Meantone family

CTE: ~2 = 1200.000, ~3/2 = 697.214
POTE: ~2 = 1200.000, ~3/2 = 696.239

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 out 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 out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a patent val tuning (corresponding to also tempering out 225/224).

Temperaments discussed elsewhere include

Plutus → Very low accuracy temperaments
Godzilla → Slendro clan
Mothra → Gamelismic clan
Mohaha → Rastmic clan

The rest are considered below.

Septimal meantone

Deutsch

Main article: Meantone § Septimal meantone

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Septimal meantone temperament

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 ]]

CTE: ~2 = 1200.000, ~3/2 = 696.952
POTE: ~2 = 1200.000, ~3/2 = 696.495

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⟩]

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 a 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 = 1200.000, ~3/2 = 697.168
POTE: ~2 = 1200.000, ~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

Twinkle canon – 74 edo by Claudi Meneghin

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 = 1200.000, ~3/2 = 696.855
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.649
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.555
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.036
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.216
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.908
POTE: ~2 = 1200.000, ~3/2 = 697.003

Optimal ET sequence: 12f, 31

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 = 1200.000, ~3/2 = 696.931
POTE: ~2 = 1200.000, ~3/2 = 697.140

Optimal ET sequence: 12f, 31

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 = 1200.000, ~3/2 = 697.258
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.300
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.327
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.516
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.508
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.485
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.610
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.701
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.538
POTE: ~2 = 1200.000, ~3/2 = 697.644

Optimal ET sequence: 12f, 43

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 = 1200.000, ~3/2 = 697.555
POTE: ~2 = 1200.000, ~3/2 = 697.715

Optimal ET sequence: 12f, 43

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 = 1200.000, ~26/15 = 948.611
POTE: ~2 = 1200.000, ~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 = 1200.000, ~26/15 = 948.617
POTE: ~2 = 1200.000, ~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 = 1200.000, ~19/11 = 948.609
POTE: ~2 = 1200.000, ~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 = 600.000, ~3/2 = 697.168
POTE: ~55/39 = 600.000, ~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 = 600.000, ~3/2 = 697.174
POTE: ~17/12 = 600.000, ~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 = 600.000, ~3/2 = 697.187
POTE: ~17/12 = 600.000, ~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 a 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 = 1200.000, ~3/2 = 696.531
POTE: ~2 = 1200.000, ~3/2 = 696.434

Minimax tuning:

11-odd-limit: ~3/2 = [0 0 1/4⟩

projection map: [[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

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 = 1200.000, ~3/2 = 696.356
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.351
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.347
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.439
POTE: ~2 = 1200.000, ~3/2 = 696.408

Optimal ET sequence: 19, 31

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 = 1200.000, ~3/2 = 696.484
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.283
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.407
POTE: ~2 = 1200.000, ~3/2 = 696.414

Optimal ET sequence: 12e, 19

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 = 1200.000, ~3/2 = 696.473
POTE: ~2 = 1200.000, ~3/2 = 696.497

Optimal ET sequence: 12e, 19

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 = 1200.000, ~3/2 = 696.661
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 697.016
POTE: ~2 = 1200.000, ~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 a 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 = 1200.000, ~3/2 = 696.153
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.098
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.216
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.277
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.450
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.793
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 695.790
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.011
POTE: ~2 = 1200.000, ~3/2 = 695.858

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 696.120
POTE: ~2 = 1200.000, ~3/2 = 696.131

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 696.059
POTE: ~2 = 1200.000, ~3/2 = 696.044

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 695.982
POTE: ~2 = 1200.000, ~3/2 = 695.913

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 695.798
POTE: ~2 = 1200.000, ~3/2 = 695.750

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 695.675
POTE: ~2 = 1200.000, ~3/2 = 695.603

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 695.724
POTE: ~2 = 1200.000, ~3/2 = 695.696

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 695.716
POTE: ~2 = 1200.000, ~3/2 = 695.688

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 695.685
POTE: ~2 = 1200.000, ~3/2 = 695.676

Optimal ET sequence: 12, 19

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 = 1200.000, ~3/2 = 696.412
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.950
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 695.620
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.279
POTE: ~2 = 1200.000, ~3/2 = 697.586

Optimal ET sequence: 7dg, 12f

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 = 1200.000, ~3/2 = 696.849
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.702
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 696.241
POTE: ~2 = 1200.000, ~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 = 600.000, ~3/2 = 696.520
POTE: ~63/44 = 600.000, ~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 = 600.000, ~3/2 = 696.341
POTE: ~55/39 = 600.000, ~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 = 600.000, ~3/2 = 696.353
POTE: ~17/12 = 600.000, ~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 = 600.000, ~3/2 = 696.384
POTE: ~17/12 = 600.000, ~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 = 1200.000, ~11/10 = 167.707
POTE: ~2 = 1200.000, ~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 = 1200.000, ~11/10 = 167.712
POTE: ~2 = 1200.000, ~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 = 1200.000, ~11/10 = 167.705
POTE: ~2 = 1200.000, ~11/10 = 167.786

