540
Meanploid
Subgroup: 2.3.5.7.11.13.17
Comma list: 51/50, 65/64, 78/77, 81/80, 85/84
Mapping: [⟨1 0 -4 -13 24 10 -7], ⟨0 1 4 10 -13 -4 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.6614
- POTE: ~2 = 1\1, ~3/2 = 696.415
Optimal ET sequence: 12e, 19g, 31fg
Badness: 0.026094
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 51/50, 57/56, 65/64, 76/75, 78/77, 81/80
Mapping: [⟨1 0 -4 -13 24 10 -7 -10], ⟨0 1 4 10 -13 -4 7 9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 697.0160
- POTE: ~2 = 1\1, ~3/2 = 696.583
Optimal ET sequence: 12e, 19gh, 31fgh
Badness: 0.023104
Meanenneadecal
Meanenneadecal maps the 11/8 to the augmented fourth (C-F♯), and tridecimal meanenneadecal still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the major second; 12/11~14/13, minor second; and 13/12, double augmented unison.
Subgroup: 2.3.5.7.11
Comma list: 45/44, 56/55, 81/80
Mapping: [⟨1 0 -4 -13 -6], ⟨0 1 4 10 6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.1527
- POTE: ~2 = 1\1, ~3/2 = 696.250
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
- 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]
Optimal ET sequence: 7d, 12, 19, 31e
Badness: 0.021423
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 56/55, 78/77, 81/80
Mapping: [⟨1 0 -4 -13 -6 -20], ⟨0 1 4 10 6 15]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.0983
- POTE: ~2 = 1\1, ~3/2 = 696.146
Optimal ET sequence: 7df, 12f, 19, 31e
Badness: 0.021182
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
Mapping: [⟨1 0 -4 -13 -6 -20 12], ⟨0 1 4 10 6 15 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.2161
- POTE: ~2 = 1\1, ~3/2 = 696.575
Optimal ET sequence: 12f, 19, 31e
Badness: 0.022980
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
Mapping: [⟨1 0 -4 -13 -6 -20 12 9], ⟨0 1 4 10 6 15 -5 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.2774
- POTE: ~2 = 1\1, ~3/2 = 696.706
Optimal ET sequence: 12f, 19, 31e
Badness: 0.020293
Meanenneadecoid
Subgroup: 2.3.5.7.11.13.17
Comma list: 34/33, 45/44, 51/50, 56/55, 78/77
Mapping: [⟨1 0 -4 -13 -6 -20 -7], ⟨0 1 4 10 6 15 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.4501
- POTE: ~2 = 1\1, ~3/2 = 696.025
Optimal ET sequence: 7dfg, 12f, 19g
Badness: 0.020171
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 78/77
Mapping: [⟨1 0 -4 -13 -6 -20 -7 -10], ⟨0 1 4 10 6 15 7 9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.7925
- POTE: ~2 = 1\1, ~3/2 = 696.121
Optimal ET sequence: 7dfgh, 12f, 19gh
Badness: 0.018045
Vincenzo
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10], ⟨0 1 4 10 6 -4]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.7897
- POTE: ~2 = 1\1, ~3/2 = 695.060
Optimal ET sequence: 7d, 12, 19
Badness: 0.024763
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12], ⟨0 1 4 10 6 -4 -5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.0114
- POTE: ~2 = 1\1, ~3/2 = 695.858
Badness: 0.025535
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12 9], ⟨0 1 4 10 6 -4 -5 -3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.1196
- POTE: ~2 = 1\1, ~3/2 = 696.131
Badness: 0.022302
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14], ⟨0 1 4 10 6 -4 -5 -3 -6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.0585
- POTE: ~2 = 1\1, ~3/2 = 696.044
Badness: 0.020139
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8], ⟨0 1 4 10 6 -4 -5 -3 -6 -2]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.9824
