Meantone family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The meantone family is the family of rank-2 temperaments that temper out the syntonic comma, 81/80, and thus can all be seen as extensions of meantone. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

Meantone

Main article: Meantone

Subgroup: 2.3.5

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

mapping generators: ~2, ~3

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 697.214

error map: ⟨0 -4.741 +2.544]

• POTE: ~2 = 1200.000, ~3/2 = 696.239

error map: ⟨0 -5.716 -1.359]

Minimax tuning:

• 5-odd-limit: ~3/2 = [0 0 1/4⟩ (1/4-comma)

unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

• 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.

• Flattertone adds [-24 17 0 -1⟩, finding the ~7/4 at the double-augmented sixth. It is well-tuned flat of 26edo.
• Flattone adds [-17 9 0 1⟩, finding the ~7/4 at the diminished seventh, It is well-tuned between 19edo and 26edo.
• Septimal meantone adds [-13 10 0 -1⟩, finding the ~7/4 at the augmented sixth (diesis-flat minor seventh). It is well-tuned between 12edo and 19edo.
• Dominant adds [6 -2 0 -1⟩, finding the ~7/4 at the minor seventh. It is well-tuned between 5edo and 12edo.
• Sharptone adds [2 -3 0 1⟩, finding the ~7/4 at the major sixth. It is "well-tuned" around 5edo; however it horribly mistunes 5/4, so it can be considered an exotemperament.

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 (+15/14) → Very low accuracy temperaments
• Godzilla (+49/48) → Semaphoresmic clan
• Mothra (+1029/1024) → Gamelismic clan
• Mohaha (+121/120) → 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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 696.952

error map: ⟨0 -5.003 +1.495 +0.695]

• POTE: ~2 = 1200.000, ~3/2 = 696.495

error map: ⟨0 -5.460 -0.334 -3.877]

Minimax tuning:

• 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⟩]

unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

• 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]]

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⟩]

unchanged-interval (eigenmonzo) 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⟩

unchanged-interval (eigenmonzo) 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⟩

unchanged-interval (eigenmonzo) 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

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⟩]

unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

• 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
• 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x³ + 6x - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Optimal ET sequence: 12e, 19, 31, 81, 112b

Badness: 0.021543

Music

• Scott Joplin's "The Entertainer" tuned into meanpop^{[dead link]}
• Twinkle canon – 50 edo by Claudi Meneghin

Tridecimal meanpop

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 24 -20], ⟨0 1 4 10 -13 15]]

Optimal tunings:

• CTE: ~2 = 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⟩

unchanged-interval (eigenmonzo) 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]]

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⟩

unchanged-interval (eigenmonzo) 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]]

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]]

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

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

See also: No-sevens subgroup temperaments § Superpine

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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 693.552

error map: ⟨0.000 -8.403 -12.106 -10.794]

• POTE: ~2 = 1200.000, ~3/2 = 693.779

error map: ⟨0.000 -8.176 -11.197 -12.838]

Minimax tuning:

• 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⟩]

unchanged-interval (eigenmonzo) 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⟩]

unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

• 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

This can also be considered a no-sevens temperament: hypnotone.

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

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 692.698

error map: ⟨0.000 -9.257 -15.520 +7.047]

• CWE: ~2 = 1200.000, ~3/2 = 692.048

error map: ⟨0.000 -9.907 -18.122 -4.012]

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 tunings:

• 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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 699.622

error map: ⟨0.000 -2.333 +12.173 +31.931]

• POTE: ~2 = 1200.000, ~3/2 = 701.573

error map: ⟨0.000 -0.382 +19.979 +28.028]

Tuning ranges:

• 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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 703.732

error map: ⟨0.000 +1.777 +28.614 -57.630]

• POTE: ~2 = 1200.000, ~3/2 = 700.140

error map: ⟨0.000 -1.815 +14.245 -68.407]

