Meantone family

The 5-limit parent comma of the meantone family is the Didymus or syntonic comma, 81/80. This is the one they all temper out. The monzo for 81/80 goes [-4 4 -1⟩, and that can be flipped around to the corresponding wedgie, <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

5-limit meantone

Comma: 81/80

POTE generator: ~3/2 = 696.239

Mapping generator: ~3

valid range: [685.714, 720.000] (7 to 5)

nice range: [694.786, 701.955] (1/3 comma to Pythagorean)

strict range: [694.786, 701.955]

Map: [<1 0 -4|, <0 1 4|]

EDOs: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb

Badness: 0.00736

Seven limit children

The 7-limit children of 81/80 are:

• Septimal meantone, with normal comma list [[-4 4 -1⟩, [-13 10 0 -1⟩],
• Flattone, with normal list [[-4 4 -1⟩, [-17 9 0 1⟩],
• Dominant, with normal list [[-4 4 -1⟩, [6 -2 0 -1⟩],
• Sharptone, with normal list [[-4 4 -1⟩, [2 -3 0 1⟩],
• Injera, with normal list [[-4 4 -1⟩, [-7 8 0 -2⟩],
• Mohajira, with normal list [[-4 4 -1⟩, [-23 11 0 2⟩],
• Godzilla, with normal list [[-4 4 -1⟩, [-4 -1 0 2⟩],
• Mothra, with normal list [[-4 4 -1⟩, [-10 1 0 3⟩],
• Squares, with normal list [[-4 4 -1⟩, [-3 9 0 -4⟩], and
• Liese, with normal list [[-4 4 -1⟩, [-9 11 0 -3⟩].

Septimal meantone

Deutsch

• Wikipedia: Septimal meantone temperament

The comma [-13 10 0 -1⟩ for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and 7/5, C-F#, the tritone. The wedgie for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and 31edo is a good tuning for it.

Commas: 81/80, 126/125

7 and 9-limit minimax

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]

Eigenmonzos: 2, 5

valid range: [694.737, 700.000] (19 to 12)

nice range: [694.786, 701.955]

strict range: [694.786, 700.000]

POTE generator: 696.495

Mapping generator: ~3

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

Map: [<1 0 -4 -13|, <0 1 4 10|]

Generators: 2, 3

Wedgie: <<1 4 10 4 13 12||

EDOs: 12, 19, 31, 43, 50, 74, 81, 105, 143b

Badness: 0.0137

Bimeantone

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

Commas: 81/80, 126/125, 245/242

POTE generator: ~3/2 = 696.016

Map: [<2 0 -8 -26 -31|, <0 1 4 10 12|]

EDOs: 12, 38d, 50

Badness: 0.0381

13-limit

Commas: 81/80, 105/104, 126/125, 245/242

POTE generator: ~3/2 = 695.836

Map: [<2 0 -8 -26 -31 -40|, <0 1 4 10 12 15|]

EDOs: 12f, 50

Badness: 0.0288

Unidecimal meantone aka Huygens

See also: Meantone vs meanpop

Commas: 81/80, 126/125, 99/98

11-limit minimax

[[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⟩]

Eigenmonzos: 2, 11/9

valid range: [696.774, 700.000] (31 to 12)

nice range: [691.202, 701.955]

strict range: [696.774, 700.000]

POTE generator: ~3/2 = 696.967

Mapping generator: ~3

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

Map: [<1 0 -4 -13 -25|, <0 1 4 10 18|]

Generators: 2, 3

EDOs: 12, 31, 43, 74, 105, 198be

Badness: 0.0170

• Twinkle canon – 74 edo by Claudi Meneghin

Tridecimal meantone

Commas: 66/65, 81/80, 99/98, 105/104

POTE generator: ~3/2 = 696.642

Mapping generator: ~3

Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]

EDOs: 12f, 31, 43f

Badness: 0.0180

Grosstone

Commas: 81/80, 99/98, 126/125, 144/143

valid range: [696.774, 700.000] (31 to 12)

nice range: [691.202, 701.955]

strict range: [696.774, 700.000]

POTE generator: ~3/2 = 697.264

Mapping generator: ~3

Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|]

EDOs: 12, 31, 43, 74, 105

Badness: 0.0259

Meridetone

Commas: 78/77, 81/80, 99/98, 126/125

POTE generator: ~3/2 = 697.529

Mapping generator: ~3

Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|]

EDOs: 12f, 31f, 43

Badness: 0.0264

Hemimeantone

Commas: 81/80, 99/98, 126/125, 169/168

POTE generator: ~52/45 = 250.304

Mapping generator: ~26/15

Map: [<1 0 -4 -13 -25 -5|, <0 2 8 20 36 11|]

