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
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
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,
- Flattertone adds [-24 17 0 -1⟩, finding the ~7/4 at the double-augmented sixth,
- 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.
Strong extensions
For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)3 as 7/5, leading to septimal meantone, a very elegant extension to the 7-limit.
For any tuning flatter than 11\19, the augmented sixth and diminished seventh swap their orders, so the diminished seventh becomes a better approximation to the interval class of 7, resulting in flattone. Likewise, for any tuning sharper than 7\12, the minor seventh is the proper approximation instead, resulting in dominant.
Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in weak extensions. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into n parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason.
Splitting the meantone fifth into two (243/242)
By tempering out 243/242 we equate the distance from 9/8 to 10/9 (= S9) with the distance between 11/10 to 12/11 (= S11), leading to mohaha which is in some sense thus a trivial tuning of rastmic (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone rastmic temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full 11-limit by finding 7/4 as the semi-diminished seventh, leading to mohajira, which inflates 64/63 to equate it with a small quarter-tone, which is characteristic. Mohajira can also be thought of as equating a slightly sharpened (5/4)2 with 11/7, which is also natural as meantone tempering usually has 5/4 slightly sharp. There is also the consideration that tempering out 121/120 leads to similarly high damage in the 11-limit as tempering 81/80 in the 5-limit, because both erase key distinctions of their respective JI subgroups.
Splitting the meantone fifth into three (1029/1024)
By tempering 1029/1024 we equate the distance from 7/6 to 8/7 (= S7) with the distance from 8/7 to 9/8 (= S8), so that (8/7)3 is equated with 3/2, because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of ultraparticular commas. This is an unusually natural extension, with a surprising coincidence: (36/35)/(64/63) = 81/80, or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want 49/48 (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)3 = 1728/1715 (S6/S7), the orwellisma.
This strategy leads to the 7-limit version of mothra, which is also sometimes called cynder, though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6~S7~S8 with S9 tempered out, we can try S8~S10 by tempering out 176/175 (S8/S10), which is (11/7)/(5/4)2]], taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, (6/5)2 = 36/25 = (3/2)/(25/24).
31edo as splitting the fifth into two, three and nine
31edo is unique as combining all aforementioned tempering strategies into one elegant 11-limit meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate 5/4 and 7/4 and an even more accurate 35/32. A tempering strategy not mentioned is splitting a flattened 3/2 into nine sharpened 25/24's, resulting in the 5-limit version of valentine so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering 225/224, which interestingly, though a rank 2 temperament, only has 31edo as a patent val tuning (corresponding to also tempering 225/224).
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
![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/63/Wikipedia-logo.png/44px-Wikipedia-logo.png)
In septimal meantone, ten 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]]
Wedgie: ⟨⟨1 4 10 18 4 13 25 12 28 16]]
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
Wedgie: ⟨⟨1 4 10 -13 4 13 -24 12 -44 -71]]
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]]
Wedgie: ⟨⟨1 4 10 -13 15 4 13 -24 20 12 -44 20 -71 5 100]]
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]]
Wedgie: ⟨⟨1 4 10 -13 -4 4 13 -24 -10 12 -44 -24 -71 -48 34]]
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]]
Wedgie: ⟨⟨1 4 10 6 4 13 6 12 0 -18]]
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]]
Wedgie: ⟨⟨1 4 10 6 15 4 13 6 20 12 0 20 -18 5 30]]
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-diminished 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 45edo, 64edo, and 71edo.
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]]
- 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
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:
- CTE: ~2 = 1\1, ~3/2 = 693.2511
- 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
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 tunings:
- CTE: ~2 = 1\1, ~3/2 = 693.0293
- 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
Flattertone
Flattertone tunings are typically at least as flat as 26edo. Here, 17 fifths get to the interval class for 7, so that 7/4 is a double-augmented sixth (C-Ax). 26edo and 33cd-edo are the two primary flattertone tunings. 1/2-comma meantone is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a 16/13 or 27/22), and deeptone temperament's mapping is more logical.
Subgroup: 2.3.5.7
Comma list: 81/80, 1875/1792
Mapping: [⟨1 0 -4 -24], ⟨0 1 4 17]]
- mapping generators: ~2, ~3
Optimal ET sequence: 7d, 19d, 26, 59bcd, 85bccd
Badness: 0.0961
11-limit
Subgroup: 2.3.5.7
Comma list: 45/44, 81/80, 1375/1344
Mapping: [⟨1 0 -4 -24 0], ⟨0 1 4 17 6]]
- mapping generators: ~2, ~3
Optimal ET sequence: 7d, 19d, 26, 59bcd, 85bccd
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
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]]
Wedgie: ⟨⟨2 8 -11 5 8 -23 1 -48 -16 52]]
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]]
Wedgie: ⟨⟨3 12 11 -1 12 9 -12 -8 -44 -41]]
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]]
Wedgie: ⟨⟨3 12 11 16 12 9 15 -8 -4 7]]
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
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]]
Wedgie: ⟨⟨4 16 9 10 16 3 2 -24 -32 -3]]
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]]
Wedgie: ⟨⟨4 16 9 -21 16 3 -47 -24 -104 -90]]
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]]
Wedgie: ⟨⟨5 20 7 4 20 -3 -11 -40 -60 -13]]
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]]
Wedgie: ⟨⟨2 8 8 12 8 7 12 -4 0 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
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
Subgroup extensions
Stützel (2.3.5.19)
Subgroup: 2.3.5.19
Comma list: 81/80, 96/95
Gencom: [2 4/3; 81/80 96/95]
Gencom mapping: [⟨1 2 4 0 0 0 0 3], ⟨0 -1 -4 0 0 0 0 3]]
Sval mapping: [⟨1 2 4 3], ⟨0 -1 -4 3]]
POL2 generator: ~3/2 = 697.867
Optimal ET sequence: 5, 7, 12, 31, 43
RMS error: 1.378 cents
Hypnotone
Subgroup: 2.3.5.11
Comma list: 45/44, 81/80
Sval mapping: [⟨1 0 -4 -6], ⟨0 1 4 6]]
- sval mapping generators: ~2, ~3
Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 694.6998
Optimal ET sequence: 7, 12, 19, 26, 45
Badness: 0.0104
2.3.5.11.13 subgroup
Subgroup: 2.3.5.11.13
Comma list: 45/44, 65/64, 81/80
Sval mapping: [⟨1 0 -4 -6 10], ⟨0 1 4 6 -4]]
- sval mapping generators: ~2, ~3
Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 693.9513
Optimal ET sequence: 7, 12, 19, 26, 45f
Badness: 0.0141
Dequarter
Subgroup: 2.3.5.11
Comma list: 33/32, 55/54
Sval mapping: [⟨1 0 -4 5], ⟨0 1 4 -1]]
- sval mapping generators: ~2, ~3
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0387
Optimal ET sequence: 5, 7, 19e, 26e
Badness: 0.0145
Dreamtone
Subgroup: 2.3.5.11.13
Comma list: 33/32, 55/54, 975/968
Sval mapping: [⟨1 0 -4 5 21], ⟨0 1 4 -1 -11]]
- sval mapping generators: ~2, ~3
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 689.6993
Optimal ET sequence: 7, 19eff, 26eff, 33ceeff, 40ceeff
Badness: 0.0353