Meantone family

valid range: [685.714, 720.000] (4\7 to 3\5)
nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
strict range: [694.786, 701.955]

Vals: 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb

Badness: 0.00736

Scales: meantone5, meantone7, meantone12

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

Main article: Meantone

See also: Wikipedia: Septimal meantone temperament

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 tempers out the common 7-limit commas 126/125 and 225/224 and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.

Subgroup: 2.3.5.7

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Wedgie: ⟨⟨1 4 10 4 13 12]]

POTE generator: ~3/2 = 696.495

7- and 9-odd-limit

[[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] (11\19 to 7\12)
nice range: [694.786, 701.955] (1/3 comma to Pythagorean)
strict range: [694.786, 700.000]

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

Vals: 12, 19, 31, 81, 112b, 143b

Badness: 0.0137

Scales: meantone5, meantone7, meantone12

Unidecimal meantone aka Huygens

See also: Meantone vs meanpop

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 99/98

Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]

POTE generator: ~3/2 = 696.967

Minimax tuning:

11-odd-limit

[[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

Tuning ranges:

valid range: [696.774, 700.000] (18\31 to 7\12)
nice range: [691.202, 701.955]
strict range: [696.774, 700.000]

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

Vals: 12, 19e, 31, 105, 136b, 167be, 198be

Badness: 0.0170

Music

Twinkle canon – 74 edo by Claudi Meneghin

Tridecimal meantone

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 105/104

Mapping: [⟨1 0 -4 -13 -25 -20], ⟨0 1 4 10 18 15]]

POTE generator: ~3/2 = 696.642

Vals: 12f, 19e, 31

Badness: 0.0180

Grosstone

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 -25 29], ⟨0 1 4 10 18 -16]]

POTE generator: ~3/2 = 697.264

Tuning ranges:

valid range: [696.774, 700.000] (18\31 to 7\12)
nice range: [691.202, 701.955]
strict range: [696.774, 700.000]

Vals: 12, 19ef, 31, 43, 74

Meridetone

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 126/125

Mapping: [⟨1 0 -4 -13 -25 -39], ⟨0 1 4 10 18 27]]

POTE generator: ~3/2 = 697.529

Vals: 12f, 31f, 43

Badness: 0.0264

Hemimeantone

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 169/168

Mapping: [⟨1 0 -4 -13 -25 -5], ⟨0 2 8 20 36 11]]

Mapping generators: ~2, ~26/15

POTE generator: ~15/13 = 250.304

Vals: 19e, 43, 62, 167bef

Badness: 0.0314

Meanpop

See also: Meantone vs meanpop

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 385/384

Mapping: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]

Mapping generator: ~2, ~3

POTE generator: ~3/2 = 696.434

Minimax tuning:

11-odd-limit: 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

Tuning ranges:

valid range: [694.737, 696.774] (11\19 to 18\31)
nice range: [691.202, 701.955]
strict range: [694.737, 696.774]

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

Vals: 12e, 19, 31, 81

Badness: 0.0215

Music

13-limit Meanpop

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 24 -20], ⟨0 1 4 10 -13 15]]

Mapping generator: ~2, ~3

POTE generator: ~3/2 = 696.211

Tuning ranges:

valid range: [694.737, 696.774] (11\19 to 18\31)
nice range: [691.202, 701.955]
strict range: [694.737, 696.774]

Vals: 12ef, 19, 31, 50, 81, 131bd, 212bbddf

Badness: 0.0209

Meanplop

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 81/80, 91/90

Mapping: [⟨1 0 -4 -13 24 10], ⟨0 1 4 10 -13 -4]]

POTE generator: ~3/2 = 696.202

Vals: 12e, 19, 31f, 50ff, 81fff

Badness: 0.0277

Meanenneadecal

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 81/80

Mapping: [⟨1 0 -4 -13 -6], ⟨0 1 4 10 6]]

POTE generator: ~3/2 = 696.250

Vals: 7d, 12, 19, 31e, 50ee

Badness: 0.0214

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 81/80

Mapping: [⟨1 0 -4 -13 -6 -20], ⟨0 1 4 10 6 15]]

POTE generator: ~3/2 = 696.146

Vals: 12f, 19, 31e, 50ee

Badness: 0.0212

Vincenzo

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 81/80

Mapping: [⟨1 0 -4 -13 -6 10], ⟨0 1 4 10 6 -4]]

POTE generator: ~3/2 = 695.060

Vals: 7d, 12, 19

Badness: 0.0248

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -4 -13 -6 10 12], ⟨0 1 4 10 6 -4 -5]]

POTE generator: ~3/2 = 695.858

Vals: 7d, 12, 19

Badness: 0.0255

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 -4 -13 -6 10 12 9], ⟨0 1 4 10 6 -4 -5 -3]]

POTE generator: ~3/2 = 696.131

Vals: 7d, 12, 19

Badness: 0.0223

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14], ⟨0 1 4 10 6 -4 -5 -3 -6]]

