Meantone family

(Redirected from Injera)

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]]

• CTE: ~2 = 1\1, ~3/2 = 697.2143
• POTE: ~2 = 1\1, ~3/2 = 696.239
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.

Septimal meantone

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

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping[1 0 -4 -13], 0 1 4 10]]

Wedgie⟨⟨1 4 10 4 13 12]]

• CTE: ~2 = 1\1, ~3/2 = 696.9521
• POTE: ~2 = 1\1, ~3/2 = 696.495
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 (huygens)

"Huygens" redirects here. For the Dutch mathematician, physicist and astronomer, see Wikipedia: Christiaan Huygens.

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 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 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

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

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

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

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

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

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

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.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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 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.)

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

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

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

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

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 diminshed fifth (C-G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. Good tunings for flattone are 45edo, 64edo, and 71edo.

Subgroup: 2.3.5.7

Comma list: 81/80, 525/512

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

Wedgie⟨⟨1 4 -9 4 -17 -32]]

• CTE: ~2 = 1\1, ~3/2 = 693.5520
• 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.

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]

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]

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. Half-comma meantone is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a 16/13 or 27/22), and deeptone temperament's mapping is more logical.

Subgroup: 2.3.5.7

Comma list: 81/80, 1875/1792

Mapping[1 0 -4 -24], 0 1 4 17]]

mapping generators: ~2, ~3
• CTE: ~2 = 1\1, ~3/2 = 692.6984
• CWE: ~2 = 1\1, ~3/2 = 692.0479

11-limit

Subgroup: 2.3.5.7

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

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

mapping generators: ~2, ~3
• CTE: ~2 = 1\1, ~3/2 = 692.642
• CWE: ~2 = 1\1, ~3/2 = 692.042

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

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

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

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

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

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

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

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