- POTE: ~2 = 1\1, ~3/2 = 695.913
Badness: 0.018168
31-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.7980
- POTE: ~2 = 1\1, ~3/2 = 695.750
Badness: 0.017069
37-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.6746
- POTE: ~2 = 1\1, ~3/2 = 695.603
Badness: 0.016129
41-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.7239
- POTE: ~2 = 1\1, ~3/2 = 695.696
Badness: 0.015356
43-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.7160
- POTE: ~2 = 1\1, ~3/2 = 695.688
Badness: 0.013906
47-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.6849
- POTE: ~2 = 1\1, ~3/2 = 695.676
Badness: 0.013818
Vincenzoid
Subgroup: 2.3.5.7.11.13.17
Comma list: 34/33, 45/44, 51/50, 56/55, 65/64
Mapping: [⟨1 0 -4 -13 -6 10 -7], ⟨0 1 4 10 6 -4 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.4125
- POTE: ~2 = 1\1, ~3/2 = 695.358
Optimal ET sequence: 7dg, 12, 19g
Badness: 0.022099
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 65/64
Mapping: [⟨1 0 -4 -13 -6 10 -7 -10], ⟨0 1 4 10 6 -4 7 9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.9500
- POTE: ~2 = 1\1, ~3/2 = 695.725
Optimal ET sequence: 7dgh, 12, 19gh
Badness: 0.019904
Meanundec
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 40/39, 45/44, 56/55
Mapping: [⟨1 0 -4 -13 -6 -1], ⟨0 1 4 10 6 3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 695.6202
- POTE: ~2 = 1\1, ~3/2 = 697.254
Optimal ET sequence: 7d, 12f, 19f
Badness: 0.024243
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 27/26, 34/33, 40/39, 45/44, 56/55
Mapping: [⟨1 0 -4 -13 -6 -1 -7], ⟨0 1 4 10 6 3 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.2789
- POTE: ~2 = 1\1, ~3/2 = 697.586
Badness: 0.021400
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 27/26, 34/33, 40/39, 45/44, 56/55, 57/55
Mapping: [⟨1 0 -4 -13 -6 -1 -7 -10], ⟨0 1 4 10 6 3 7 9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.8486
- POTE: ~2 = 1\1, ~3/2 = 698.118
Optimal ET sequence: 7dgh, 12f
Badness: 0.018996
Meanundeci
Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C-F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C-A♭).
Subgroup: 2.3.5.7.11
Comma list: 33/32, 55/54, 77/75
Mapping: [⟨1 0 -4 -13 5], ⟨0 1 4 10 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.7022
- POTE: ~2 = 1\1, ~3/2 = 694.689
Optimal ET sequence: 7d, 12e, 19e
Badness: 0.031539
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 33/32, 55/54, 65/64, 77/75
Mapping: [⟨1 0 -4 -13 5 10], ⟨0 1 4 10 -1 -4]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 696.2408
- POTE: ~2 = 1\1, ~3/2 = 694.764
Optimal ET sequence: 7d, 12e, 19e
Badness: 0.026288
Bimeantone
11/8 is mapped to half octave minus the meantone diesis.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 245/242
Mapping: [⟨2 0 -8 -26 -31], ⟨0 1 4 10 12]]
- mapping generators: ~63/44, ~3
Optimal tunings:
- CTE: ~63/44 = 1\2, ~3/2 = 696.5199
- POTE: ~63/44 = 1\2, ~3/2 = 696.016
Optimal ET sequence: 12, 26de, 38d, 50
Badness: 0.038122
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 126/125, 245/242
Mapping: [⟨2 0 -8 -26 -31 -40], ⟨0 1 4 10 12 15]]
Optimal tunings:
- CTE: ~55/39 = 1\2, ~3/2 = 696.3410
- POTE: ~55/39 = 1\2, ~3/2 = 695.836
Optimal ET sequence: 12f, 26deff, 38df, 50
Badness: 0.028817
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