EDOs: 19e, 43, 62, 167bef

Badness: 0.0314

Meanpop

See also: Meantone vs meanpop

Commas: 81/80, 126/125, 385/384

11-limit minimax 1/4 comma

[[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⟩]

Eigenmonzos: 2, 5

valid range: [694.737, 696.774] (19 to 31)

nice range: [691.202, 701.955]

strict range: [694.737, 696.774]

POTE generator: 696.434

Mapping generator: ~3

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

Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]

Generators: 2, 3

EDOs: 12, 19, 31, 50, 81

Badness: 0.0215

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

13-limit Meanpop

Commas: 81/80, 105/104, 126/125, 144/143

valid range: [694.737, 696.774] (19 to 31)

nice range: [691.202, 701.955]

strict range: [694.737, 696.774]

POTE generator: ~3/2 = 696.211

Mapping generator: ~3

Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|]

EDOs: 12ef, 19, 31, 50, 81

Badness: 0.0209

Meanplop

Commas: 65/64, 78/77, 81/80, 91/90

POTE generator: ~3/2 = 696.202

Mapping generator: ~3

Map: [<1 0 -4 -13 24 10|, <0 1 4 10 -13 -4|]

EDOs: 12e, 19, 31f, 50ff

Badness: 0.0277

Meanenneadecal

Commas: 45/44, 56/55, 81/80

POTE generator: ~3/2 = 696.250

Mapping generator: ~3

Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]

EDOs: 7d, 12, 19, 31e, 50ee

Badness: 0.0214

13-limit

Commas: 45/44, 56/55, 78/77, 81/80

POTE generator: ~3/2 = 696.146

Mapping generator: ~3

Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]

EDOs: 12f, 19, 31e, 50ee

Badness: 0.0212

Vincenzo

Commas: 45/44, 56/55, 65/64, 81/80

POTE generator: ~3/2 = 695.060

Mapping generator: ~3

Map: [<1 0 -4 -13 -6 10|, <0 1 4 10 6 -4|]

EDOs: 7d, 12, 19, 26d

Badness: 0.0248

17-limit

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

POTE generator: ~3/2 = 695.858

Map: [<1 0 -4 -13 -6 10 12|, <0 1 4 10 6 -4 -5|]

EDOs: 7d, 12, 19

Badness: 0.0255

19-limit

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

POTE generator: ~3/2 = 696.131

Map: [<1 0 -4 -13 -6 10 12 9|, <0 1 4 10 6 -4 -5 -3|]

EDOs: 7d, 12, 19

Badness: 0.0223

23-limit

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

POTE generator: ~3/2 = 696.044

Map: [<1 0 -4 -13 -6 10 12 9 14|, <0 1 4 10 6 -4 -5 -3 -6|]

EDOs: 7d, 12, 19

Badness: 0.0201

29-limit

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

POTE generator: ~3/2 = 695.913

Map: [<1 0 -4 -13 -6 10 12 9 14 8|, <0 1 4 10 6 -4 -5 -3 -6 -2|]

EDOs: 7d, 12, 19

Badness: 0.0182

31-limit

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

POTE generator: ~3/2 = 695.750

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

EDOs: 7d, 12, 19

Badness: 0.0171

37-limit

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

POTE generator: ~3/2 = 695.603

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

EDOs: 7d, 12, 19

Badness: 0.0161

41-limit

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

POTE generator: ~3/2 = 695.696

Map: [<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|]

EDOs: 7d, 12, 19

Badness: 0.0154

43-limit

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

POTE generator: ~3/2 = 695.688

Map: [<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|]

EDOs: 7d, 12, 19

Badness: 0.0139

47-limit

Commas: 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

POTE generator: ~3/2 = 695.676

Map: [<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|]

EDOs: 7d, 12, 19

Badness: 0.0138

Meanundeci

Commas: 33/32, 55/54, 77/75

POTE generator: ~3/2 = 694.689

Mapping generator: ~3

Map: [<1 0 -4 -13 5|, <0 1 4 10 -1|]

EDOs: 5d, 7d, 12e, 19e

Badness: 0.0315

13-limit

Commas: 33/32, 55/54, 65/64, 77/75

POTE generator: ~3/2 = 694.764

Mapping generator: ~3

Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|]

EDOs: 7d, 12e, 19e

Badness: 0.0263

Meanundec

Commas: 27/26, 40/39, 45/44, 56/55

POTE generator: ~3/2 = 697.254

Mapping generator: ~3

Map: [<1 0 -4 -13 -6 -1|, <0 1 4 10 6 3|]

EDOs: 7d, 12f, 19f, 31eff

Badness: 0.0242

Flattone

Commas: 81/80, 525/512

The wedgie for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished seventh interval. Other intervals are 7/6, a diminished third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are 26edo, 45edo and 64edo.