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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 703.811

error map: ⟨0.000 +1.856 +28.929 -11.665]

• POTE: ~2 = 1200.000, ~3/2 = 704.889

error map: ⟨0.000 +2.934 +33.242 +4.507]

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 half a chroma flat of 16/9 to 7/4 so that it's mapped to a semidiminished seventh, where a chroma is equated to two quarter-tones, 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

Optimal tunings:

• CTE: ~2 = 1200.000, ~128/105 = 348.437

error map: ⟨0.000 -5.080 +1.186 -1.637]

• POTE: ~2 = 1200.000, ~128/105 = 348.415

error map: ⟨0.000 -5.125 +1.005 -1.390]

Minimax tuning:

• 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⟩]

Unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:

• 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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~11/9 = 348.561
• 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⟩]

unchanged-interval (eigenmonzo) 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 tunings:

• CTE: ~2 = 1200.000, ~11/9 = 348.587
• 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 tunings:

• CTE: ~2 = 1200.000, ~11/9 = 348.735
• POTE: ~2 = 1200.000, ~11/9 = 348.736

Optimal ET sequence: 7, 24, 31

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 tunings:

• CTE: ~2 = 1200.000, ~11/9 = 348.821
• POTE: ~2 = 1200.000, ~11/9 = 348.810

Optimal ET sequence: 7, 24, 31, 55

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 tunings:

• CTE: ~2 = 1200.000, ~25/21 = 350.352

error map: ⟨0.000 -1.250 +16.505 -14.245]

• POTE: ~2 = 1200.000, ~25/21 = 350.586

error map: ⟨0.000 -0.784 +18.370 -11.214]

Optimal ET sequence: 7d, 17c, 24

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 tunings:

• CTE: ~2 = 1200.000, ~11/9 = 350.347
• POTE: ~2 = 1200.000, ~11/9 = 350.565

Optimal ET sequence: 7d, 17c, 24

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 tunings:

• CTE: ~2 = 1200.000, ~11/9 = 350.365
• POTE: ~2 = 1200.000, ~11/9 = 350.745

Optimal ET sequence: 7d, 17c, 24, 41c

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. Liese is a very natural 13-limit tuning, given the generator is so near 13/9. 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

Optimal tunings:

• CTE: ~2 = 1200.000, ~10/7 = 632.783

error map: ⟨0.000 -3.606 +7.084 -8.212]

• POTE: ~2 = 1200.000, ~10/7 = 632.406

error map: ⟨0.000 -4.738 +2.554 -12.363]

Minimax tuning:

• 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⟩]

unchanged-interval (eigenmonzo) basis: 2.5

Algebraic generator: Radix, the real root of x⁵ - 2x⁴ + 2x³ - 2x² + 2x - 2, also a root of x⁶ - x⁵ - 2. The recurrence converges.

Optimal ET sequence: 17c, 19, 55, 74d

Badness: 0.046706

Liesel

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 540/539

Mapping: [⟨1 0 -4 -3 4], ⟨0 3 12 11 -1]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~10/7 = 632.812
• POTE: ~2 = 1200.000, ~10/7 = 633.073

Optimal ET sequence: 17c, 19, 36

Badness: 0.040721

13-limit

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 tunings:

• CTE: ~2 = 1200.000, ~10/7 = 632.925
• 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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~10/7 = 633.317
• 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 tunings:

• CTE: ~2 = 1200.000, ~10/7 = 633.370
• 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 tunings:

• CTE: ~2 = 1200.000, ~10/7 = 631.981
• 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 tunings:

• CTE: ~2 = 1200.000, ~10/7 = 632.093
• POTE: ~2 = 1200.000, ~10/7 = 631.221

Optimal ET sequence: 17cee, 19

Badness: 0.036144

Superpine

See also: No-sevens subgroup temperaments § 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 tunings:

• CTE: ~2 = 1200.000, ~35/32 = 167.279

error map: ⟨0.000 -3.793 +6.336 +5.804]