Mapping: [⟨2 0 -8 -26 -31 -40 5], ⟨0 1 4 10 12 15 1]]
Optimal tunings:
- CTE: ~17/12 = 1\2, ~3/2 = 696.3526
- POTE: ~17/12 = 1\2, ~3/2 = 695.783
Optimal ET sequence: 12f, 38df, 50
Badness: 0.022666
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
Mapping: [⟨2 0 -8 -26 -31 -40 5 -1], ⟨0 1 4 10 12 15 1 3]]
Optimal tunings:
- CTE: ~17/12 = 1\2, ~3/2 = 696.3837
- POTE: ~17/12 = 1\2, ~3/2 = 695.752
Optimal ET sequence: 12f, 26deff, 38df, 50
Badness: 0.017785
Trimean
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 1344/1331
Mapping: [⟨1 2 4 7 5], ⟨0 -3 -12 -30 -11]]
- mapping generators: ~2, ~11/10
Optimal tunings:
- CTE: ~2 = 1\1, ~11/10 = 167.7074
- POTE: ~2 = 1\1, ~11/10 = 167.805
Optimal ET sequence: 7d, 36d, 43, 50, 93
Badness: 0.050729
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 126/125, 144/143, 364/363
Mapping: [⟨1 2 4 7 5 3], ⟨0 -3 -12 -30 -11 5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/10 = 167.7121
- POTE: ~2 = 1\1, ~11/10 = 167.790
Optimal ET sequence: 7d, 43, 50, 93
Badness: 0.035445
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
Mapping: [⟨1 2 4 7 5 3 8], ⟨0 -3 -12 -30 -11 5 -28]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/10 = 167.7047
- POTE: ~2 = 1\1, ~11/10 = 167.786
Optimal ET sequence: 7dg, 43, 50, 93
Badness: 0.025221
Flattone
In flattone tunings, the fifth is typically even flatter than that of 19edo. Here, 9 fourths get to the interval class for 7, so that 7/4 is a diminished seventh (C-B𝄫), 7/6 is a diminished third (C-E𝄫), and 7/5 is a doubly diminshed fifth (C-G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are 26edo, 45edo, and 64edo.
Subgroup: 2.3.5.7
Comma list: 81/80, 525/512
Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]
Wedgie: ⟨⟨1 4 -9 4 -17 -32]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.779
- 7-odd-limit: ~3/2 = [8/13 0 1/13 -1/13⟩
- [[1 0 0 0⟩, [21/13 0 1/13 -1/13⟩, [32/13 0 4/13 -4/13⟩, [32/13 0 -9/13 9/13⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/5
- 9-odd-limit: ~3/2 = [6/11 2/11 0 -1/11⟩
- [[1 0 0 0⟩, [17/11 2/11 0 -1/11⟩, [24/11 8/11 0 -4/11⟩, [34/11 -18/11 0 9/11⟩]
- eigenmonzo (unchanged-interval) basis: 2.9/7
- 7- and 9-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
- 7-odd-limit diamond tradeoff: ~3/2 = [692.353, 701.955]
- 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
Algebraic generator: Squarto, the positive root of 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Optimal ET sequence: 7, 19, 26, 45
Badness: 0.038553
Scales: flattone12
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 385/384
Mapping: [⟨1 0 -4 17 -6], ⟨0 1 4 -9 6]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.126
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
- 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
Optimal ET sequence: 7, 19, 26, 45, 71bc, 116bcde
Badness: 0.033839
Scales: flattone12
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 65/64, 78/77, 81/80
Mapping: [⟨1 0 -4 17 -6 10], ⟨0 1 4 -9 6 -4]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.058
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
Optimal ET sequence: 7, 19, 26, 45f, 71bcf, 116bcdef
Badness: 0.022260
Scales: flattone12
Dominant
- See also: Archytas clan
The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is 12edo, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with 29edo, 41edo, or 53edo.