Optimal ET sequence: 7dg, 43, 50, 93

Badness: 0.025221

Flattone

Main article: 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 double-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 ]]

CTE: ~2 = 1200.000, ~3/2 = 693.552
POTE: ~2 = 1200.000, ~3/2 = 693.779

7-odd-limit: ~3/2 = [8/13 0 1/13 -1/13⟩

projection map: [[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⟩

projection map: [[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 = 1200.000, ~3/2 = 693.251
POTE: ~2 = 1200.000, ~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 = 1200.000, ~3/2 = 693.029
POTE: ~2 = 1200.000, ~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

CTE: ~2 = 1200.000, ~3/2 = 692.698
CWE: ~2 = 1200.000, ~3/2 = 692.0479

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

CTE: ~2 = 1200.000, ~3/2 = 692.642
CWE: ~2 = 1200.000, ~3/2 = 692.042

Optimal ET sequence: 7d, 19d, 26, 59bcd, 85bccd

Music

Music in 33EDO (33-Tone Equal Temperament) - Feb 2024 - Budjarn Lambeth (2024)

Dominant

Main article: Dominant (temperament)

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.

Because dominant entails a near-pure perfect fifth, a small number of generators will not land on an interval close to prime 11. The canonical 11-limit extension takes the tritone as 16/11, which it barely sounds like. The first alternative, domineering, takes the same step as 11/8, which it barely sounds like either. Domination tempers out 77/75 and identifies 11/8 with the augmented third; arnold tempers out 33/32 and identifies 11/8 with the perfect fourth. None of them are nearly as good as the weak extension neutrominant, splitting the fifth as well as the chromatic semitone in two like in all rastmic temperaments.

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 ]]

CTE: ~2 = 1200.000, ~3/2 = 699.622
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 703.334
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 704.847
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 704.034
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 696.240
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 695.315
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 695.894
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 696.139
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 694.840
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 703.268
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 703.719
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 698.546
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 695.929
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 696.683
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 696.996
POTE: ~2 = 1200.000, ~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 ]]

CTE: ~2 = 1200.000, ~3/2 = 703.732
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 702.730
POTE: ~2 = 1200.000, ~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]]

CTE: ~2 = 1200.000, ~3/2 = 703.811
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 704.016
POTE: ~2 = 1200.000, ~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 tunings:

CTE: ~2 = 1200.000, ~3/2 = 704.121
POTE: ~2 = 1200.000, ~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 = 1200.000, ~128/105 = 348.415

7- and 9-odd-limit: ~128/105 = [0 0 1/8⟩

projection map: [[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

Scales: mohaha7, mohaha10

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 = 1200.000, ~11/9 = 348.477

Minimax tuning:

11-odd-limit: ~11/9 = [0 0 1/8⟩

projection map: [[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

Scales: mohaha7, mohaha10

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 = 1200.000, ~11/9 = 348.558

Optimal ET sequence: 7, 24, 31

Badness: 0.023388

Scales: mohaha7, mohaha10

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 = 1200.000, ~11/9 = 348.736

Optimal ET sequence: 7, 24, 31, 86ef

Badness: 0.020576

Scales: mohaha7, mohaha10

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 = 1200.000, ~11/9 = 348.810

Optimal ET sequence: 7, 24, 31, 55, 86efh

Badness: 0.017302

Scales: mohaha7, mohaha10

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 = 1200.000, ~25/21 = 350.586

Optimal ET sequence: 7d, 17c, 24, 65cc, 89ccd

Badness: 0.077734

Scales: mohaha7, mohaha10

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 = 1200.000, ~11/9 = 350.565

Optimal ET sequence: 7d, 17c, 24, 65cc, 89ccd

Badness: 0.036207

Scales: mohaha7, mohaha10

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 = 1200.000, ~11/9 = 350.745

Optimal ET sequence: 7d, 17c, 24, 41c, 65cc

Badness: 0.028738

Scales: mohaha7, mohaha10

Liese

Deutsch

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 scales: 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 = 1200.000, ~10/7 = 632.406

7- and 9-odd-limit: ~10/7 = [1/3 0 1/12⟩

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]

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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~11/10 = 167.427

Optimal ET sequence: 7, 36, 43

Badness: 0.0368

Lithium

Lithium is named after the 3rd element for having a 3rd-octave period, 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 = 400.000, ~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 scales 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 = 1200.000, ~9/7 = 425.942

7- and 9-odd-limit: ~9/7 = [1/2 0 -1/16⟩

projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]

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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 5^1/20, 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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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 = 1200.000, ~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.

Origin of the name

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 = 600.000, ~3/2 = 694.375