7-limit minimax

[[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⟩]

Eigenmonzos: 2, 7/5

9-limit minimax

[[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⟩]

Eigenmonzos: 2, 9/7

valid range: [692.308, 694.737] (26 to 19)

nice range: [692.353, 701.955]

strict range: [692.353, 694.737]

POTE generator: 693.779

Mapping generator: ~3

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

Map: [<1 0 -4 17|, <0 1 4 -9|]

Wedgie: <<1 4 -9 4 -17 -32||

Generators: 2, 3

EDOs: 7, 19, 26, 45

Badness: 0.0386

11-limit

Commas: 45/44, 81/80, 385/384

valid range: [692.308, 694.737] (26 to 19)

nice range: [682.502, 701.955]

strict range: [692.308, 694.737]

POTE generator: ~3/2 = 693.126

Mapping generator: ~3

Map: [<1 0 -4 17 -6|, <0 1 4 -9 6|]

EDOs: 7, 19, 26, 45, 71bc, 116bcde

Badness: 0.0338

13-limit

45/44, 65/64, 78/77, 81/80

valid range: [692.308, 694.737] (26 to 19)

nice range: [682.502, 701.955]

strict range: [692.308, 694.737]

POTE generator: ~3/2 = 693.058

Mapping generator: ~3

Map: [<1 0 -4 17 -6 10|, <0 1 4 -9 6 -4|]

EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef

Badness: 0.0223

Dominant

Commas: 36/35, 64/63

The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now 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.

valid range: [700.000, 720.000] (12 to 5)

nice range: [694.786, 715.587]

strict range: [700.000, 715.587]

POTE generator: 701.573

Mapping generator: ~3

Map: [<1 0 -4 6|, <0 1 4 -2|]

Wedgie: <<1 4 -2 4 -6 -16||

EDOs: 5, 7, 12, 17c, 29cd

Badness: 0.0207

11-limit

Commas: 36/35, 64/63, 56/55

valid range: [700.000, 705.882] (12 to 17)

nice range: [691.202, 715.587]

strict range: [700.000, 705.882]

POTE generator: ~3/2 = 703.254

Mapping generator: ~3

Map: [<1 0 -4 6 13|, <0 1 4 -2 -6|]

EDOs: 5, 12, 17c, 29cde

Badness: 0.0242

13-limit

Commas: 36/35, 56/55, 64/63, 66/65

valid range: 705.882 (17)

nice range: [691.202, 715.587]

strict range:705.882

POTE generator: ~3/2 = 703.636

Map: [<1 0 -4 6 13 18|, <0 1 4 -2 -6 -9|]

EDOs: 12f, 17c, 29cdef

Badness: 0.0241

Dominion

Commas: 26/25, 36/35, 56/55, 64/63

POTE generator: ~3/2 = 704.905

Map: [<1 0 -4 6 13 -9|, <0 1 4 -2 -6 8|]

EDOs: 5, 12, 17c, 46cde

Badness: 0.0273

Domineering

Commas: 36/35, 45/44, 64/63

POTE generator: ~3/2 = 698.776

Mapping generator: ~3

Map: [<1 0 -4 6 -6|, <0 1 4 -2 6|]

EDOs: 5e, 7, 12, 19d, 43de

Badness: 0.0220

13-limit

Commas: 36/35, 45/44, 52/49, 64/63

POTE generator: ~3/2 = 695.762

Mapping generator: ~3

Map: [<1 0 -4 6 -6 10|, <0 1 4 -2 6 -4|]

EDOs: 5ef, 7, 12, 19d, 31def

Badness: 0.0270

17-limit

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

POTE generator: ~3/2 = 696.115

Mapping generator: ~3

Map: [<1 0 -4 6 -6 10 12|, <0 1 4 -2 6 -4 -5|]

EDOs: 5ef, 7, 12, 19d, 31def

Badness: 0.0245

19-limit

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

POTE generator: ~3/2 = 696.217

Mapping generator: ~3

Map: [<1 0 -4 6 -6 10 12 9|, <0 1 4 -2 6 -4 -5 -3|]