Subgroup: 2.3.5.7
Comma list: 36/35, 64/63
Mapping: [⟨1 0 -4 6], ⟨0 1 4 -2]]
Wedgie: ⟨⟨1 4 -2 4 -6 -16]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.573
- 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
- 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 715.587]
- 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
Optimal ET sequence: 5, 7, 12, 41cd, 53cdd, 65ccddd
Badness: 0.020690
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 64/63
Mapping: [⟨1 0 -4 6 13], ⟨0 1 4 -2 -6]]
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
- 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.254
Optimal ET sequence: 5, 12, 17c, 29cde
Badness: 0.024180
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 56/55, 64/63, 66/65
Mapping: [⟨1 0 -4 6 13 18], ⟨0 1 4 -2 -6 -9]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.636
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
Optimal ET sequence: 12f, 17c, 29cdef
Badness: 0.024108
Dominion
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 56/55, 64/63
Mapping: [⟨1 0 -4 6 13 -9], ⟨0 1 4 -2 -6 8]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.905
Optimal ET sequence: 5, 12, 17c, 46cde
Badness: 0.027295
Domineering
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 64/63
Mapping: [⟨1 0 -4 6 -6], ⟨0 1 4 -2 6]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.776
Optimal ET sequence: 5e, 7, 12, 19d, 43de
Badness: 0.021978
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 45/44, 52/49, 64/63
Mapping: [⟨1 0 -4 6 -6 10], ⟨0 1 4 -2 6 -4]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.762
Optimal ET sequence: 5ef, 7, 12, 19d, 31def
Badness: 0.027039
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
Mapping: [⟨1 0 -4 6 -6 10 12], ⟨0 1 4 -2 6 -4 -5]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.115
Optimal ET sequence: 5ef, 7, 12, 19d, 31def
Badness: 0.024539
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
Mapping: [⟨1 0 -4 6 -6 10 12 9], ⟨0 1 4 -2 6 -4 -5 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.217
Optimal ET sequence: 5ef, 7, 12, 19d, 31def
Badness: 0.020398
Dominatrix
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 36/35, 45/44, 64/63
Mapping: [⟨1 0 -4 6 -6 -1], ⟨0 1 4 -2 6 3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.544
Optimal ET sequence: 5e, 7, 12f, 19df
Badness: 0.018289
Domination
Subgroup: 2.3.5.7.11
Comma list: 36/35, 64/63, 77/75
Mapping: [⟨1 0 -4 6 -14], ⟨0 1 4 -2 11]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.004
Optimal ET sequence: 5e, 12e, 17c, 46cd
Badness: 0.036562
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 64/63, 66/65
Mapping: [⟨1 0 -4 6 -14 -9], ⟨0 1 4 -2 11 8]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.496
Optimal ET sequence: 5e, 12e, 17c
Badness: 0.027435
Arnold
Subgroup: 2.3.5.7.11
Comma list: 22/21, 33/32, 36/35
Mapping: [⟨1 0 -4 6 5], ⟨0 1 4 -2 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.491
Optimal ET sequence: 5, 7, 12e
Badness: 0.026141
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 22/21, 27/26, 33/32, 36/35
Mapping: [⟨1 0 -4 6 5 -1], ⟨0 1 4 -2 -1 3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.743
Optimal ET sequence: 5, 7, 12ef, 19def
Badness: 0.023300
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49
Mapping: [⟨1 0 -4 6 5 -1 12], ⟨0 1 4 -2 -1 3 -5]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.978
Optimal ET sequence: 5, 7, 12ef, 19def
Badness: 0.024535
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
Mapping: [⟨1 0 -4 6 5 -1 12 9], ⟨0 1 4 -2 -1 3 -5 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 697.068
Optimal ET sequence: 5, 7, 12ef, 19def
Badness: 0.021098
Sharptone
Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. 12edo tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
Subgroup: 2.3.5.7
Comma list: 21/20, 28/27
Mapping: [⟨1 0 -4 -2], ⟨0 1 4 3]]
Wedgie: ⟨⟨1 4 3 4 2 -4]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.140
Optimal ET sequence: 5, 7d, 12d
Badness: 0.024848
Meanertone
Subgroup: 2.3.5.7.11
Comma list: 21/20, 28/27, 33/32
Mapping: [⟨1 0 -4 -2 5], ⟨0 1 4 3 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.615
Optimal ET sequence: 5, 7d, 12de
Badness: 0.025167
Supermean
Subgroup: 2.3.5.7
Comma list: 81/80, 672/625
Mapping: [⟨1 0 -4 -21], ⟨0 1 4 15]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.889
Optimal ET sequence: 5d, 12d, 17c, 29c
Badness: 0.134204
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 81/80, 132/125
Mapping: [⟨1 0 -4 -21 -14], ⟨0 1 4 15 11]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.096
Optimal ET sequence: 5de, 12de, 17c, 29c
Badness: 0.063262
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 56/55, 66/65, 81/80
Mapping: [⟨1 0 -4 -21 -14 -9], ⟨0 1 4 15 11 8]]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.094
Optimal ET sequence: 5de, 12de, 17c, 29c
Badness: 0.040324
Mohajira
- Main article: Mohajira
Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. 31edo makes for an excellent (7-limit) mohajira tuning, with generator 9\31.