• CWE: ~2 = 1200.000, ~35/32 = 167.256

error map: ⟨0.000 -3.723 +6.613 +5.503]

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 tunings:

• CTE: ~2 = 1200.000, ~11/10 = 167.407
• CWE: ~2 = 1200.000, ~11/10 = 167.338

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 tunings:

• CTE: ~2 = 1200.000, ~11/10 = 167.427
• CWE: ~2 = 1200.000, ~11/10 = 167.396

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 tunings:

• CTE: ~56/45 = 400.000, ~3/2 = 695.827 (~15/14 = 104.173)

error map: ⟨0.000 -6.128 -3.007 +6.135]

• CWE: ~56/45 = 400.000, ~3/2 = 695.141 {~15/14 = 104.859)

error map: ⟨0.000 -6.814 -5.748 +2.022]

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

Optimal tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.695

error map: ⟨0.000 -4.734 +2.570 -0.079]

• POTE: ~2 = 1200.000, ~9/7 = 425.942

error map: ⟨0.000 -5.721 -1.378 -2.300]

Minimax tuning:

• 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⟩]

unchanged-interval (eigenmonzo) 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, 169b, 200b

Badness: 0.045993

Scales: skwares8, skwares11, skwares14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 121/120

Mapping: [⟨1 3 8 6 7], ⟨0 -4 -16 -9 -10]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.599
• POTE: ~2 = 1200.000, ~9/7 = 425.957

Optimal ET sequence: 14c, 17c, 31, 130bee, 169beee

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 tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.578
• 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 tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.297
• POTE: ~2 = 1200.000, ~9/7 = 425.752

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 tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.870
• 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 tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.919
• POTE: ~2 = 1200.000, ~9/7 = 426.187

Optimal ET sequence: 14cf, 31

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 tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.984
• POTE: ~2 = 1200.000, ~9/7 = 426.225

Optimal ET sequence: 14cf, 31

Badness: 0.018839

Cuboctahedra

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 1375/1372

Mapping: [⟨1 3 8 6 -4], ⟨0 -4 -16 -9 21]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~9/7 = 425.899
• POTE: ~2 = 1200.000, ~9/7 = 425.993

Optimal ET sequence: 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

Optimal tunings:

• CTE: ~2 = 1200.000, ~54/49 = 139.371

error map: ⟨0.000 -5.098 +1.114 +6.774]

• POTE: ~2 = 1200.000, ~54/49 = 139.343

error map: ⟨0.000 -5.238 +0.553 +6.577]

Optimal ET sequence: 17c, 26, 43

Badness: 0.108656

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 864/847

Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~12/11 = 139.349
• POTE: ~2 = 1200.000, ~12/11 = 139.428

Optimal ET sequence: 17c, 26, 43

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 tunings:

• CTE: ~2 = 1200.000, ~13/12 = 139.369
• POTE: ~2 = 1200.000, ~13/12 = 139.387

Optimal ET sequence: 17c, 26, 43

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 tunings:

• CTE: ~2 = 1200.000, ~13/12 = 139.334
• POTE: ~2 = 1200.000, ~13/12 = 139.362

Optimal ET sequence: 17cg, 26, 43

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 tunings:

• CTE: ~2 = 1200.000, ~13/12 = 139.296
• POTE: ~2 = 1200.000, ~13/12 = 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]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~7/5 = 580.568

error map: ⟨0.000 -4.795 +2.325 +0.381]

• POTE: ~2 = 1200.000, ~7/5 = 580.766

error map: ⟨0.000 -5.785 -1.634 -3.380]

Optimal ET sequence: 29cd, 31, 188bcd, 219bbcd

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 tunings:

• CTE: ~2 = 1200.000, ~7/5 = 580.482
• POTE: ~2 = 1200.000, ~7/5 = 580.647

Optimal ET sequence: 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

Optimal tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 695.330 (~21/20 = 95.330)

error map: ⟨0.000 -6.625 -4.993 +12.495]

