# Meantone family

The 5-limit parent comma of the **meantone family** is the syntonic comma, 81/80. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

## Meantone

Subgroup: 2.3.5

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

- mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 4]]

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

- 5-odd-limit diamond monotone: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
- 5-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)

Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b

Badness: 0.007381

### Extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

- Septimal meantone adds [-13 10 0 -1⟩, finding the ~7/4 at the augmented sixth,
- Flattone adds [-17 9 0 1⟩, finding the ~7/4 at the diminished seventh,
- Dominant adds [6 -2 0 -1⟩, finding the ~7/4 at the minor seventh,
- Flattertone adds [-24 17 0 -1⟩, finding the ~7/4 at the double-diminished seventh,
- Sharptone adds [2 -3 0 1⟩, finding the ~7/4 at the major sixth,

Those all have a fifth as generator.

- Injera adds [-7 8 0 -2⟩ with a half-octave period.
- Mohajira adds [-23 11 0 2⟩ and splits the fifth in two.
- Godzilla adds [-4 -1 0 2⟩ with an ~8/7 generator, two of which give the fourth.
- Mothra adds [-10 1 0 3⟩ with an ~8/7 generator, three of which give the fifth.
- Liese adds [-9 11 0 -3⟩ with a ~10/7 generator, three of which give the twelfth.
- Squares adds [-3 9 0 -4⟩ with a ~9/7 generator, four of which give the eleventh.
- Jerome adds [3 7 0 -5⟩ and slices the fifth in five.

#### Strong extensions

For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)^{3} as 7/5, leading to septimal meantone, a very elegant extension to the 7-limit.

For any tuning flatter than 11\19, the augmented sixth and diminished seventh swap their orders, so the diminished seventh becomes a better approximation to the interval class of 7, resulting in flattone. Likewise, for any tuning sharper than 7\12, the minor seventh is the proper approximation instead, resulting in dominant.

Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in weak extensions. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into *n* parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason.

#### Splitting the meantone fifth into two (243/242)

By tempering out 243/242 we equate the distance from 9/8 to 10/9 (= S9) with the distance between 11/10 to 12/11 (= S11), leading to mohaha which is in some sense thus a trivial tuning of rastmic (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone rastmic temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full 11-limit by finding 7/4 as the semi-diminished seventh, leading to mohajira, which inflates 64/63 to equate it with a small quarter-tone, which is characteristic. Mohajira can also be thought of as equating a slightly sharpened (5/4)^{2} with 11/7, which is also natural as meantone tempering usually has 5/4 slightly sharp. There is also the consideration that tempering out 121/120 leads to similarly high damage in the 11-limit as tempering 81/80 in the 5-limit, because both erase key distinctions of their respective JI subgroups.

#### Splitting the meantone fifth into three (1029/1024)

By tempering 1029/1024 we equate the distance from 7/6 to 8/7 (= S7) with the distance from 8/7 to 9/8 (= S8), so that (8/7)^{3} is equated with 3/2, because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of ultraparticular commas. This is an unusually natural extension, with a surprising coincidence: (36/35)/(64/63) = 81/80, or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want 49/48 (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)^{3} = 1728/1715 (S6/S7), the orwellisma.

This strategy leads to the 7-limit version of mothra, which is also sometimes called cynder, though confusingly cynder has a different mapping for 11 in the 11-limit. Though mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6~S7~S8 with S9 tempered out, we can try S8~S10 by tempering out 176/175 (S8/S10), which is (11/7)/(5/4)^{2}]], taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, (6/5)^{2} = 36/25 = (3/2)/(25/24).

#### 31edo as splitting the fifth into two, three and nine

31edo is unique as combining all aforementioned tempering strategies into one elegant 11-limit meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate 5/4 and 7/4 and an even more accurate 35/32. A tempering strategy not mentioned is splitting a flattened 3/2 into nine sharpened 25/24's, resulting in the 5-limit version of valentine so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering 225/224, which interestingly, though a rank 2 temperament, only has 31edo as a patent val tuning (corresponding to also tempering 225/224).

Temperaments discussed elsewhere include

*Plutus*→ Very low accuracy temperaments- Godzilla → Slendro clan
- Mothra → Gamelismic clan
- Mohaha → Rastmic clan
- Dequarter → No-sevens subgroup temperaments

The rest are considered below.

## Septimal meantone

In septimal meantone, ten fifths get to the interval class for 7, so that 7/4 is an augmented sixth (C-A♯), 7/6 is an augmented second (C-D♯), 7/5 is an augmented fourth (C-F♯), and 21/16 is an augmented third (C-E♯). Septimal meantone tempers out the common 7-limit commas 126/125, 225/224, and 3136/3125 and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Wedgie: ⟨⟨1 4 10 4 13 12]]

- 7- and 9-odd-limit: ~3/2 = [0 0 1/4⟩ (1/4-comma)

- projection map: [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
- eigenmonzo (unchanged-interval) basis: 2.5

- 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
- 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
- 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Cybozem, the real root of 15*x*^{3} - 10*x*^{2} - 18, 503.4257 cents. The recurrence converges quickly.

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness: 0.013707

### Undecimal meantone (huygens)

*"Huygens" redirects here. For the Dutch mathematician, physicist and astronomer, see Wikipedia: Christiaan Huygens.**See also: Meantone vs meanpop*

Undecimal meantone maps the 11/8 to the double augmented third (C-E𝄪), and tridecimal meantone maps the 13/8 to the double augmented fifth (C-G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is the double augmented unison; 12/11 is a double diminished third; and 14/13 is a minor second.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 126/125

Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.1676
- POTE: ~2 = 1\1, ~3/2 = 696.967

Minimax tuning:

- 11-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩

- projection map: [[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]
- eigenmonzo (unchanged-interval) basis: 2.11/9

Tuning ranges:

- 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
- 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Traverse, the positive real root of *x*^{4} + 2*x* - 13, or 696.9529 cents.

