# Meantone family

(Redirected from Meantritone)

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

## Meantone

Main article: Meantone

Subgroup: 2.3.5

Comma list: 81/80

Mapping[1 0 -4], 0 1 4]]

mapping generators: ~2, ~3

Wedgie⟨⟨1 4 4]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2143

eigenmonzo (unchanged-interval) basis: 2.5

### Overview to extensions

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

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

Those all have a fifth as generator.

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

Temperaments discussed elsewhere include

The rest are considered below.

### Dequarter

Subgroup: 2.3.5.11

Comma list: 33/32, 55/54

Sval mapping: [1 0 -4 5], 0 1 4 -1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0387

Optimal ET sequence: 5, 7, 19e, 26e

## Septimal meantone

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Main article: Meantone
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The 7/4 in septimal meantone is the augmented sixth (C-A#), and other septimal intervals are 7/6, the augmented second (C-D#), 7/5, the augmented fourth (C-F#), and 21/16, the augmented third (C-E#). Septimal meantone tempers out the common 7-limit commas 126/125 and 225/224 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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9521

projection map: [[1 0 0 0, [1 0 1/4 0, [0 0 1 0, [-3 0 5/2 0]
eigenmonzo (unchanged-interval) basis: 2.5
• 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
• 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
• 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, 503.4257 cents. The recurrence converges quickly.

### Undecimal meantone aka huygens

Undecimal meantone maps the 11/8 to the double augmented third (C-Ex), and tridecimal meantone maps the 13/8 to the double augmented fifth (C-Gx). Note 13/11 is conflated with 6/5, represented by the minor third. Note also 11/10~13/12 is the double augmented unison; 12/11, double diminished third; and 14/13, 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.1676

Minimax tuning:

• 11-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16
projection map: [[1 0 0 0 0, [25/16 -1/8 0 0 1/16, [9/4 -1/2 0 0 1/4, [21/8 -5/4 0 0 5/8, [25/8 -9/4 0 0 9/8]
eigenmonzo (unchanged-interval) basis: 2.11/9

Tuning ranges:

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

Algebraic generator: Traverse, the positive real root of x4 + 2x - 13, or 696.9529 cents.

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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.8552

Minimax tuning:

• 13- and 15-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16
eigenmonzo (unchanged-interval) basis: 2.11/9

##### Meantonic

Dubbed meantonic here, this extension maps the 17/16 to the triple augmented seventh (C-Bx#) octave reduced, and 19/16 to the quadruple augmented unison (C-Cxx). 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.6486

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.5551

##### 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 inverse, 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.0360

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.2161

##### Huygens

Dubbed huygens here, this extension is perhaps the most practical, as it maps 17/16 to the minor second (C-Db), and 19/16 to the minor third (C-Eb), 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.9080

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.9308

#### Grosstone

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

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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.2582

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.)

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.2996

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.3271

#### Meridetone

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

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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.5155

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

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.5076

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.4848

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.6098

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.7012

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.5384

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.5550

#### 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 tuning (CTE): ~2 = 1\1, ~26/15 = 948.6109

##### 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 tuning (CTE): ~2 = 1\1, ~26/15 = 948.6173

##### 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 tuning (CTE): ~2 = 1\1, ~19/11 = 948.6088

#### 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 tuning (CTE): ~55/39 = 1\2, ~3/2 = 697.1678

##### 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 tuning (CTE): ~17/12 = 1\2, ~3/2 = 697.1740

##### 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 tuning (CTE): ~17/12 = 1\2, ~3/2 = 697.1871

### Meanpop

Meanpop maps the 11/8 to the double diminished fifth (C-Gbb), and tridecimal meanpop still maps the 13/8 to the double augmented fifth (C-Gx). 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.5311

Minimax tuning:

• 11-odd-limit: ~3/2 = [0 0 1/4
[[1 0 0 0 0, [1 0 1/4 0 0, [0 0 1 0 0, [-3 0 5/2 0 0, [11 0 -13/4 0 0]
eigenmonzo (unchanged-interval) basis: 2.5

Tuning ranges:

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

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

Music

#### Tridecimal meanpop

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3563

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.)

