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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.

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 (patent val edo list is complete): 5, 7, 12, 19, 24, 26, 31, 36, 38, 43, 45, 50, 55, 57, 62, 67, 69, 74, 76, 81, 86, 88, 93, 98, 100, 105, 117, 129, 212b
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

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, 81, 143b
Badness: 0.0137

Bimeantone

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: 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: 7, 12, 31, 105, 198be
Badness: 0.0170

Twinkle canon – 74 edo by Claudi Meneghin

Tridecimal meantone

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

valid range: 697.674 (43)
nice range: [691.202, 701.955]
strict range: 697.674

POTE generator: ~3/2 = 696.642
Mapping generator: ~3

Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]
EDOs: 12, 19, 31, 267, 298
Badness: 0.0180

Grosstone

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

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
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: 43, 117df, 160bdf, 203bcdef
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: 43, 62, 167bef, 229bef
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 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
EDOs: 12, 19, 31, 81, 112
Badness: 0.0215

Twinkle canon – 50 edo by Claudi Meneghin

13-limit Meanpop

Commas: 81/80, 105/104, 144/143, 196/195

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: 19, 31, 50, 81, 131bd, 212bdf
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, 50f
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: 7, 12, 19, 31e, 50e
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: 19, 31e, 50e]
Badness: 0.0212

Vincenzo

Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22

POTE generator: ~3/2
Mapping generator: ~3

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

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: 12e, 19e
Badness: 0.0315

13-limit

Commas: 33/32, 55/54, 77/75, 729/728

POTE generator: ~3/2 = 694.764
Mapping generator: ~3

Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|]
EDOs: 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: 12f, 19f, 31ef
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
EDOs: 7, 19, 45, 64
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, 53, 65
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: 7, 12, 43de
Badness: 0.0220

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: 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: 17c
Badness: 0.0274

Twelve

Commas: 81/80 64/63 45/44 65/64 256/255 153/152

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: 7, 12, 19d, 31def
Badness: 0.0204

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, 40/39

POTE generator: ~3/2 = 696.743
Mapping generator: ~3

Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|]
EDOs: 5, 7, 12ef, 19def, 31def
Badness: 0.0233

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: 7, 12f
Badness: 0.0183

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.

POTE generator: 700.140
Mapping generator: ~3

Map: [<1 0 -4 -2|, <0 1 4 3|]
Wedgie: <<1 4 3 4 2 -4||
EDOs: 5, 12
Badness: 0.0248

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: 7
Badness: 0.0453

11-limit

Commas: 15/14, 22/21, 125/121

POTE generator: ~3/2 = 685.234
Mapping generator: ~3

Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|]
EDOs: 7
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: 17c, 46c
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: 17c, 46c
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: 17c, 46c

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, 140bcd, 178bcd
Badness: 0.0311

Two Pairs of Socks (in 26edo) by Igliashon Calvin Jones-Coolidge
Injera Jam (in 26edo) by Zach Curley

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, 116bce
Badness: 0.0231

13-limit

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

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: 26, 104bcf
Badness: 0.0216

Enjera

Commas: 27/26, 40/39, 45/44, 99/98

POTE generator: ~3/2 = 694.121
Mapping generator: ~3

Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|]
EDOs: 12f, 26f, 38ef
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, 40ce
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: 12
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

Deutsch

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||
EDOs: 7, 24, 31
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
Badness: 0.0261

13-limit

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

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, 117ef, 148bef
Badness: 0.0234

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, 38d, 45e, 83bcde
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, 38df, 45ef, 83bcdef
Badness: 0.0343

Maqamic

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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: 31, 100de, 131bde, 162bde
Badness: 0.0255

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||
EDOs: 5, 26, 31
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, 88, 150, 181
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: 26, 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: 26, 57e, 83bce
Badness: 0.0341

Mosura

Commas: 81/80, 176/175, 1029/1024

POTE generator: ~8/7 = 232.419
Mapping generator: ~8/7

Map: [<1 1 0 3 -1|, <0 3 12 -1 23|]
EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce
Badness: 0.0313

13-limit

Commas: 81/80, 144/143, 176/175, 1029/1024

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, 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
EDOs: 14, 31, 262, 293
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: 5, 8, 11, 14, 17, 31
Badness: 0.0216

13-limit

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

POTE generator: ~9/7 = 425.550
Mapping generator: ~9/7

Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]
EDOs: 17c, 31, 79cf, 110cef, 141cef
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: 31, 45ef, 76e
Badness: 0.0245

Cuboctahedra

11-limit

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: 14, 31, 45, 200
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
EDOs: 17, 19, 55, 74
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, 91ce
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, 91cef
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: 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
Mapping generator: ~10/7

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: 8, 9, 17, 26, 43, 112
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: 8, 9, 17, 26, 43, 241
Badness: 0.0479

13-limit

Commas: 77/78, 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: 8, 9, 17, 26, 43, 155, 198
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: 8, 9, 17, 26, 43, 155
Badness: 0.0209

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, 57, 76, 171bcd
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