Optimal ET sequence: 12, 19e, 31, 105, 136b

Badness: 0.017027

- Music

#### Tridecimal meantone

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 105/104

Mapping: [⟨1 0 -4 -13 -25 -20], ⟨0 1 4 10 18 15]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.8552
- POTE: ~2 = 1\1, ~3/2 = 696.642

Minimax tuning:

- 13- and 15-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩

- eigenmonzo (unchanged-interval) basis: 2.11/9

Optimal ET sequence: 12f, 19e, 31

Badness: 0.018048

##### Meantonic

Dubbed *meantonic* here, this extension maps the 17/16 to the octave-reduced triple augmented seventh (C-B𝄪♯), and 19/16 to the quadruple augmented unison (C-C𝄪𝄪). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 99/98, 105/104, 121/119

Mapping: [⟨1 0 -4 -13 -25 -20 -37], ⟨0 1 4 10 18 15 26]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.6486
- POTE: ~2 = 1\1, ~3/2 = 696.377

Optimal ET sequence: 12fg, 19eg, 31, 50e

Badness: 0.019037

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 99/98, 105/104, 121/119

Mapping: [⟨1 0 -4 -13 -25 -20 -37 -40], ⟨0 1 4 10 18 15 26 28]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.5551
- POTE: ~2 = 1\1, ~3/2 = 696.273

Optimal ET sequence: 12fghh, 19egh, 31, 50e

Badness: 0.017846

##### Meantoid

Dubbed *meantoid* here, this extension maps 17/16~19/18 to the augmented unison (C-C♯) and 19/16 to the augmented second (C-D♯). For any tuning flatter than 12edo, the sizes of 17/16 (augmented unison) and 18/17 (minor second) are inverted, so genuine septendecimal and undevicesimal harmony cannot be expected.

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 66/65, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -20 -7], ⟨0 1 4 10 18 15 7]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.0360
- POTE: ~2 = 1\1, ~3/2 = 696.448

Optimal ET sequence: 12f, 19eg, 31g

Badness: 0.019433

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 57/56, 66/65, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -20 -7 -10], ⟨0 1 4 10 18 15 7 9]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.2161
- POTE: ~2 = 1\1, ~3/2 = 696.394

Optimal ET sequence: 12f, 19egh, 31gh

Badness: 0.017437

##### Huygens

Dubbed *huygens* here, this extension is perhaps the most practical, as it maps 17/16 to the minor second (C-D♭), and 19/16 to the minor third (C-E♭), suitable for a system generated by a mildly tempered fifth.

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 99/98, 105/104, 120/119

Mapping: [⟨1 0 -4 -13 -25 -20 12], ⟨0 1 4 10 18 15 -5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.9080
- POTE: ~2 = 1\1, ~3/2 = 697.003

Badness: 0.019982

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119

Mapping: [⟨1 0 -4 -13 -25 -20 12 9], ⟨0 1 4 10 18 15 -5 -3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.9308
- POTE: ~2 = 1\1, ~3/2 = 697.140

Badness: 0.018047

#### Grosstone

Grosstone maps 13/8 to the double diminished seventh (C-B♭♭♭).

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 -25 29], ⟨0 1 4 10 18 -16]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.2582
- POTE: ~2 = 1\1, ~3/2 = 697.264

Minimax tuning:

- 13- and 15-odd-limit: ~3/2 = [8/13 0 0 1/26 0 -1/26⟩

- eigenmonzo basis (unchanged-interval basis): 2.13/7

Tuning ranges:

- 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Optimal ET sequence: 12, 31, 43, 74

Badness: 0.025899

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 120/119, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 -25 29 12], ⟨0 1 4 10 18 -16 -5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.2996
- POTE: ~2 = 1\1, ~3/2 = 697.335

Optimal ET sequence: 12, 31, 43, 74g

Badness: 0.020889

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 -25 29 12 9], ⟨0 1 4 10 18 -16 -5 -3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.3271
- POTE: ~2 = 1\1, ~3/2 = 697.380

Optimal ET sequence: 12, 31, 43, 74gh

Badness: 0.017611

#### Meridetone

Meridetone maps the 13/8 to the quadruple augmented fourth (C-F𝄪𝄪).

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 126/125

Mapping: [⟨1 0 -4 -13 -25 -39], ⟨0 1 4 10 18 27]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.5155
- POTE: ~2 = 1\1, ~3/2 = 697.529

Minimax tuning:

- 13- and 15-odd-limit: ~3/2 = [14/25 -2/25 0 0 0 1/25⟩

- eigenmonzo (unchanged-interval) basis: 2.13/9

Optimal ET sequence: 12f, 31f, 43

Badness: 0.026421

##### Meridetonic

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 126/125, 273/272

Mapping: [⟨1 0 -4 -13 -25 -39 -56], ⟨0 1 4 10 18 27 38]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.5076
- POTE: ~2 = 1\1, ~3/2 = 697.514

Optimal ET sequence: 12fg, 31fg, 43

Badness: 0.027706

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 126/125, 153/152, 273/272

Mapping: [⟨1 0 -4 -13 -25 -39 -56 -59], ⟨0 1 4 10 18 27 38 40]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.4848
- POTE: ~2 = 1\1, ~3/2 = 697.481

Optimal ET sequence: 12fghh, 31fgh, 43

Badness: 0.025315

##### Meridetoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 78/77, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -39 -7], ⟨0 1 4 10 18 27 7]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.6098
- POTE: ~2 = 1\1, ~3/2 = 697.376

Optimal ET sequence: 12f, 31fg, 43g

Badness: 0.027518

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 57/56, 78/77, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -39 -7 -10], ⟨0 1 4 10 18 27 7 9]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.7012
- POTE: ~2 = 1\1, ~3/2 = 697.316

Optimal ET sequence: 12f, 19effgh, 31fgh, 43gh

Badness: 0.023613

##### Sauveuric

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 -25 -39 12], ⟨0 1 4 10 18 27 -5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.5384
- POTE: ~2 = 1\1, ~3/2 = 697.644

Badness: 0.023881

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 -25 -39 12 9], ⟨0 1 4 10 18 27 -5 -3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.5550
- POTE: ~2 = 1\1, ~3/2 = 697.715

Badness: 0.020540

#### Hemimeantone

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 169/168

Mapping: [⟨1 0 -4 -13 -25 -5], ⟨0 2 8 20 36 11]]

- mapping generators: ~2, ~26/15

Optimal tunings:

- CTE: ~2 = 1\1, ~26/15 = 948.6109
- POTE: ~2 = 1\1, ~26/15 = 948.465

Optimal ET sequence: 19e, 43, 62

Badness: 0.031433

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 169/168, 221/220

Mapping: [⟨1 0 -4 -13 -25 -5 -22], ⟨0 2 8 20 36 11 33]]

Optimal tunings:

- CTE: ~2 = 1\1, ~26/15 = 948.6173
- POTE: ~2 = 1\1, ~26/15 = 948.477

Optimal ET sequence: 19eg, 43, 62

Badness: 0.023380

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220

Mapping: [⟨1 0 -4 -13 -25 -5 -22 -25], ⟨0 2 8 20 36 11 33 37]]

Optimal tunings:

- CTE: ~2 = 1\1, ~19/11 = 948.6088
- POTE: ~2 = 1\1, ~19/11 = 948.473

Optimal ET sequence: 19egh, 43, 62

Badness: 0.018952

#### Semimeantone

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 847/845

Mapping: [⟨2 0 -8 -26 -50 -59], ⟨0 1 4 10 18 21]]

- mapping generators: ~55/39, ~3

Optimal tunings:

- CTE: ~55/39 = 1\2, ~3/2 = 697.1678
- POTE: ~55/39 = 1\2, ~3/2 = 697.005

Optimal ET sequence: 12f, 38deefff, 50eff, 62, 136b

Badness: 0.040668

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 221/220, 289/288

Mapping: [⟨2 0 -8 -26 -50 -59 5], ⟨0 1 4 10 18 21 1]]

Optimal tunings:

- CTE: ~17/12 = 1\2, ~3/2 = 697.1740
- POTE: ~17/12 = 1\2, ~3/2 = 696.927

Optimal ET sequence: 12f, 50eff, 62, 136bg

Badness: 0.031491

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220

Mapping: [⟨2 0 -8 -26 -50 -59 5 -1], ⟨0 1 4 10 18 21 1 3]]

Optimal tunings:

- CTE: ~17/12 = 1\2, ~3/2 = 697.1871
- POTE: ~17/12 = 1\2, ~3/2 = 696.906

Optimal ET sequence: 12f, 50eff, 62

Badness: 0.024206

### Meanpop

*See also: Meantone vs meanpop*

Meanpop maps the 11/8 to the double diminished fifth (C-G𝄫), and tridecimal meanpop still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the double diminished third; 12/11~13/12, double augmented unison; and 14/13, minor second.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 385/384

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

- mapping generator: ~2, ~3

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.5311
- POTE: ~2 = 1\1, ~3/2 = 696.434

Minimax tuning:

- 11-odd-limit: ~3/2 = [0 0 1/4⟩

- [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [-3 0 5/2 0 0⟩, [11 0 -13/4 0 0⟩]
- eigenmonzo (unchanged-interval) basis: 2.5

Tuning ranges:

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

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

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

Badness: 0.021543

- Music

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

#### Tridecimal meanpop

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.3563
- POTE: ~2 = 1\1, ~3/2 = 696.211

Minimax tuning:

- 13- and 15-odd-limit: ~3/2 = [4/7 0 0 0 -1/28 1/28⟩

- eigenmonzo (unchanged-interval) basis: 2.13/11

Tuning ranges:

- 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Optimal ET sequence: 19, 31, 50, 81

Badness: 0.020883

##### Meanpoppic

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 144/143, 273/272

Mapping: [⟨1 0 -4 -13 24 -20 -37], ⟨0 1 4 10 -13 15 26]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.3508
- POTE: ~2 = 1\1, ~3/2 = 696.194

Optimal ET sequence: 19g, 31, 50, 81, 131bd

Badness: 0.019953

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272

Mapping: [⟨1 0 -4 -13 24 -20 -37 -40], ⟨0 1 4 10 -13 15 26 28]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.3471
- POTE: ~2 = 1\1, ~3/2 = 696.188

Optimal ET sequence: 19gh, 31, 50, 81

Badness: 0.017791

##### Meanpoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 120/119, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 24 -20 12], ⟨0 1 4 10 -13 15 -5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.4388
- POTE: ~2 = 1\1, ~3/2 = 696.408

Badness: 0.022870

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 24 -20 12 9], ⟨0 1 4 10 -13 15 -5 -3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.4838
- POTE: ~2 = 1\1, ~3/2 = 696.499

Optimal ET sequence: 12ef, 19, 31

Badness: 0.020488

#### Meanplop

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 81/80, 91/90

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

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.2827
- POTE: ~2 = 1\1, ~3/2 = 696.202

Minimax tuning:

- 13- and 15-odd-limit: ~3/2 = [11/13 0 0 0 -1/13⟩

- Eigenmonzo (unchanged-interval) basis: 2.11

Optimal ET sequence: 12e, 19, 31f

Badness: 0.027666

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 52/51, 65/64, 78/77, 81/80, 91/90

Mapping: [⟨1 0 -4 -13 24 10 12], ⟨0 1 4 10 -13 -4 -5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.4069
- POTE: ~2 = 1\1, ~3/2 = 696.414

Badness: 0.026836

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 39/38, 52/51, 65/64, 77/76, 81/80, 91/90

Mapping: [⟨1 0 -4 -13 24 10 12 9], ⟨0 1 4 10 -13 -4 -5 -3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.4731
- POTE: ~2 = 1\1, ~3/2 = 696.497

Badness: 0.023540

##### Meanploid

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 65/64, 78/77, 81/80, 85/84

Mapping: [⟨1 0 -4 -13 24 10 -7], ⟨0 1 4 10 -13 -4 7]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.6614
- POTE: ~2 = 1\1, ~3/2 = 696.415

Optimal ET sequence: 12e, 19g, 31fg

Badness: 0.026094

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 57/56, 65/64, 76/75, 78/77, 81/80

Mapping: [⟨1 0 -4 -13 24 10 -7 -10], ⟨0 1 4 10 -13 -4 7 9]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 697.0160
- POTE: ~2 = 1\1, ~3/2 = 696.583

Optimal ET sequence: 12e, 19gh, 31fgh

Badness: 0.023104

### Meanenneadecal

Meanenneadecal maps the 11/8 to the augmented fourth (C-F♯), and tridecimal meanenneadecal still maps the 13/8 to the double augmented fifth (C-G𝄪). Note also 11/10 is the major second; 12/11~14/13, minor second; and 13/12, double augmented unison.