Subgroup: 2.3.5.7
Comma list: 81/80, 6144/6125
Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]
- mapping generators: ~2, ~128/105
Wedgie: ⟨⟨2 8 -11 8 -23 -48]]
Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 348.415
- 7- and 9-odd-limit: ~128/105 = [0 0 1/8⟩
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5
- 7- and 9-odd-limit diamond monotone: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
- 7-odd-limit diamond tradeoff: ~128/105 = [347.393, 350.978]
- 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]
Algebraic generator: Mohabis, real root of 3x3 - 3x2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Optimal ET sequence: 7, 24, 31
Badness: 0.055714
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120, 176/175
Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.477
Minimax tuning:
- 11-odd-limit: ~11/9 = [0 0 1/8⟩
- [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [6 0 -11/8 0 0⟩, [2 0 5/8 0 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5
Tuning ranges:
- 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
- 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
Optimal ET sequence: 7, 24, 31
Badness: 0.026064
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 105/104, 121/120
Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.558
Optimal ET sequence: 7, 24, 31
Badness: 0.023388
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.736
Optimal ET sequence: 7, 24, 31, 86ef
Badness: 0.020576
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
Mapping: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.810
Optimal ET sequence: 7, 24, 31, 55, 86efh
Badness: 0.017302
Mohamaq
Subgroup: 2.3.5.7
Comma list: 81/80, 392/375
Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]
- mapping generators: ~2, ~25/21
Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 350.586
Optimal ET sequence: 7d, 17c, 24, 65cc, 89ccd
Badness: 0.077734
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 243/242
Mapping: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.565
Optimal ET sequence: 7d, 17c, 24, 65cc, 89ccd
Badness: 0.036207
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 66/65, 77/75, 243/242
Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.745
Optimal ET sequence: 7d, 17c, 24, 41c, 65cc
Badness: 0.028738
Liese
Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. 74edo makes for a good liese tuning, though 19edo can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Subgroup: 2.3.5.7
Comma list: 81/80, 686/675
Mapping: [⟨1 0 -4 -3], ⟨0 3 12 11]]
- mapping generators: ~2, ~10/7
Wedgie: ⟨⟨3 12 11 12 9 -8]]
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.406
- 7- and 9-odd-limit: ~10/7 = [1/3 0 1/12⟩
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5
Algebraic generator: Radix, the real root of x5 - 2x4 + 2x3 - 2x2 + 2x - 2, also a root of x6 - x5 - 2. The recurrence converges.
Optimal ET sequence: 17c, 19, 55, 74d
Badness: 0.046706
Liesel
Subgroup: 2.3.5.7.11
Comma list: 56/55, 81/80, 540/539
Mapping: [⟨1 0 -4 -3 4], ⟨0 3 12 11 -1]]
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.073
Optimal ET sequence: 17c, 19, 36, 91cee
Badness: 0.040721
13-limit
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 78/77, 81/80, 91/90
Mapping: [⟨1 0 -4 -3 4 0], ⟨0 3 12 11 -1 7]]
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.042
Optimal ET sequence: 17c, 19, 36, 91ceef
Badness: 0.027304
Elisa
Subgroup: 2.3.5.7.11
Comma list: 77/75, 81/80, 99/98
Mapping: [⟨1 0 -4 -3 -5], ⟨0 3 12 11 16]]
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.061
Optimal ET sequence: 17c, 19e, 36e
Badness: 0.041592
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 77/75, 81/80, 99/98
Mapping: [⟨1 0 -4 -3 -5 0], ⟨0 3 12 11 16 7]]
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.991
Optimal ET sequence: 17c, 19e, 36e
Badness: 0.026922
Lisa
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 343/330
Mapping: [⟨1 0 -4 -3 -6], ⟨0 3 12 11 18]]
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.370
Optimal ET sequence: 17cee, 19
Badness: 0.054829
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 81/80, 91/88, 147/143
Mapping: [⟨1 0 -4 -3 -6 0], ⟨0 3 12 11 18 7]]
Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.221
Optimal ET sequence: 17cee, 19
Badness: 0.036144
Superpine
The superpine temperament is generated by 1/3 of a fourth, represented by 35/32, which resembles porcupine, but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent 6/5–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as 11/10 as in porcupine, which makes 11/8 high-complexity like the other harmonics, but in the 13-limit 5 generators up closely approximates 13/8. 43edo is a good tuning especially for the higher-limit extensions.