EDOs: 5ef, 7, 12, 19d, 31def

Badness: 0.0204

Dominatrix

Commas: 27/26, 36/35, 45/44, 64/63

POTE generator: ~3/2 = 698.544

Mapping generator: ~3

Map: [<1 0 -4 6 -6 -1|, <0 1 4 -2 6 3|]

EDOs: 5e, 7, 12f, 19df

Domination

Commas: 36/35, 64/63, 77/75

POTE generator: ~3/2 = 705.004

Mapping generator: ~3

Map: [<1 0 -4 6 -14|, <0 1 4 -2 11|]

EDOs: 5e, 12e, 17c, 46cd

Badness: 0.0366

13-limit

Commas: 26/25, 36/35, 64/63, 66/65

POTE generator: ~3/2 = 705.496

Mapping generator: ~3

Map: [<1 0 -4 6 -14 -9|, <0 1 4 -2 11 8|]

EDOs: 5e, 12e, 17c

Badness: 0.0274

Arnold

Commas: 22/21, 33/32, 36/35

POTE generator: ~3/2 = 698.491

Mapping generator: ~3

Map: [<1 0 -4 6 5|, <0 1 4 -2 -1|]

EDOs: 5, 7, 12e

Badness: 0.0261

13-limit

Commas: 22/21, 27/26, 33/32, 36/35

POTE generator: ~3/2 = 696.743

Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|]

EDOs: 5, 7, 12ef, 19def

Badness: 0.0233

17-limit

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

POTE generator: ~3/2 = 696.978

Map: [<1 0 -4 6 5 -1 12|, <0 1 4 -2 -1 3 -5|]

EDOs: 5, 7, 12ef, 19def

Badness: 0.0245

19-limit

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

POTE generator: ~3/2 = 697.068

Map: [<1 0 -4 6 5 -1 12 9|, <0 1 4 -2 -1 3 -5 -3|]

EDOs: 5, 7, 12ef, 19def

Badness: 0.0211

Sharptone

Commas: 21/20, 28/27

Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 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.

POTE generator: ~3/2 = 700.140

Mapping generator: ~3

Map: [<1 0 -4 -2|, <0 1 4 3|]

Wedgie: <<1 4 3 4 2 -4||

EDOs: 5, 7d, 12d

Badness: 0.0248

Meanertone

Commas: 21/20, 28/27, 33/32

POTE generator: ~3/2 = 696.615

Map: [<1 0 -4 -2 5|, <0 1 4 3 -1|]

EDOs: 5, 7d, 12de

Badness: 0.0252

Meansept

Commas: 15/14, 81/80

POTE generator: ~3/2 = 682.895

Mapping generator: ~3

Map: [<1 0 -4 -5|, <0 1 4 5|]

Wedgie: <<1 4 5 4 5 0||

EDOs: 5d, 7, 12dd

Badness: 0.0453

11-limit

Commas: 15/14, 22/21, 81/80

POTE generator: ~3/2 = 685.234

Mapping generator: ~3

Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|]

EDOs: 5de, 7, 12dd

Badness: 0.0325

Supermean

Commas: 81/80, 672/625

POTE generator: ~3/2 = 704.889

Map: [<1 0 -4 -21|, <0 1 4 15|]

EDOs: 5d, 12d, 17c, 29c

Badness: 0.1342

11-limit

Commas: 56/55, 81/80, 132/125

POTE generator: ~3/2 = 705.096

Map: [<1 0 -4 -21 -14|, <0 1 4 15 11|]

EDOs: 5de, 12de, 17c, 29c

Badness: 0.0633

13-limit

Commas: 26/25, 56/55, 66/65, 81/80

POTE generator: ~3/2 = 705.094

Map: [<1 0 -4 -21 -14 -9|, <0 1 4 15 11 8|]

EDOs: 5de, 12de, 17c, 29c

Injera

Commas: 50/49, 81/80

The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. 38edo, which is two parallel 19edos, is an excellent tuning for injera.