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 81/80

Mapping: [⟨1 0 -4 -13 -6], ⟨0 1 4 10 6]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.1527
- POTE: ~2 = 1\1, ~3/2 = 696.250

Tuning ranges:

- 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
- 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]

Optimal ET sequence: 7d, 12, 19, 31e

Badness: 0.021423

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 81/80

Mapping: [⟨1 0 -4 -13 -6 -20], ⟨0 1 4 10 6 15]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.0983
- POTE: ~2 = 1\1, ~3/2 = 696.146

Optimal ET sequence: 7df, 12f, 19, 31e

Badness: 0.021182

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 81/80, 120/119

Mapping: [⟨1 0 -4 -13 -6 -20 12], ⟨0 1 4 10 6 15 -5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.2161
- POTE: ~2 = 1\1, ~3/2 = 696.575

Optimal ET sequence: 12f, 19, 31e

Badness: 0.022980

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119

Mapping: [⟨1 0 -4 -13 -6 -20 12 9], ⟨0 1 4 10 6 15 -5 -3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.2774
- POTE: ~2 = 1\1, ~3/2 = 696.706

Optimal ET sequence: 12f, 19, 31e

Badness: 0.020293

##### Meanenneadecoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 34/33, 45/44, 51/50, 56/55, 78/77

Mapping: [⟨1 0 -4 -13 -6 -20 -7], ⟨0 1 4 10 6 15 7]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.4501
- POTE: ~2 = 1\1, ~3/2 = 696.025

Optimal ET sequence: 7dfg, 12f, 19g

Badness: 0.020171

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 78/77

Mapping: [⟨1 0 -4 -13 -6 -20 -7 -10], ⟨0 1 4 10 6 15 7 9]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.7925
- POTE: ~2 = 1\1, ~3/2 = 696.121

Optimal ET sequence: 7dfgh, 12f, 19gh

Badness: 0.018045

#### Vincenzo

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 81/80

Mapping: [⟨1 0 -4 -13 -6 10], ⟨0 1 4 10 6 -4]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.7897
- POTE: ~2 = 1\1, ~3/2 = 695.060

Optimal ET sequence: 7d, 12, 19

Badness: 0.024763

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: [⟨1 0 -4 -13 -6 10 12], ⟨0 1 4 10 6 -4 -5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.0114
- POTE: ~2 = 1\1, ~3/2 = 695.858

Badness: 0.025535

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: [⟨1 0 -4 -13 -6 10 12 9], ⟨0 1 4 10 6 -4 -5 -3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.1196
- POTE: ~2 = 1\1, ~3/2 = 696.131

Badness: 0.022302

###### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14], ⟨0 1 4 10 6 -4 -5 -3 -6]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.0585
- POTE: ~2 = 1\1, ~3/2 = 696.044

Badness: 0.020139

###### 29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8], ⟨0 1 4 10 6 -4 -5 -3 -6 -2]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.9824
- POTE: ~2 = 1\1, ~3/2 = 695.913

Badness: 0.018168

###### 31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.7980
- POTE: ~2 = 1\1, ~3/2 = 695.750

Badness: 0.017069

###### 37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.6746
- POTE: ~2 = 1\1, ~3/2 = 695.603

Badness: 0.016129

###### 41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.7239
- POTE: ~2 = 1\1, ~3/2 = 695.696

Badness: 0.015356

###### 43-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43

Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.7160
- POTE: ~2 = 1\1, ~3/2 = 695.688

Badness: 0.013906

###### 47-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47

Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.6849
- POTE: ~2 = 1\1, ~3/2 = 695.676

Badness: 0.013818

##### Vincenzoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 34/33, 45/44, 51/50, 56/55, 65/64

Mapping: [⟨1 0 -4 -13 -6 10 -7], ⟨0 1 4 10 6 -4 7]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.4125
- POTE: ~2 = 1\1, ~3/2 = 695.358

Optimal ET sequence: 7dg, 12, 19g

Badness: 0.022099

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 65/64

Mapping: [⟨1 0 -4 -13 -6 10 -7 -10], ⟨0 1 4 10 6 -4 7 9]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.9500
- POTE: ~2 = 1\1, ~3/2 = 695.725

Optimal ET sequence: 7dgh, 12, 19gh

Badness: 0.019904

#### Meanundec

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 40/39, 45/44, 56/55

Mapping: [⟨1 0 -4 -13 -6 -1], ⟨0 1 4 10 6 3]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 695.6202
- POTE: ~2 = 1\1, ~3/2 = 697.254

Optimal ET sequence: 7d, 12f, 19f

Badness: 0.024243

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 27/26, 34/33, 40/39, 45/44, 56/55

Mapping: [⟨1 0 -4 -13 -6 -1 -7], ⟨0 1 4 10 6 3 7]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.2789
- POTE: ~2 = 1\1, ~3/2 = 697.586

Badness: 0.021400

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 27/26, 34/33, 40/39, 45/44, 56/55, 57/55

Mapping: [⟨1 0 -4 -13 -6 -1 -7 -10], ⟨0 1 4 10 6 3 7 9]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.8486
- POTE: ~2 = 1\1, ~3/2 = 698.118

Optimal ET sequence: 7dgh, 12f

Badness: 0.018996

### Meanundeci

Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C-F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C-A♭).

Subgroup: 2.3.5.7.11

Comma list: 33/32, 55/54, 77/75

Mapping: [⟨1 0 -4 -13 5], ⟨0 1 4 10 -1]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.7022
- POTE: ~2 = 1\1, ~3/2 = 694.689

Optimal ET sequence: 7d, 12e, 19e

Badness: 0.031539

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 55/54, 65/64, 77/75

Mapping: [⟨1 0 -4 -13 5 10], ⟨0 1 4 10 -1 -4]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 696.2408
- POTE: ~2 = 1\1, ~3/2 = 694.764

Optimal ET sequence: 7d, 12e, 19e

Badness: 0.026288

### Bimeantone

11/8 is mapped to half octave minus the meantone diesis.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 245/242

Mapping: [⟨2 0 -8 -26 -31], ⟨0 1 4 10 12]]

- mapping generators: ~63/44, ~3

Optimal tunings:

- CTE: ~63/44 = 1\2, ~3/2 = 696.5199
- POTE: ~63/44 = 1\2, ~3/2 = 696.016

Optimal ET sequence: 12, 26de, 38d, 50

Badness: 0.038122

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 245/242

Mapping: [⟨2 0 -8 -26 -31 -40], ⟨0 1 4 10 12 15]]

Optimal tunings:

- CTE: ~55/39 = 1\2, ~3/2 = 696.3410
- POTE: ~55/39 = 1\2, ~3/2 = 695.836

Optimal ET sequence: 12f, 26deff, 38df, 50

Badness: 0.028817

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 189/187, 221/220

Mapping: [⟨2 0 -8 -26 -31 -40 5], ⟨0 1 4 10 12 15 1]]

Optimal tunings:

- CTE: ~17/12 = 1\2, ~3/2 = 696.3526
- POTE: ~17/12 = 1\2, ~3/2 = 695.783

Optimal ET sequence: 12f, 38df, 50

Badness: 0.022666

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220

Mapping: [⟨2 0 -8 -26 -31 -40 5 -1], ⟨0 1 4 10 12 15 1 3]]

Optimal tunings:

- CTE: ~17/12 = 1\2, ~3/2 = 696.3837
- POTE: ~17/12 = 1\2, ~3/2 = 695.752

Optimal ET sequence: 12f, 26deff, 38df, 50

Badness: 0.017785

### Trimean

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 1344/1331

Mapping: [⟨1 2 4 7 5], ⟨0 -3 -12 -30 -11]]

- mapping generators: ~2, ~11/10

Optimal tunings:

- CTE: ~2 = 1\1, ~11/10 = 167.7074
- POTE: ~2 = 1\1, ~11/10 = 167.805

Optimal ET sequence: 7d, 36d, 43, 50, 93

Badness: 0.050729

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 144/143, 364/363

Mapping: [⟨1 2 4 7 5 3], ⟨0 -3 -12 -30 -11 5]]

Optimal tunings:

- CTE: ~2 = 1\1, ~11/10 = 167.7121
- POTE: ~2 = 1\1, ~11/10 = 167.790

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

Badness: 0.035445

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 126/125, 144/143, 189/187, 221/220

Mapping: [⟨1 2 4 7 5 3 8], ⟨0 -3 -12 -30 -11 5 -28]]

Optimal tunings:

- CTE: ~2 = 1\1, ~11/10 = 167.7047
- POTE: ~2 = 1\1, ~11/10 = 167.786

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

Badness: 0.025221

## Flattone

In flattone tunings, the fifth is typically even flatter than that of 19edo. Here, 9 fourths get to the interval class for 7, so that 7/4 is a diminished seventh (C-B𝄫), 7/6 is a diminished third (C-E𝄫), and 7/5 is a doubly-diminished fifth (C-G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are 45edo, 64edo, and 71edo.

Subgroup: 2.3.5.7

Comma list: 81/80, 525/512

Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]

Wedgie: ⟨⟨1 4 -9 4 -17 -32]]

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

- [[1 0 0 0⟩, [21/13 0 1/13 -1/13⟩, [32/13 0 4/13 -4/13⟩, [32/13 0 -9/13 9/13⟩]
- eigenmonzo (unchanged-interval) basis: 2.7/5

- 9-odd-limit: ~3/2 = [6/11 2/11 0 -1/11⟩

- [[1 0 0 0⟩, [17/11 2/11 0 -1/11⟩, [24/11 8/11 0 -4/11⟩, [34/11 -18/11 0 9/11⟩]
- eigenmonzo (unchanged-interval) basis: 2.9/7

- 7- and 9-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
- 7-odd-limit diamond tradeoff: ~3/2 = [692.353, 701.955]
- 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]

Algebraic generator: Squarto, the positive root of 8*x*^{2} - 4*x* - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

Optimal ET sequence: 7, 19, 26, 45

Badness: 0.038553

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 385/384

Mapping: [⟨1 0 -4 17 -6], ⟨0 1 4 -9 6]]

Optimal tuning:

- CTE: ~2 = 1\1, ~3/2 = 693.2511
- POTE: ~2 = 1\1, ~3/2 = 693.126

Tuning ranges:

- 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
- 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]

Optimal ET sequence: 7, 19, 26, 45, 71bc, 116bcde

Badness: 0.033839

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 81/80

Mapping: [⟨1 0 -4 17 -6 10], ⟨0 1 4 -9 6 -4]]

Optimal tunings:

- CTE: ~2 = 1\1, ~3/2 = 693.0293
- POTE: ~2 = 1\1, ~3/2 = 693.058

Tuning ranges:

- 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]

Optimal ET sequence: 7, 19, 26, 45f, 71bcf, 116bcdef

Badness: 0.022260

## Flattertone

Flattertone tunings are typically at least as flat as 26edo. Here, 17 fifths get to the interval class for 7, so that 7/4 is a double-augmented sixth (C-Ax). 26edo and 33cd-edo are the two primary flattertone tunings. 1/2-comma meantone is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a 16/13 or 27/22), and deeptone temperament's mapping is more logical.

Subgroup: 2.3.5.7

Comma list: 81/80, 1875/1792

Mapping: [⟨1 0 -4 -24], ⟨0 1 4 17]]

- mapping generators: ~2, ~3

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

Badness: 0.0961

#### 11-limit

Subgroup: 2.3.5.7

Comma list: 45/44, 81/80, 1375/1344

Mapping: [⟨1 0 -4 -24 0], ⟨0 1 4 17 6]]

- mapping generators: ~2, ~3

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

## Dominant

*See also: Archytas clan*

The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is 12edo, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with 29edo, 41edo, or 53edo.

Subgroup: 2.3.5.7

Comma list: 36/35, 64/63

Mapping: [⟨1 0 -4 6], ⟨0 1 4 -2]]

Wedgie: ⟨⟨1 4 -2 4 -6 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.573

- 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
- 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 715.587]
- 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

Optimal ET sequence: 5, 7, 12, 41cd, 53cdd, 65ccddd

Badness: 0.020690

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 64/63

Mapping: [⟨1 0 -4 6 13], ⟨0 1 4 -2 -6]]

Tuning ranges:

- 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
- 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.254

Optimal ET sequence: 5, 12, 17c, 29cde

Badness: 0.024180

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 64/63, 66/65

Mapping: [⟨1 0 -4 6 13 18], ⟨0 1 4 -2 -6 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.636

Tuning ranges:

- 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

Optimal ET sequence: 12f, 17c, 29cdef

Badness: 0.024108

#### Dominion

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 64/63

Mapping: [⟨1 0 -4 6 13 -9], ⟨0 1 4 -2 -6 8]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.905

Optimal ET sequence: 5, 12, 17c, 46cde

Badness: 0.027295

### Domineering

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 64/63

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.776

Optimal ET sequence: 5e, 7, 12, 19d, 43de

Badness: 0.021978

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 45/44, 52/49, 64/63

Mapping: [⟨1 0 -4 6 -6 10], ⟨0 1 4 -2 6 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.762

Optimal ET sequence: 5ef, 7, 12, 19d, 31def

Badness: 0.027039

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 36/35, 45/44, 51/49, 52/49, 64/63