Subgroup: 2.3.5.7
Comma list: 81/80, 1119744/1071875
Mapping: [⟨1 2 4 1], ⟨0 -3 -12 13]]
Optimal tuning (CTE): ~2 = 1\1, ~35/32 = 167.279
Optimal ET sequence: 7, 36, 43, 79c
Badness: 0.137
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 176/175, 864/847
Mapping: [⟨1 2 4 1 5], ⟨0 -3 -12 13 -11]]
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.407
Optimal ET sequence: 7, 36, 43
Badness: 0.0576
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 81/80, 144/143, 176/175
Mapping: [⟨1 2 4 1 5 3], ⟨0 -3 -12 13 -11 5]]
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.427
Optimal ET sequence: 7, 36, 43
Badness: 0.0368
Lithium
Lithium is named after the 3rd element for being period-3, and also for lithium's molar mass of 6.9 g/mol since 69edo supports it. It supports a 3L 6s scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.
Subgroup: 2.3.5.7
Comma list: 81/80, 3125/3087
Mapping: [⟨3 0 -12 -20], ⟨0 1 4 6]]
- mapping generators: ~56/45, ~3
Optimal tuning (CTE): ~56/45 = 1\3, ~3/2 = 695.827
Optimal ET sequence: 12, 33cd, 45, 57
Badness: 0.0692
Squares
- Main article: Squares
Squares splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. 31edo, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.
Subgroup: 2.3.5.7
Comma list: 81/80, 2401/2400
Mapping: [⟨1 3 8 6], ⟨0 -4 -16 -9]]
- mapping generators: ~2, ~9/7
Wedgie: ⟨⟨4 16 9 16 3 -24]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.942
- 7- and 9-odd-limit: ~9/7 = [1/2 0 -1/16⟩
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5
Algebraic generator: Sceptre2, the positive root of 9x2 + x - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
Optimal ET sequence: 14c, 17c, 31
Badness: 0.045993
Scales: skwares8, skwares11, skwares14
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 121/120
Mapping: [⟨1 3 8 6 7], ⟨0 -4 -16 -9 -10]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.957
Optimal ET sequence: 14c, 17c, 31
Badness: 0.021636
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 99/98, 121/120
Mapping: [⟨1 3 8 6 7 3], ⟨0 -4 -16 -9 -10 2]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.550
Optimal ET sequence: 14c, 17c, 31, 79cf
Badness: 0.025514
Squad
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 81/80, 91/90, 99/98
Mapping: [⟨1 3 8 6 7 9], ⟨0 -4 -16 -9 -10 -15]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.7516
Optimal ET sequence: 14cf, 17c, 31f
Badness: 0.026877
Agora
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 105/104, 121/120
Mapping: [⟨1 3 8 6 7 14], ⟨0 -4 -16 -9 -10 -29]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.276
Optimal ET sequence: 14cf, 31, 45ef, 76e
Badness: 0.024522
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
Mapping: [⟨1 3 8 6 7 14 8], ⟨0 -4 -16 -9 -10 -29 -11]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.187
Optimal ET sequence: 14cf, 31, 76e
Badness: 0.022573
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
Mapping: [⟨1 3 8 6 7 14 8 11], ⟨0 -4 -16 -9 -10 -29 -11 -19]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.225
Optimal ET sequence: 14cf, 31, 76e
Badness: 0.018839
Cuboctahedra
Subgroup: 2.3.5.7.11
Comma list: 81/80, 385/384, 1375/1372
Mapping: [⟨1 3 8 6 -4], ⟨0 -4 -16 -9 21]]
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.993
Optimal ET sequence: 14ce, 17ce, 31, 107b, 138b, 169be, 200be
Badness: 0.056826
Jerome
Jerome is related to Hieronymus' tuning; the Hieronymus generator is 51/20, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
Subgroup: 2.3.5.7
Comma list: 81/80, 17280/16807
Mapping: [⟨1 1 0 2], ⟨0 5 20 7]]
- mapping generators: ~2, ~54/49
Wedgie: ⟨⟨5 20 7 20 -3 -40]]
Optimal tuning (POTE): ~2 = 1\1, ~54/49 = 139.343
Optimal ET sequence: 17c, 26, 43, 69, 112bd
Badness: 0.108656
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 864/847
Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.428
Optimal ET sequence: 17c, 26, 43, 69
Badness: 0.047914
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 81/80, 99/98, 144/143
Mapping: [⟨1 1 0 2 3 3], ⟨0 5 20 7 4 6]]
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.387
Optimal ET sequence: 17c, 26, 43, 69
Badness: 0.029285
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
Mapping: [⟨1 1 0 2 3 3 2], ⟨0 5 20 7 4 6 18]]
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.362
Optimal ET sequence: 17cg, 26, 43, 69
Badness: 0.020878
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
Mapping: [⟨1 1 0 2 3 3 2 1], ⟨0 5 20 7 4 6 18 28]]
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.313
Optimal ET sequence: 17cgh, 26, 43, 69
Badness: 0.018229
Meantritone
The meantritone temperament tempers out the mirkwai comma (16875/16807) and trimyna comma (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus 15/11-wide super-fourth) and five of them equals ~16/3 (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.