Origin of the name

valid range: [685.714, 700.000] (14c to 12)

nice range: [688.957, 701.955]

strict range: [688.957, 700.000]

POTE generator: 694.375

Mapping generator: ~3

Map: [<2 0 -8 -7|, <0 1 4 4|]

Wedgie: <<2 8 8 8 7 -4||

EDOs: 12, 26, 38, 102bcd, 140bccd, 178bbccdd

Badness: 0.0311

Music

• Two Pairs of Socks (in 26edo) by Igliashon Jones

11-limit

Commas: 45/44, 50/49, 81/80

valid range: [685.714, 700.000] (14c to 12)

nice range: [682.458, 701.955]

strict range: [685.714, 700.000]

POTE generator: ~3/2 = 692.840

Mapping generator: ~3

Map: [<2 0 -8 -7 -12|, <0 1 4 4 6|]

EDOs: 12, 14c, 26, 90bce, 116bcce

Badness: 0.0231

13-limit

Commas: 45/44, 50/49, 78/77, 81/80

valid range: 692.308 (26)

nice range: [682.458, 701.955]

strict range: 692.308 (26)

POTE generator: ~3/2 = 692.673

Mapping generator: ~3

Map: [<2 0 -8 -7 -12 -21|, <0 1 4 4 6 9|]

EDOs: 12f, 14cf, 26, 38e

Badness: 0.0216

Enjera

Commas: 27/26, 40/39, 45/44, 50/49

POTE generator: ~3/2 = 694.121

Mapping generator: ~3

Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|]

EDOs: 12f, 14c, 26f, 38eff

Badness: 0.0265

Injerous

Commas: 33/32, 50/49, 55/54

POTE generator: ~3/2 = 690.548

Mapping generator: ~3

Map: [<2 0 -8 -7 10|, <0 1 4 4 -1|]

EDOs: 12e, 14c, 26e, 40cee

Badness: 0.0386

Lahoh

Commas: 50/49, 56/55, 81/77

POTE generator: ~3/2 = 699.001

Mapping generator: ~3

Map: [<2 0 -8 -7 7|, <0 1 4 4 0|]

EDOs: 2cd, 12, 14ce

Badness: 0.0431

Godzilla

Deutsch

Main article: Semaphore and Godzilla

Commas: 49/48, 81/80

Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. 19edo is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.

valid range: [240.000, 257.143] (5 to 14c)

nice range: [231.174, 266.871]

strict range: [240.000, 257.143]

POTE generator: ~8/7 = 252.635

Mapping generator: ~7/4

Map: [<1 0 -4 2|, <0 2 8 1|]

Wedgie: <<2 8 1 8 -4 -20||

EDOs: 5, 9c, 14c, 19, 62d, 81d, 143bd

Badness: 0.0267

11-limit

Commas: 45/44, 49/48, 81/80

valid range: [252.632, 257.143] (19 to 14c)

nice range: [231.174, 266.871]

strict range: [252.632, 257.143]

POTE generator: ~8/7 = 254.027

Mapping generator: ~7/4

Map: [<1 0 -4 2 -6|, <0 2 8 1 12|]

EDOs: 14c, 19, 33cd, 52cd

Badness: 0.0290

13-limit

Commas: 45/44, 49/48, 78/77, 81/80

valid range: 694.737 (19)

nice range: [621.581, 737.652]

strict range: 694.737

POTE generator: ~8/7 = 253.603

Mapping generator: ~7/4

Map: [<1 0 -4 2 -6 -5|, <0 2 8 1 12 11|]

EDOs: 14cf, 19, 33cdf, 52cdf

Badness: 0.0225

Semafour

Commas: 33/32, 49/48, 55/54

POTE generator: ~8/7 = 254.042

Mapping generator: ~7/4

Map: [<1 0 -4 2 5|, <0 2 8 1 -2|]

EDOs: 5, 14c, 19e, 33cde

Badness: 0.0285

Varan

Commas: 49/48, 77/75, 81/80

POTE generator: ~8/7 = 251.079

Mapping generator: ~7/4

Map: [<1 0 -4 2 -10|, <0 2 8 1 17|]

EDOs: 19e, 24, 43de

Badness: 0.0396

13-limit

Commas: 49/48, 66/65, 77/75, 81/80

POTE generator: ~8/7 = 251.165

Mapping generator: ~7/4

Map: [<1 0 -4 2 -10 -5|, <0 2 8 1 17 11|]

EDOs: 19e, 24, 43de

Badness: 0.0257

Baragon

Commas: 49/48, 56/55, 81/80

POTE generator: ~8/7 = 251.173

Mapping generator: ~7/4

Map: [<1 0 -4 2 9|, <0 2 8 1 -7|]

EDOs: 19, 24, 43d

Badness: 0.0357

Music

• Godzilla Example by Cameron Bobro
• "Change is on the Wind" in Godzilla[9] by Igliashon Jones

Mohajira

Main article: Mohajira

Commas: 81/80, 6144/6125

Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. 31edo makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.

Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.