Mapping: [⟨1 0 -4 6 -6 10 12], ⟨0 1 4 -2 6 -4 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.115

Optimal ET sequence: 5ef, 7, 12, 19d, 31def

Badness: 0.024539

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56

Mapping: [⟨1 0 -4 6 -6 10 12 9], ⟨0 1 4 -2 6 -4 -5 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.217

Optimal ET sequence: 5ef, 7, 12, 19d, 31def

Badness: 0.020398

#### Dominatrix

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 64/63

Mapping: [⟨1 0 -4 6 -6 -1], ⟨0 1 4 -2 6 3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.544

Optimal ET sequence: 5e, 7, 12f, 19df

Badness: 0.018289

### Domination

Subgroup: 2.3.5.7.11

Comma list: 36/35, 64/63, 77/75

Mapping: [⟨1 0 -4 6 -14], ⟨0 1 4 -2 11]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.004

Optimal ET sequence: 5e, 12e, 17c, 46cd

Badness: 0.036562

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 64/63, 66/65

Mapping: [⟨1 0 -4 6 -14 -9], ⟨0 1 4 -2 11 8]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.496

Optimal ET sequence: 5e, 12e, 17c

Badness: 0.027435

### Arnold

Subgroup: 2.3.5.7.11

Comma list: 22/21, 33/32, 36/35

Mapping: [⟨1 0 -4 6 5], ⟨0 1 4 -2 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.491

Optimal ET sequence: 5, 7, 12e

Badness: 0.026141

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 27/26, 33/32, 36/35

Mapping: [⟨1 0 -4 6 5 -1], ⟨0 1 4 -2 -1 3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.743

Optimal ET sequence: 5, 7, 12ef, 19def

Badness: 0.023300

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 22/21, 27/26, 33/32, 36/35, 51/49

Mapping: [⟨1 0 -4 6 5 -1 12], ⟨0 1 4 -2 -1 3 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.978

Optimal ET sequence: 5, 7, 12ef, 19def

Badness: 0.024535

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56

Mapping: [⟨1 0 -4 6 5 -1 12 9], ⟨0 1 4 -2 -1 3 -5 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 697.068

Optimal ET sequence: 5, 7, 12ef, 19def

Badness: 0.021098

## Sharptone

Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. 12edo tuning does sharptone about as well as such a thing can be done, of course not in its patent val.

Subgroup: 2.3.5.7

Comma list: 21/20, 28/27

Mapping: [⟨1 0 -4 -2], ⟨0 1 4 3]]

Wedgie: ⟨⟨1 4 3 4 2 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.140

Optimal ET sequence: 5, 7d, 12d

Badness: 0.024848

### Meanertone

Subgroup: 2.3.5.7.11

Comma list: 21/20, 28/27, 33/32

Mapping: [⟨1 0 -4 -2 5], ⟨0 1 4 3 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 696.615

Optimal ET sequence: 5, 7d, 12de

Badness: 0.025167

## Supermean

Subgroup: 2.3.5.7

Comma list: 81/80, 672/625

Mapping: [⟨1 0 -4 -21], ⟨0 1 4 15]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.889

Optimal ET sequence: 5d, 12d, 17c, 29c

Badness: 0.134204

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 132/125

Mapping: [⟨1 0 -4 -21 -14], ⟨0 1 4 15 11]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.096

Optimal ET sequence: 5de, 12de, 17c, 29c

Badness: 0.063262

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 66/65, 81/80

Mapping: [⟨1 0 -4 -21 -14 -9], ⟨0 1 4 15 11 8]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 705.094

Optimal ET sequence: 5de, 12de, 17c, 29c

Badness: 0.040324

## Mohajira

Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. 31edo makes for an excellent (7-limit) mohajira tuning, with generator 9\31.

Subgroup: 2.3.5.7

Comma list: 81/80, 6144/6125

Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]

- mapping generators: ~2, ~128/105

Wedgie: ⟨⟨2 8 -11 8 -23 -48]]

Optimal tuning (POTE): ~2 = 1\1, ~128/105 = 348.415

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

- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5

- 7- and 9-odd-limit diamond monotone: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
- 7-odd-limit diamond tradeoff: ~128/105 = [347.393, 350.978]
- 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]

Algebraic generator: Mohabis, real root of 3*x*^{3} - 3*x*^{2} - 1, 348.6067 cents. Corresponding recurrence converges quickly.

Optimal ET sequence: 7, 24, 31

Badness: 0.055714

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 176/175

Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.477

Minimax tuning:

- 11-odd-limit: ~11/9 = [0 0 1/8⟩

- [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [6 0 -11/8 0 0⟩, [2 0 5/8 0 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5

Tuning ranges:

- 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
- 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]

Optimal ET sequence: 7, 24, 31

Badness: 0.026064

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 105/104, 121/120

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.558

Optimal ET sequence: 7, 24, 31

Badness: 0.023388

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 105/104, 121/120, 154/153

Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.736

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

Badness: 0.020576

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

Mapping: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 348.810

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

Badness: 0.017302

## Mohamaq

Subgroup: 2.3.5.7

Comma list: 81/80, 392/375

Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]

- mapping generators: ~2, ~25/21

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 350.586

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

Badness: 0.077734

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 243/242

Mapping: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.565

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

Badness: 0.036207

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 243/242

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.745

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

Badness: 0.028738

## Liese

Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. 74edo makes for a good liese tuning, though 19edo can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.

Subgroup: 2.3.5.7

Comma list: 81/80, 686/675

Mapping: [⟨1 0 -4 -3], ⟨0 3 12 11]]

- mapping generators: ~2, ~10/7

Wedgie: ⟨⟨3 12 11 12 9 -8]]

Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.406

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

- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5

Algebraic generator: Radix, the real root of *x*^{5} - 2*x*^{4} + 2*x*^{3} - 2*x*^{2} + 2*x* - 2, also a root of *x*^{6} - *x*^{5} - 2. The recurrence converges.

