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 period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
5-limit meantone (12&19, 2.3.5)
Period: 1\1
Optimal (POTE) generator: ~3/2 = 696.239
EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50
Scales (Scala files): Meantone5, Meantone7, Meantone12
Period-generator mapping: [⟨1 0 -4], ⟨0 1 4]]
Comma: 81/80
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]
EDOs: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb
Wedgie: ⟨⟨1 4 4]]
Badness: 0.00736
Seven-limit extensions
The 7-limit extensions of meantone 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 (19&31, 2.3.5.7)
The 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, 7/5, C-F#, the tritone, and 21/16, C-E#, the augmented third. Septimal meantone also tempers out the common 7-limit comma 225/224 and is in fact can be defined as the 7-limit temperament that tempers out 81/80 and 225/224.
Period: 1\1
Optimal (POTE) generator: 696.495
EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50
Scales (Scala files): Meantone5, Meantone7, Meantone12
Period-generator mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]
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]
Mapping generator: ~3
Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Wedgie: ⟨⟨1 4 10 4 13 12]]
Vals: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb
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]]
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|]
Badness: 0.0288
Unidecimal meantone aka Huygens
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
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|]
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|]
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|]
Badness: 0.0314
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
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
Badness: 0.0242
Flattone (19&26, 2.3.5.7)
In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished seventh interval (C-Bbb). Other intervals are 7/6, a diminished third (C-Ebb), and 7/5, a doubly diminshed fifth (C-Gbb). Good tunings for flattone are 26edo, 45edo and 64edo.
Period: 1\1
Optimal (POTE) generator: ~3/2 = 693.779
EDO generators: 11\19, 15\26, 26\45, 37\64
Scales (Scala files):
Period-generator mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]
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]
Mapping generator: ~3
Algebraic generator: Squarto, the positive root of 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
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
Godzilla (19&24, 2.3.5.7)
Godzilla 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 close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
Period: 1\1
Optimal (POTE) generator: ~8/7 = 252.635
EDO generators: 4\19, 5\24, 9\43, 13\62
Scales (Scala files):
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|]
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|]
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|]
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|]
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|]
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|]
Badness: 0.0357
Music
- Godzilla Example by Cameron Bobro
- "Change is on the Wind" in Godzilla[9] by Igliashon Jones
Mohajira (24&31, 2.3.5.7)
Commas: 81/80, 6144/6125
Mohajira 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||
Badness: 0.0557
11-limit
Period: 1\1
Optimal (POTE) generator: ~11/9 = 348.477
Scales (Scala files):
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|]
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|]
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|]
Badness: 0.0173
Dominant (12&17c, 2.3.5.7)
Commas: 36/35, 64/63
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
Badness: 0.0211
Sharptone
Commas: 21/20, 28/27
Sharptone 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|]
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||
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|]
Badness: 0.0325
Supermean
Commas: 81/80, 672/625
POTE generator: ~3/2 = 704.889
Map: [<1 0 -4 -21|, <0 1 4 15|]
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|]
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|]
Injera (12&26, 2.3.5.7)
Commas: 50/49, 81/80
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.
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|]
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|]
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|]
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|]
Badness: 0.0431
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
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|]
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|]
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|]
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|]
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||
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 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||
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|]
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|]
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|]
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|]
Badness: 0.0369
Squares
Commas: 81/80, 2401/2400
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.
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
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|]
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|]
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|]
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|]
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 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
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|]
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|]
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|]
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|]
Badness: 0.0361
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.
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|]
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|]
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|]
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|]
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||
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|]
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|]
Badness: 0.0229