7 and 9-limit minimax 1/4 comma

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]

Eigenmonzos: 2, 5

POTE generator: ~128/105 = 348.415

Mapping generator: ~128/105

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

Map: [<1 1 0 6|, <0 2 8 -11|]

Generators: 2, 128/105

Wedgie: <<2 8 -11 8 -23 -48||

EDOs: 7, 24, 31, 38, 55, 69

Badness: 0.0557

11-limit

Commas: 81/80, 121/120, 176/175

11-limit minimax 1/4 comma

[[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⟩]

Eigenmonzos: 2, 5

POTE generator: ~11/9 = 348.477

Mapping generator: ~11/9

Map: [<1 1 0 6 2|, <0 2 8 -11 5|]

Generators: 2, 11/9

EDOs: 7, 24, 31, 38, 55

Badness: 0.0261

13-limit

Commas: 66/65, 81/80, 105/104, 121/120

POTE generator: ~11/9 = 348.558

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|]

EDOs: 7, 24, 31, 38, 55

Badness: 0.0234

17-limit

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

POTE generator: ~11/9 = 348.736

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4 7|, <0 2 8 -11 5 -1 -10|]

EDOs: 7, 24, 31, 38g, 55

Badness: 0.0206

19-limit

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

POTE generator: ~11/9 = 348.810

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4 7 6|, <0 2 8 -11 5 -1 -10 -6|]

EDOs: 7, 24, 31, 38gh, 55

Badness: 0.0173

Ptolemy

Commas: 81/80, 121/120, 525/512

POTE generator: ~11/9 = 346.922

Map: [<1 1 0 8 2|, <0 2 8 -18 5|]

EDOs: 7, 31dd, 38d, 45e, 83bcddee

Badness: 0.0588

13-limit

Commas: 65/64, 81/80, 105/104, 121/120

POTE generator: ~11/9 = 346.910

Map: [<1 1 0 8 2 6|, <0 2 8 -18 5 -8|]

EDOs: 7, 31ddf, 38df, 45ef, 83bcddeeff

Badness: 0.0343

Maqamic

Deutsch

Main article: Maqamic

Commas: 81/80, 36/35, 121/120

Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.

POTE generator: ~11/9 = 350.934

Mapping generator: ~11/9

Map: [<1 1 0 4 2|, <0 2 8 -4 5|]

Generators: 2, 11/9

EDOs: 7, 10c, 17c, 24d, 31d

13-limit

Commas: 81/80, 36/35, 121/120, 144/143

POTE generator: ~11/9 = 350.816

Mapping generator: ~11/9

Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|]

Generators: 2, 11/9

EDOs: 7, 10c, 17c, 24d, 31d

Migration

Commas: 81/80, 121/120, 126/125

POTE generator: ~11/9 = 348.182

Mapping generator: ~11/9

Map: [<1 1 0 -3 2|, <0 2 8 20 5|]

EDOs: 7d, 31, 100de, 131bdee, 162bdee

Badness: 0.0255

13-limit

Commas: 66/65, 81/80, 121/120, 126/125

POTE generator: ~11/9 = 348.490

Map: [<1 1 0 -3 2 4|, <0 2 8 20 5 -1|]

EDOs: 7d, 24d, 31, 55d

Badness: 0.0281

Mohamaq

Commas: 81/80, 392/375

POTE generator: ~25/21 = 350.586

Mapping generator: ~25/21

Map: [<1 1 0 -1|, <0 2 8 13|]

EDOs: 17c, 24, 65c, 89cd

Badness: 0.0777

11-limit

Commas: 56/55, 77/75, 243/242

POTE generator: ~11/9 = 350.565

Mapping generator: ~11/9

Map: [<1 1 0 -1 2|, <0 2 8 13 5|]

EDOs: 17c, 24, 65c, 89cd

Badness: 0.0362

13-limit

Commas: 56/55, 66/65, 77/75, 243/242

POTE generator: ~11/9 = 350.745

Mapping generator: ~11/9

Map: [<1 1 0 -1 2 4|, <0 2 8 13 5 -1|]

EDOs: 17c, 24, 41c, 65c

Badness: 0.0287

Orphic

Commas: 81/80, 5898240/5764801

POTE generator: ~7/6 = 275.794

Mapping generator: ~343/288

Map: [<2 1 -4 4|, <0 4 16 3|]

Wedgie: <<8 32 6 32 -13 -76||

EDOs: 26, 74, 174bd, 248bd

Badness: 0.2588

11-limit

Commas: 81/80, 99/98, 73728/73205

POTE generator: ~7/6 = 275.762

Mapping generator: ~77/64

Map: [<2 1 -4 4 8|, <0 4 16 3 -2|]

EDOs: 26, 48c, 74, 248bd, 322bd

Badness: 0.1015

13-limit

Commas: 81/80, 99/98, 144/143, 2200/2197

POTE generator: ~7/6 = 275.774

Mapping generator: ~63/52

Map: [<2 1 -4 4 8 2|, <0 4 16 3 -2 10|]

EDOs: 26, 48c, 74, 174bd, 248bd, 322bd

Badness: 0.0535

Mothra

Commas: 81/80, 1029/1024

Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using 31edo with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to slendric.

Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.

7 and 9-limit minimax 1/4 comma

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3 0 -1/12 0⟩]

Eigenmonzos: 2, 5

POTE generator: ~8/7 = 232.193

Mapping generator: ~8/7

Algebraic generator: Rabrindanath, largest real root of x⁸ - 3x² + 1, or 232.0774 cents.

Map: [<1 1 0 3|, <0 3 12 -1|]

Generators: 2, 8/7

Wedgie: <<3 12 -1 12 -10 -36||

EDOs: 5, 26, 31, 57, 88

Badness: 0.0371

11-limit

Commas: 81/80, 99/98, 385/384

POTE generator: ~8/7 = 232.031

Mapping generator: ~8/7

Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]

EDOs: 5, 26, 31, 57, 88, 150be, 181bee

Badness: 0.0256

13-limit

Commas: 81/80, 99/98, 105/104, 144/143

POTE generator: ~8/7 = 231.811

Mapping generator: ~8/7

Map: [<1 1 0 3 5 1|, <0 3 12 -1 -8 14|]

EDOs: 5, 26, 31, 57, 88

Badness: 0.0240

Cynder

Commas: 45/44, 81/80, 1029/1024

POTE generator: ~8/7 = 231.317

Mapping generator: ~8/7

Map: [<1 1 0 3 0|, <0 3 12 -1 18|]

EDOs: 5e, 26, 31e, 57e, 83bce

Badness: 0.0557

13-limit

Commas: 45/44, 78/77, 81/80, 640/637

POTE generator: ~8/7 = 231.293

Mapping generator: ~8/7

Map: [<1 1 0 3 0 1|, <0 3 12 -1 18 14|]

EDOs: 5e, 26, 31e, 57e, 83bce

Badness: 0.0341

Mosura

Commas: 81/80, 176/175, 540/539

POTE generator: ~8/7 = 232.419

Mapping generator: ~8/7

Map: [<1 1 0 3 -1|, <0 3 12 -1 23|]

EDOs: 31, 36, 67, 98, 129, 160be, 191bce, 222bce, 253bcee

Badness: 0.0313

13-limit

Commas: 81/80, 144/143, 176/175, 196/195

POTE generator: ~8/7 = 232.640

Mapping generator: ~8/7

Map: [<1 1 0 3 -1 7|, <0 3 12 -1 23 -17|]

EDOs: 31, 36, 67, 98

Badness: 0.0369

Squares

Commas: 81/80, 2401/2400

Squares, with wedgie <<4 16 9 16 3 -24||, 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.

7 and 9 limit minimax 1/4 comma

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

Eigenmonzos: 2, 5

POTE generator: ~9/7 = 425.942

Mapping generator: ~9/7

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

Map: [<1 3 8 6|, <0 -4 -16 -9|]

Generators: 2, 9/7

EDOs: 14c, 17c, 31, 45, 76

Badness: 0.0460

Music:

By Chris Vaisvil

• Square 8

11-limit

Commas: 81/80, 99/98, 121/120

POTE generator: ~9/7 = 425.957

Mapping generator: ~9/7

Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|]

EDOs: 14c, 17c, 31, 45e, 76e

Badness: 0.0216

13-limit

Commas: 66/65, 81/80, 99/98, 121/120

POTE generator: ~9/7 = 425.550

Mapping generator: ~9/7

Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]

EDOs: 14c, 17c, 31, 45e, 79cf

Badness: 0.0255

Agora

Commas: 81/80, 99/98, 105/104, 121/120

POTE generator: ~9/7 = 426.276

Mapping generator: ~9/7

Map: [<1 3 8 6 7 14|, <0 -4 -16 -9 -10 -29|]

EDOs: 14cf, 31, 45ef, 76e

Badness: 0.0245

17-limit

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

POTE generator: ~9/7 = 426.187

Mapping generator: ~9/7

Map: [<1 3 8 6 7 14 8|, <0 -4 -16 -9 -10 -29 -11|]

EDOs: 14cf, 31, 45ef, 76e

19-limit

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

POTE generator: ~9/7 = 426.225

Mapping generator: ~9/7

Map: [<1 3 8 6 7 14 8 11|, <0 -4 -16 -9 -10 -29 -11 -19|]

EDOs: 14cf, 31, 45ef, 76e

Cuboctahedra

Commas: 81/80, 385/384, 1375/1372

POTE generator: ~9/7 = 425.993

Mapping generator: ~9/7

Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]

EDOs: 14ce, 17ce, 31, 45, 76, 107b

Badness: 0.0568

Liese

Deutsch

Commas: 81/80, 686/675

Liese, with wedgie <<3 12 11 12 9 -8||, 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.