• POTE: ~7/5 = 600.000, ~3/2 = 694.375 (~21/20 = 94.375)

error map: ⟨0.000 -7.580 -8.813 +8.675]

Tuning ranges:

• 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

Badness: 0.031130

Music

• Two Pairs of Socks by Igliashon Jones – in 26edo tuning

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 81/80

Mapping: [⟨2 0 -8 -7 -12], ⟨0 1 4 4 6]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 694.098 (~21/20 = 94.098)
• POTE: ~7/5 = 600.000, ~3/2 = 692.840 (~21/20 = 92.840)

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, 26

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 tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 693.806 (~21/20 = 93.806)
• POTE: ~7/5 = 600.000, ~3/2 = 692.673 (~21/20 = 92.673)

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

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 tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 693.852 (~21/20 = 93.852)
• POTE: ~7/5 = 600.000, ~3/2 = 692.487 (~21/20 = 92.487)

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 tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 694.031 (~21/20 = 94.031)
• POTE: ~7/5 = 600.000, ~3/2 = 692.299 (~21/20 = 92.299)

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 tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 693.088 (~21/20 = 93.088)
• POTE: ~7/5 = 600.000, ~3/2 = 694.121 (~21/20 = 94.121)

Optimal ET sequence: 10cdeef, 12f

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 tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 694.619 (~21/20 = 94.619)
• POTE: ~7/5 = 600.000, ~3/2 = 690.548 (~21/20 = 90.548)

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 tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 695.330 (~21/20 = 95.330)
• POTE: ~7/5 = 600.000, ~3/2 = 699.001 (~21/20 = 99.001)

Optimal ET sequence: 10cd, 12

Badness: 0.043062

Teff

Main article: Teff

Teff, found and named 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 tunings:

• CTE: ~7/5 = 600.000, ~11/8 = 552.331 (~33/32 = 47.669)
• POTE: ~7/5 = 600.000, ~11/8 = 552.530 (~33/32 = 47.470)

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 tunings:

• CTE: ~7/5 = 600.000, ~11/8 = 552.297 (~33/32 = 47.703)
• POTE: ~7/5 = 600.000, ~11/8 = 552.532 (~33/32 = 47.468)

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 tunings:

• CTE: ~7/5 = 600.000, ~11/8 = 552.249 (~34/33 = 47.751)
• POTE: ~7/5 = 600.000, ~11/8 = 552.656 (~34/33 = 47.344)

Optimal ET sequence: 24d, 26

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 tunings:

• CTE: ~7/5 = 600.000, ~11/8 = 552.107 (~34/33 = 47.893)
• POTE: ~7/5 = 600.000, ~11/8 = 552.638 (~34/33 = 47.362)

Optimal ET sequence: 24d, 26

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

Optimal tunings:

• CTE: ~735/512 = 600.000, ~48/35 = 551.845 (~36/35 = 48.155)

error map: ⟨0.000 -5.645 -1.075 -9.600]

• POTE: ~735/512 = 600.000, ~48/35 = 552.221 (~36/35 = 48.779)

error map: ⟨0.000 -6.229 -3.411 -8.140]

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 tunings:

• CTE: ~99/70 = 600.000, ~11/8 = 551.843 (~36/35 = 48.157)
• POTE: ~99/70 = 600.000, ~11/8 = 552.093 (~36/35 = 47.907)

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 tunings:

• CTE: ~99/70 = 600.000, ~11/8 = 551.848 (~36/35 = 48.152)
• POTE: ~99/70 = 600.000, ~11/8 = 552.150 (~36/35 = 47.850)

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 tunings:

• CTE: ~17/12 = 600.000, ~11/8 = 551.793 (~36/35 = 48.207)
• POTE: ~17/12 = 600.000, ~11/8 = 552.158 (~36/35 = 47.842)