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.3508

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.3471

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.4388

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.4838

#### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.2827

Minimax tuning:

• 13- and 15-odd-limit: ~3/2 = [11/13 0 0 0 -1/13
Eigenmonzo (unchanged-interval) basis: 2.11

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.4069

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.4731

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.6614

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 697.0160

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-Gx). 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.1527

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]

#### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.0983

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.2161

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.2774

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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.4501

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.7925

#### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.7897

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.0114

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.1196

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.0585

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.9824

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.7980

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.6746

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.7239

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.7160

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.6849

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.4125

###### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.9500

#### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 695.6202

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.2789

##### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.8486

### 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-Ab).

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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.7022

#### 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 tuning (CTE): ~2 = 1\1, ~3/2 = 696.2408

### 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 tuning (CTE): ~63/44 = 1\2, ~3/2 = 696.5199

#### 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 tuning (CTE): ~55/39 = 1\2, ~3/2 = 696.3410

#### 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 tuning (CTE): ~17/12 = 1\2, ~3/2 = 696.3526

#### 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 tuning (CTE): ~17/12 = 1\2, ~3/2 = 696.3837

### 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 tuning (CTE): ~2 = 1\1, ~11/10 = 167.7074

#### 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 tuning (CTE): ~2 = 1\1, ~11/10 = 167.7121

#### 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 tuning (CTE): ~2 = 1\1, ~11/10 = 167.7047

## Flattone

In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished seventh interval (C-Bbb). Other intervals are 7/6, a diminished third (C-Ebb), and 7/5, a doubly diminshed fifth (C-Gbb). In general, most septimal subminor intervals are diminished and most septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are 26edo, 45edo and 64edo.

Subgroup: 2.3.5.7

Comma list: 81/80, 525/512

Mapping[1 0 -4 17], 0 1 4 -9]]

Wedgie⟨⟨1 4 -9 4 -17 -32]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.779

[[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
[[1 0 0 0, [17/11 2/11 0 -1/11, [24/11 8/11 0 -4/11, [34/11 -18/11 0 9/11]
eigenmonzo (unchanged-interval) basis: 2.9/7
• 7- and 9-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
• 7-odd-limit diamond tradeoff: ~3/2 = [692.353, 701.955]
• 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]

Algebraic generator: Squarto, the positive root of 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

Scales: flattone12

### 11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.126

Tuning ranges:

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

Scales: flattone12

#### 13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 693.058

Tuning ranges:

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

Scales: flattone12

## Dominant

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

### 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

#### 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]

#### 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

### 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

#### 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

##### 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

##### 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

#### 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

### 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

#### 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

### 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

#### 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

#### 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

#### 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

## 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

### 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

## 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

### 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

### 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

## Mohajira

Main article: Mohajira

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

Subgroup: 2.3.5.7

Comma list: 81/80, 6144/6125

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

mapping generators: ~2, ~128/105

Wedgie⟨⟨2 8 -11 8 -23 -48]]

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

[[1 0 0 0, [1 0 1/4 0, [0 0 1 0, [6 0 -11/8 0]
Eigenmonzo (unchanged-interval) basis: 2.5
• 7- and 9-odd-limit diamond monotone: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
• 7-odd-limit diamond tradeoff: ~128/105 = [347.393, 350.978]
• 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]

Algebraic generator: Mohabis, real root of 3x3 - 3x2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.

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 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]

Scales: mohaha7, mohaha10

### 13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Scales: mohaha7, mohaha10

### 17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

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

Scales: mohaha7, mohaha10

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

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

Scales: mohaha7, mohaha10

## Mohamaq

Subgroup: 2.3.5.7

Comma list: 81/80, 392/375

Mapping: [1 1 0 -1], 0 2 8 13]]

mapping generators: ~2, ~25/21

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

Scales: mohaha7, mohaha10

### 11-limit

Subgroup: 2.3.5.7.11

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

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

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

Scales: mohaha7, mohaha10

### 13-limit

Subgroup: 2.3.5.7.11.13

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

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

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

Scales: mohaha7, mohaha10

## 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

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

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

### 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

#### 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

### 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

#### 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

### 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

#### 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

## 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

## Squares

Main article: Squares

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

Subgroup: 2.3.5.7

Comma list: 81/80, 2401/2400

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

mapping generators: ~2, ~9/7

Wedgie⟨⟨4 16 9 16 3 -24]]

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

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

Algebraic generator: Sceptre2, the positive root of 9x2 + x - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.

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

#### 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

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

#### 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

##### 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

##### 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

### 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

## Jerome

Jerome is related to Hieronymus' tuning; the Hieronymus generator is 51/20, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.

Subgroup: 2.3.5.7

Comma list: 81/80, 17280/16807

Mapping: [1 1 0 2], 0 5 20 7]]

mapping generators: ~2, ~54/49

Wedgie⟨⟨5 20 7 20 -3 -40]]

Optimal tuning (POTE): ~2 = 1\1, ~54/49 = 139.343

### 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

### 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

### 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

### 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

## 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

### 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

## 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]

Music

### 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]

#### 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]

##### 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

##### 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

#### 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

### 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

### 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

### Teff

Main article: Teff

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

Subgroup: 2.3.5.7.11

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

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

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

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

#### 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

#### 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

#### 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

## 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

### 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

### 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

### 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

### 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

## 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

### 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

### 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

## 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

### 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