7 and 9 limit minimax 1/4 comma

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

Eigenmonzos: 2, 5

POTE generator: ~10/7 = 632.406

Mapping generator: ~10/7

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

Map: [<1 0 -4 -3|, <0 3 12 11|]

Generators: 2, 10/7

EDOs: 17c, 19, 36, 55, 74d

Badness: 0.0467

Liesel

Commas: 56/55, 81/80, 540/539

POTE generator: ~10/7 = 633.073

Mapping generator: ~10/7

Map: [<1 0 -4 -3 4|, <0 3 12 11 -1|]

EDOs: 17c, 19, 36, 55e, 91cee

Badness: 0.0407

13-limit

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

Commas: 56/55, 78/77, 81/80, 91/90

POTE generator: ~10/7 = ~13/9 = 633.042

Mapping generator: ~10/7

Map: [<1 0 -4 -3 4 0|, <0 3 12 11 -1 7|]

EDOs: 17c, 19, 36, 55ef, 91ceef

Badness: 0.0273

Elisa

Commas: 77/75, 81/80, 99/98

POTE generator: ~10/7 = 633.061

Mapping generator: ~10/7

Map: [<1 0 -4 -3 -5|, <0 3 12 11 16|]

EDOs: 17c, 19e, 36e

Badness: 0.0416

Lisa

Commas: 45/44, 81/80, 343/330

POTE generator: ~10/7 = 631.370

Mapping generator: ~10/7

Map: [<1 0 -4 -3 -6|, <0 3 12 11 18|]

EDOs: 19

Badness: 0.0548

13-limit

Commas: 45/44, 81/80, 91/88, 147/143

POTE generator: ~10/7 = 631.221

Map: [<1 0 -4 -3 -6 0|, <0 3 12 11 18 7|]

EDOs: 19

Badness: 0.0361

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.

Commas: 81/80, 17280/16807

POTE generator: ~54/49 = 139.343

Mapping generator: ~54/49

Map: [<1 1 0 2|, <0 5 20 7|]

Wedgie: <<5 30 7 20 -3 -40||

EDOs: 9c, 17c, 26, 43, 69, 112bd

Badness: 0.1087

11-limit

Commas: 81/80, 99/98, 864/847

POTE generator: ~12/11 = 139.428

Mapping generator: ~12/11

Map: [<1 1 0 2 3|, <0 5 20 7 4|]

EDOs: 9c, 17c, 26, 43, 69

Badness: 0.0479

13-limit

Commas: 78/77, 81/80, 99/98, 144/143

POTE generator: ~13/12 = 139.387

Mapping generator: ~12/11

Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|]

EDOs: 9c, 17c, 26, 43, 69

Badness: 0.0293

17-limit

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

POTE generator: ~13/12 = 139.362

Mapping generator: ~12/11

Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|]

EDOs: 26, 43, 69

Badness: 0.0209

19-limit

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

POTE generator: ~13/12 = 139.313

Mapping generator: ~12/11

Map: [<1 1 0 2 3 3 2 1|, <0 5 20 7 4 6 18 28|]

EDOs: 26, 43, 69

Badness: 0.0182

Meanmag

Commas: 81/80, 3125/3072

POTE generator: ~8/7 = 238.396

Mapping generator: ~7

Map: [<19 30 44 0|, <0 0 0 1|]

Wedgie: <<0 0 19 0 30 44||

EDOs: 19, 38, 57, 76, 95bc

Badness: 0.0770

Undevigintone

Commas: 49/48, 81/80, 126/125

POTE generator: ~11/8 = 538.047

Mapping generator: ~11

Map: [<19 30 44 53 0|, <0 0 0 0 1|]

EDOs: 19, 38d

Badness: 0.0364

13-limit

Commas: 49/48, 65/64, 81/80, 126/125

POTE generator: ~11/8 = 537.061

Map: [<19 30 44 53 0 70|, <0 0 0 0 1 0|]

EDOs: 19, 38d

Badness: 0.0229