Optimal ET sequence: 17c, 19, 55, 74d

Badness: 0.046706

### Liesel

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.073

Optimal ET sequence: 17c, 19, 36, 91cee

Badness: 0.040721

#### 13-limit

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

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 81/80, 91/90

Mapping: [⟨1 0 -4 -3 4 0], ⟨0 3 12 11 -1 7]]

Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.042

Optimal ET sequence: 17c, 19, 36, 91ceef

Badness: 0.027304

### Elisa

Subgroup: 2.3.5.7.11

Comma list: 77/75, 81/80, 99/98

Mapping: [⟨1 0 -4 -3 -5], ⟨0 3 12 11 16]]

Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 633.061

Optimal ET sequence: 17c, 19e, 36e

Badness: 0.041592

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 77/75, 81/80, 99/98

Mapping: [⟨1 0 -4 -3 -5 0], ⟨0 3 12 11 16 7]]

Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 632.991

Optimal ET sequence: 17c, 19e, 36e

Badness: 0.026922

### Lisa

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 343/330

Mapping: [⟨1 0 -4 -3 -6], ⟨0 3 12 11 18]]

Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.370

Optimal ET sequence: 17cee, 19

Badness: 0.054829

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 81/80, 91/88, 147/143

Mapping: [⟨1 0 -4 -3 -6 0], ⟨0 3 12 11 18 7]]

Optimal tuning (POTE): ~2 = 1\1, ~10/7 = 631.221

Optimal ET sequence: 17cee, 19

Badness: 0.036144

## Superpine

The superpine temperament is generated by 1/3 of a fourth, represented by 35/32, which resembles porcupine, but it favors flat fifths instead of sharp ones. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent 6/5–harmonics other than 3 all require the 15-tone mos to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as 11/10 as in porcupine, which makes 11/8 high-complexity like the other harmonics, but in the 13-limit 5 generators up closely approximates 13/8. 43edo is a good tuning especially for the higher-limit extensions.

Subgroup: 2.3.5.7

Comma list: 81/80, 1119744/1071875

Mapping: [⟨1 2 4 1], ⟨0 -3 -12 13]]

Optimal tuning (CTE): ~2 = 1\1, ~35/32 = 167.279

Optimal ET sequence: 7, 36, 43, 79c

Badness: 0.137

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 864/847

Mapping: [⟨1 2 4 1 5], ⟨0 -3 -12 13 -11]]

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.407

Optimal ET sequence: 7, 36, 43

Badness: 0.0576

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 144/143, 176/175

Mapping: [⟨1 2 4 1 5 3], ⟨0 -3 -12 13 -11 5]]

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.427

Optimal ET sequence: 7, 36, 43

Badness: 0.0368

## Lithium

Lithium is named after the 3rd element for being period-3, and also for lithium's molar mass of 6.9 g/mol since 69edo supports it. It supports a 3L 6s scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.

Subgroup: 2.3.5.7

Comma list: 81/80, 3125/3087

Mapping: [⟨3 0 -12 -20], ⟨0 1 4 6]]

- mapping generators: ~56/45, ~3

Optimal tuning (CTE): ~56/45 = 1\3, ~3/2 = 695.827

Optimal ET sequence: 12, 33cd, 45, 57

Badness: 0.0692

## Squares

Squares splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. 31edo, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.

Subgroup: 2.3.5.7

Comma list: 81/80, 2401/2400

Mapping: [⟨1 3 8 6], ⟨0 -4 -16 -9]]

- mapping generators: ~2, ~9/7

Wedgie: ⟨⟨4 16 9 16 3 -24]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.942

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

- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]
- Eigenmonzo (unchanged-interval) basis: 2.5

Algebraic generator: Sceptre2, the positive root of 9*x*^{2} + *x* - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.

Optimal ET sequence: 14c, 17c, 31

Badness: 0.045993

Scales: skwares8, skwares11, skwares14

### 11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.957

Optimal ET sequence: 14c, 17c, 31

Badness: 0.021636

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 121/120

Mapping: [⟨1 3 8 6 7 3], ⟨0 -4 -16 -9 -10 2]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.550

Optimal ET sequence: 14c, 17c, 31, 79cf

Badness: 0.025514

#### Squad

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 91/90, 99/98

Mapping: [⟨1 3 8 6 7 9], ⟨0 -4 -16 -9 -10 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.7516

Optimal ET sequence: 14cf, 17c, 31f

Badness: 0.026877

#### Agora

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 121/120

Mapping: [⟨1 3 8 6 7 14], ⟨0 -4 -16 -9 -10 -29]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.276

Optimal ET sequence: 14cf, 31, 45ef, 76e

Badness: 0.024522

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: [⟨1 3 8 6 7 14 8], ⟨0 -4 -16 -9 -10 -29 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.187

Optimal ET sequence: 14cf, 31, 76e

Badness: 0.022573

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: [⟨1 3 8 6 7 14 8 11], ⟨0 -4 -16 -9 -10 -29 -11 -19]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 426.225

Optimal ET sequence: 14cf, 31, 76e

Badness: 0.018839

### Cuboctahedra

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.993

Optimal ET sequence: 14ce, 17ce, 31, 107b, 138b, 169be, 200be

Badness: 0.056826

## Jerome

Jerome is related to Hieronymus' tuning; the Hieronymus generator is 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 = 1\1, ~54/49 = 139.343

Optimal ET sequence: 17c, 26, 43, 69, 112bd

Badness: 0.108656

### 11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.428

Optimal ET sequence: 17c, 26, 43, 69

Badness: 0.047914

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 144/143

Mapping: [⟨1 1 0 2 3 3], ⟨0 5 20 7 4 6]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.387

Optimal ET sequence: 17c, 26, 43, 69

Badness: 0.029285

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 144/143, 189/187

Mapping: [⟨1 1 0 2 3 3 2], ⟨0 5 20 7 4 6 18]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.362

Optimal ET sequence: 17cg, 26, 43, 69

Badness: 0.020878

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143

Mapping: [⟨1 1 0 2 3 3 2 1], ⟨0 5 20 7 4 6 18 28]]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 139.313

Optimal ET sequence: 17cgh, 26, 43, 69

Badness: 0.018229

## Meantritone

The *meantritone* temperament tempers out the mirkwai comma (16875/16807) and trimyna comma (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus 15/11-wide super-fourth) and five of them equals ~16/3 (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.

Subgroup: 2.3.5.7

Comma list: 81/80, 16875/16807

Mapping: [⟨1 4 12 12], ⟨0 -5 -20 -19]]

Wedgie: ⟨⟨5 20 19 20 16 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.766

Optimal ET sequence: 2cd, 29cd, 31

Badness: 0.082239

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 2541/2500

Mapping: [⟨1 4 12 12 17], ⟨0 -5 -20 -19 -28]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.647

Optimal ET sequence: 2cde, 29cde, 31

Badness: 0.042869

## Injera

Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. 38EDO, which is two parallel 19edos, is an excellent tuning for injera.