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 tunings:

• CTE: ~17/12 = 600.000, ~11/8 = 551.660 (~36/35 = 48.340)
• POTE: ~17/12 = 600.000, ~11/8 = 552.120 (~36/35 = 47.880)

Optimal ET sequence: 24, 26, 50

Badness: 0.016548

Orphic

Subgroup: 2.3.5.7

Comma list: 81/80, 5898240/5764801

Mapping: [⟨2 1 -4 4], ⟨0 4 16 3]]

mapping generators: ~2401/1728, ~343/288

Optimal tunings:

• CTE: ~2401/1728 = 600.000, ~343/288 = 324.275 (~7/6 = 275.725)

error map: ⟨0.000 -4.854 +2.091 +4.000]

• POTE: ~2401/1728 = 600.000, ~343/288 = 324.206 (~7/6 = 275.794)

error map: ⟨0.000 -5.131 +0.983 +3.792]

Optimal ET sequence: 26, 48c, 74

Badness: 0.258825

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 73728/73205

Mapping: [⟨2 1 -4 4 8], ⟨0 -4 -16 -3 2]]

Optimal tunings:

• CTE: ~363/256 = 600.000, ~77/64 = 324.276 (~7/6 = 275.724)
• POTE: ~363/256 = 600.000, ~77/64 = 324.238 (~7/6 = 275.762)

Optimal ET sequence: 26, 48c, 74

Badness: 0.101499

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 144/143, 2200/2197

Mapping: [⟨2 1 -4 4 8 2], ⟨0 -4 -16 -3 2 -10]]

Optimal tunings:

• CTE: ~55/39 = 600.000, ~77/64 = 324.250 (~7/6 = 275.750)
• POTE: ~55/39 = 600.000, ~77/64 = 324.226 (~7/6 = 275.774)

Optimal ET sequence: 26, 48c, 74

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

Optimal tunings:

• CTE: ~8/7 = 240.000, ~3/2 = 697.214 (~49/48 = 22.786)

error map: ⟨0.000 -4.741 +2.544 -8.826]

• POTE: ~8/7 = 240.000, ~3/2 = 695.720 (~49/48 = 24.280)

error map: ⟨0.000 -6.235 -3.435 -8.826]

Optimal ET sequence: 5, 40c, 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 tunings:

• CTE: ~8/7 = 240.000, ~3/2 = 697.034 (~56/55 = 22.966)
• POTE: ~8/7 = 240.000, ~3/2 = 696.536 (~56/55 = 23.464)

Optimal ET sequence: 5, 45, 50

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 tunings:

• CTE: ~8/7 = 240.000, ~3/2 = 696.749 (~91/90 = 23.251)
• POTE: ~8/7 = 240.000, ~3/2 = 696.162 (~91/90 = 23.838)

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

Sval mapping: [⟨1 0 -4 9], ⟨0 1 4 -3]]

sval mapping generators: ~2, ~3

Gencom mapping: [⟨1 0 -4 0 0 0 0 9], ⟨0 1 4 0 0 0 0 -3]]

gencom: [2 3; 81/80 96/95]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 697.681
• POTE: ~2 = 1200.000, ~3/2 = 697.867

Optimal ET sequence: 5, 7, 12, 31, 43

RMS error: 1.378 cents

Hypnotone

Hypnotone is no-sevens flattone.

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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 694.700
• CWE: ~2 = 1200.000, ~3/2 = 693.908

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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 693.951
• CTE: ~2 = 1200.000, ~3/2 = 693.087

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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 696.039
• CWE: ~2 = 1200.000, ~3/2 = 693.121

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 tunings:

• CTE: ~2 = 1200.000, ~3/2 = 689.699
• CWE: ~2 = 1200.000, ~3/2 = 690.183

Optimal ET sequence: 7, 19eff, 26eff, 33ceeff, 40ceeff

Badness: 0.0353