Subgroup: 2.3.5.7
Comma list: 81/80, 16875/16807
Mapping: [⟨1 4 12 12], ⟨0 -5 -20 -19]]
Wedgie: ⟨⟨5 20 19 20 16 -12]]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.766
Optimal ET sequence: 2cd, 29cd, 31
Badness: 0.082239
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 2541/2500
Mapping: [⟨1 4 12 12 17], ⟨0 -5 -20 -19 -28]]
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.647
Optimal ET sequence: 2cde, 29cde, 31
Badness: 0.042869
Injera
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. 38EDO, which is two parallel 19edos, is an excellent tuning for injera.
Subgroup: 2.3.5.7
Comma list: 50/49, 81/80
Mapping: [⟨2 0 -8 -7], ⟨0 1 4 4]]
- mapping generators: ~7/5, ~3
Wedgie: ⟨⟨2 8 8 8 7 -4]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.375
- 7- and 9-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
- 7-odd-limit diamond tradeoff: ~3/2 = [688.957, 701.955]
- 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
Optimal ET sequence: 12, 26, 38, 102bcd, 140bccd, 178bbccdd
Badness: 0.031130
- Music
- Two Pairs of Socks (in 26EDO) by Igliashon Jones
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 50/49, 81/80
Mapping: [⟨2 0 -8 -7 -12], ⟨0 1 4 4 6]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.840
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
- 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
Optimal ET sequence: 12, 14c, 26, 90bce, 116bcce
Badness: 0.023124
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 50/49, 78/77, 81/80
Mapping: [⟨2 0 -8 -7 -12 -21], ⟨0 1 4 4 6 9]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.673
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
Optimal ET sequence: 12f, 14cf, 26, 38e
Badness: 0.021565
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 45/44, 50/49, 78/77, 81/80, 85/84
Mapping: [⟨2 0 -8 -7 -12 -21 5], ⟨0 1 4 4 6 9 1]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.487
Optimal ET sequence: 12f, 14cf, 26
Badness: 0.018358
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84
Mapping: [⟨2 0 -8 -7 -12 -21 5 -1], ⟨0 1 4 4 6 9 1 3]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.299
Optimal ET sequence: 12f, 14cf, 26
Badness: 0.015118
Enjera
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 40/39, 45/44, 50/49
Mapping: [⟨2 0 -8 -7 -12 -2], ⟨0 1 4 4 6 3]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.121
Optimal ET sequence: 12f, 14c, 26f, 38eff
Badness: 0.026542
Injerous
Subgroup: 2.3.5.7.11
Comma list: 33/32, 50/49, 55/54
Mapping: [⟨2 0 -8 -7 10], ⟨0 1 4 4 -1]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 690.548
Optimal ET sequence: 12e, 14c, 26e, 40cee
Badness: 0.038577
Lahoh
Subgroup: 2.3.5.7.11
Comma list: 50/49, 56/55, 81/77
Mapping: [⟨2 0 -8 -7 7], ⟨0 1 4 4 0]]
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 699.001
Optimal ET sequence: 2cd, 10cd, 12
Badness: 0.043062
Teff
- Main article: Teff
Teff (found by Mason Green) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.