Subgroup: 2.3.5.7

Comma list: 50/49, 81/80

Mapping: [⟨2 0 -8 -7], ⟨0 1 4 4]]

- mapping generators: ~7/5, ~3

Wedgie: ⟨⟨2 8 8 8 7 -4]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.375

- 7- and 9-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
- 7-odd-limit diamond tradeoff: ~3/2 = [688.957, 701.955]
- 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

Optimal ET sequence: 12, 26, 38, 102bcd, 140bccd, 178bbccdd

Badness: 0.031130

- Music

- Two Pairs of Socks (in 26EDO) by Igliashon Jones

### 11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.840

Tuning ranges:

- 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
- 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

Optimal ET sequence: 12, 14c, 26, 90bce, 116bcce

Badness: 0.023124

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 81/80

Mapping: [⟨2 0 -8 -7 -12 -21], ⟨0 1 4 4 6 9]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.673

Tuning ranges:

- 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
- 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

Optimal ET sequence: 12f, 14cf, 26, 38e

Badness: 0.021565

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 50/49, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5], ⟨0 1 4 4 6 9 1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.487

Optimal ET sequence: 12f, 14cf, 26

Badness: 0.018358

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5 -1], ⟨0 1 4 4 6 9 1 3]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 692.299

Optimal ET sequence: 12f, 14cf, 26

Badness: 0.015118

#### Enjera

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 40/39, 45/44, 50/49

Mapping: [⟨2 0 -8 -7 -12 -2], ⟨0 1 4 4 6 3]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 694.121

Optimal ET sequence: 12f, 14c, 26f, 38eff

Badness: 0.026542

### Injerous

Subgroup: 2.3.5.7.11

Comma list: 33/32, 50/49, 55/54

Mapping: [⟨2 0 -8 -7 10], ⟨0 1 4 4 -1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 690.548

Optimal ET sequence: 12e, 14c, 26e, 40cee

Badness: 0.038577

### Lahoh

Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 81/77

Mapping: [⟨2 0 -8 -7 7], ⟨0 1 4 4 0]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 699.001

Optimal ET sequence: 2cd, 10cd, 12

Badness: 0.043062

### Teff

Teff (found by Mason Green) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.

Subgroup: 2.3.5.7.11

Comma list: 50/49, 81/80, 864/847

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

- mapping generators: ~7/5, ~16/11

Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5303

Optimal ET sequence: 24d, 26, 50d

Badness: 0.070689

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 81/80, 144/143

Mapping: [⟨2 1 -4 -3 8 2], ⟨0 2 8 8 -1 5]]

Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.5324

Optimal ET sequence: 24d, 26, 50d

Badness: 0.040047

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6], ⟨0 2 8 8 -1 5 2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6558

Badness: 0.029499

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6 2], ⟨0 2 8 8 -1 5 2 6]]

Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 552.6382

Badness: 0.023133

## Pombe

Pombe (named after the African millet beer) is a variant of #Teff by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.

Subgroup: 2.3.5.7

Comma list: 81/80, 300125/294912

Mapping: [⟨2 1 -4 11], ⟨0 2 8 -5]]

- mapping generators: ~735/512, ~35/24

Wedgie: ⟨⟨4 16 -10 16 -27 -68]]

Optimal tuning (POTE): ~735/512 = 1\2, ~48/35 = 552.2206

Optimal ET sequence: 24, 26, 50, 126bcd, 176bcdd, 226bbcdd

Badness: 0.116104

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 245/242, 385/384

Mapping: [⟨2 1 -4 11 8], ⟨0 2 8 -5 -1]]

Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.0929

Optimal ET sequence: 24, 26, 50

Badness: 0.052099

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 245/242

Mapping: [⟨2 1 -4 11 8 2], ⟨0 2 8 -5 -1 5]]

Optimal tuning (POTE): ~99/70 = 1\2, ~11/8 = 552.1498

Optimal ET sequence: 24, 26, 50

Badness: 0.031039

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 144/143, 245/242, 273/272

Mapping: [⟨2 1 -4 11 8 2 6], ⟨0 2 8 -5 -1 5 2]]

Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1579

Optimal ET sequence: 24, 26, 50

Badness: 0.021260

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209

Mapping: [⟨2 1 -4 11 8 2 6 2], ⟨0 2 8 -5 -1 5 2 6]]

Optimal tuning (POTE): ~17/12 = 1\2, ~11/8 = 552.1196

Optimal ET sequence: 24, 26, 50

Badness: 0.016548

## Orphic

Subgroup: 2.3.5.7

Comma list: 81/80, 5898240/5764801

Mapping: [⟨2 5 12 7], ⟨0 -4 -16 -3]]

Mapping generators: ~2401/1728, ~7/6

Wedgie: ⟨⟨8 32 6 32 -13 -76]]

Optimal tuning (POTE): ~2401/1728 = 1\2, ~7/6 = 275.794

Optimal ET sequence: 26, 48c, 74, 174bd, 248bbd

Badness: 0.258825

### 11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 5 12 7 6], ⟨0 -4 -16 -3 2]]

Optimal tuning (POTE): ~363/256 = 1\2, ~7/6 = 275.762

Optimal ET sequence: 26, 48c, 74, 248bbd, 322bbdd

Badness: 0.101499

### 13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 5 12 7 6 12], ⟨0 -4 -16 -3 2 -10]]

Optimal tuning (POTE): ~55/39 = 1\2, ~7/6 = 275.774

Optimal ET sequence: 26, 48c, 74, 174bd, 248bbd, 322bbdd

Badness: 0.053482

## Cloudtone

The *cloudtone* temperament (5&50) tempers out the cloudy comma, 16807/16384 and the syntonic comma, 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 81/80, 16807/16384

Mapping: [⟨5 0 -20 14], ⟨0 1 4 0]]

- mapping generators: ~8/7, ~3

Wedgie: ⟨⟨5 20 0 20 -14 -56]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 695.720

Optimal ET sequence: 5, 45, 50

Badness: 0.102256

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 2401/2376

Mapping: [⟨5 0 -20 14 41], ⟨0 1 4 0 -3]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.536

Optimal ET sequence: 5, 45, 50, 155bdd, 205bddd

Badness: 0.070378

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 2401/2376

Mapping: [⟨5 0 -20 14 41 -21], ⟨0 1 4 0 -3 5]]

Optimal tuning (POTE): ~8/7 = 1\5, ~3/2 = 696.162

Optimal ET sequence: 5, 45f, 50

Badness: 0.048829