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.
Contents
- 1 5-limit meantone
- 2 Septimal meantone
- 3 Flattone
- 4 Dominant
- 5 Sharptone
- 6 Meansept
- 7 Supermean
- 8 Injera
- 9 Godzilla
- 10 Mohajira
- 11 Ptolemy
- 12 Maqamic
- 13 Migration
- 14 Mohamaq
- 15 Orphic
- 16 Mothra
- 17 Squares
- 18 Cuboctahedra
- 19 Liese
- 20 Jerome
- 21 Meanmag
- 22 Undevigintone
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
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 15x^3-10x^2-18, 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 x^4+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|]
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><nowiki>]</nowiki>
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 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Scott Joplin's "The Entertainer" tuned into meanpop
Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]
Generators: 2, 3
Badness: 0.0215
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|]
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|]
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 8x^2-4x-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
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||
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||
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.
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
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
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 3x^3-3x^2-1, 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||
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
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
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
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 x^8-3x^2+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||
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 9x^2+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
Badness: 0.0460
Music:
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|]
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
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 x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.
Map: [<1 0 -4 -3|, <0 3 12 11|]
Generators: 2, 10/7
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