Subgroup: 2.3.5.7.11
Comma list: 50/49, 81/80, 864/847
Mapping: [⟨2 1 -4 -3 8], ⟨0 2 8 8 -1]]
- mapping generators: ~7/5, ~16/11
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5303
Optimal ET sequence: 24d, 26, 50d
Badness: 0.070689
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 78/77, 81/80, 144/143
Mapping: [⟨2 1 -4 -3 8 2], ⟨0 2 8 8 -1 5]]
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5324
Optimal ET sequence: 24d, 26, 50d
Badness: 0.040047
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
Mapping: [⟨2 1 -4 -3 8 2 6], ⟨0 2 8 8 -1 5 2]]
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6558
Badness: 0.029499
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
Mapping: [⟨2 1 -4 -3 8 2 6 2], ⟨0 2 8 8 -1 5 2 6]]
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6382
Badness: 0.023133
Pombe
Pombe (named after the African millet beer) is a variant of #Teff by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.
Subgroup: 2.3.5.7
Comma list: 81/80, 300125/294912
Mapping: [⟨2 1 -4 11], ⟨0 2 8 -5]]
- mapping generators: ~735/512, ~35/24
Wedgie: ⟨⟨4 16 -10 16 -27 -68]]
Optimal tuning (POTE): ~735/512 = 1\2, ~48/35 = 552.2206
Optimal ET sequence: 24, 26, 50, 126bcd, 176bcdd, 226bbcdd
Badness: 0.116104
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 245/242, 385/384
Mapping: [⟨2 1 -4 11 8], ⟨0 2 8 -5 -1]]
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.0929
Optimal ET sequence: 24, 26, 50
Badness: 0.052099
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 144/143, 245/242
Mapping: [⟨2 1 -4 11 8 2], ⟨0 2 8 -5 -1 5]]
Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.1498
Optimal ET sequence: 24, 26, 50
Badness: 0.031039
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
Mapping: [⟨2 1 -4 11 8 2 6], ⟨0 2 8 -5 -1 5 2]]
Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1579
Optimal ET sequence: 24, 26, 50
Badness: 0.021260
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
Mapping: [⟨2 1 -4 11 8 2 6 2], ⟨0 2 8 -5 -1 5 2 6]]
Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1196
Optimal ET sequence: 24, 26, 50
Badness: 0.016548
Orphic
Subgroup: 2.3.5.7
Comma list: 81/80, 5898240/5764801
Mapping: [⟨2 5 12 7], ⟨0 -4 -16 -3]]
Mapping generators: ~2401/1728, ~7/6
Wedgie: ⟨⟨8 32 6 32 -13 -76]]
Optimal tuning (POTE): ~2401/1728 = 1\2, ~7/6 = 275.794
Optimal ET sequence: 26, 48c, 74, 174bd, 248bbd
Badness: 0.258825
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 73728/73205
Mapping: [⟨2 5 12 7 6], ⟨0 -4 -16 -3 2]]
Optimal tuning (POTE): ~363/256 = 1\2, ~7/6 = 275.762
Optimal ET sequence: 26, 48c, 74, 248bbd, 322bbdd
Badness: 0.101499
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 144/143, 2200/2197
Mapping: [⟨2 5 12 7 6 12], ⟨0 -4 -16 -3 2 -10]]
Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 275.774
Optimal ET sequence: 26, 48c, 74, 174bd, 248bbd, 322bbdd
Badness: 0.053482
Cloudtone
The cloudtone temperament (5&50) tempers out the cloudy comma, 16807/16384 and the syntonic comma, 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
Subgroup: 2.3.5.7
Comma list: 81/80, 16807/16384
Mapping: [⟨5 0 -20 14], ⟨0 1 4 0]]
- mapping generators: ~8/7, ~3
Wedgie: ⟨⟨5 20 0 20 -14 -56]]
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 695.720
Optimal ET sequence: 5, 45, 50
Badness: 0.102256
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 385/384, 2401/2376
Mapping: [⟨5 0 -20 14 41], ⟨0 1 4 0 -3]]
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.536
Optimal ET sequence: 5, 45, 50, 155bdd, 205bddd
Badness: 0.070378
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 144/143, 2401/2376
Mapping: [⟨5 0 -20 14 41 -21], ⟨0 1 4 0 -3 5]]
Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.162
Optimal ET sequence: 5, 45f, 50
Badness: 0.048829