POTE generator: ~3/2 = 696.044

Vals: 7d, 12, 19

Badness: 0.0201

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

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

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8], ⟨0 1 4 10 6 -4 -5 -3 -6 -2]]

POTE generator: ~3/2 = 695.913

Vals: 7d, 12, 19

Badness: 0.0182

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

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

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

POTE generator: ~3/2 = 695.750

Vals: 7d, 12, 19

Badness: 0.0171

37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37

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

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

POTE generator: ~3/2 = 695.603

Vals: 7d, 12, 19

Badness: 0.0161

41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41

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

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8]]

POTE generator: ~3/2 = 695.696

Vals: 7d, 12, 19

Badness: 0.0154

43-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43

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

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1]]

POTE generator: ~3/2 = 695.688

Vals: 7d, 12, 19

Badness: 0.0139

47-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47

Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123

Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1]]

POTE generator: ~3/2 = 695.676

Vals: 7d, 12, 19

Badness: 0.0138

Meanundec

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 40/39, 45/44, 56/55

Mapping: [⟨1 0 -4 -13 -6 -1], ⟨0 1 4 10 6 3]]

POTE generator: ~3/2 = 697.254

Vals: 7d, 12f, 19f, 31eff

Badness: 0.0242

Meanundeci

Subgroup: 2.3.5.7.11

Comma list: 33/32, 55/54, 77/75

Mapping: [⟨1 0 -4 -13 5], ⟨0 1 4 10 -1]]

POTE generator: ~3/2 = 694.689

Vals: 7d, 12e, 19e

Badness: 0.0315

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 55/54, 65/64, 77/75

Mapping: [⟨1 0 -4 -13 5 10], ⟨0 1 4 10 -1 -4]]

POTE generator: ~3/2 = 694.764

Vals: 7d, 12e, 19e

Badness: 0.0263

Bimeantone

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

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 245/242

Mapping: [⟨2 0 -8 -26 -31], ⟨0 1 4 10 12]]

Mapping generators: ~63/44, ~3

POTE generator: ~3/2 = 696.016

Vals: 12, 26de, 38d, 50

Badness: 0.0381

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 245/242

Mapping: [⟨2 0 -8 -26 -31 -40], ⟨0 1 4 10 12 15]]

Mapping generators: ~55/39, ~3

POTE generator: ~3/2 = 695.836

Vals: 12f, 26deff, 38df, 50

Badness: 0.0288

Flattone

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.

Subgroup: 2.3.5.7

Comma list: 81/80, 525/512

Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]

Wedgie: ⟨⟨1 4 -9 4 -17 -32]]

POTE generator: ~3/2 = 693.779

7-odd-limit

[[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-odd-limit

[[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] (15\26 to 11\19)
nice range: [692.353, 701.955]
strict range: [692.353, 694.737]

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

Vals: 7, 19, 26, 45

Badness: 0.0386

Scales: flattone12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 385/384

Mapping: [⟨1 0 -4 17 -6], ⟨0 1 4 -9 6]]

POTE generator: ~3/2 = 693.126

Tuning ranges:

valid range: [692.308, 694.737] (15\26 to 11\19)
nice range: [682.502, 701.955]
strict range: [692.308, 694.737]

Vals: 7, 19, 26, 45, 71bc, 116bcde

Badness: 0.0338

Scales: flattone12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 81/80

Mapping: [⟨1 0 -4 17 -6 10], ⟨0 1 4 -9 6 -4]]

POTE generator: ~3/2 = 693.058

Tuning ranges:

valid range: [692.308, 694.737] (15\26 to 11\19)
nice range: [682.502, 701.955]
strict range: [692.308, 694.737]

Vals: 7, 19, 26, 45f, 71bcf, 116bcdef

Badness: 0.0223

Scales: flattone12

Dominant

The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is 12edo, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with 29edo, 41edo, or 53edo.

Subgroup: 2.3.5.7

Comma list: 36/35, 64/63

Mapping: [⟨1 0 -4 6], ⟨0 1 4 -2]]

Wedgie: ⟨⟨1 4 -2 4 -6 -16]]

POTE generator: ~3/2 = 701.573

valid range: [700.000, 720.000] (7\12 to 3\5)
nice range: [694.786, 715.587]
strict range: [700.000, 715.587]

Vals: 5, 7, 12, 41cd, 53cdd, 65ccddd

Badness: 0.0207

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 64/63

Mapping: [⟨1 0 -4 6 13], ⟨0 1 4 -2 -6]]

Tuning ranges:

valid range: [700.000, 705.882] (7\12 to 10\17)
nice range: [691.202, 715.587]
strict range: [700.000, 705.882]

POTE generator: ~3/2 = 703.254

Vals: 5, 12, 17c, 29cde

Badness: 0.0242

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 64/63, 66/65

Mapping: [⟨1 0 -4 6 13 18], ⟨0 1 4 -2 -6 -9]]

POTE generator: ~3/2 = 703.636

Tuning ranges:

valid range: 705.882 (10\17)
nice range: [691.202, 715.587]
strict range: 705.882

Vals: 12f, 17c, 29cdef

Badness: 0.0241

Dominion

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 64/63

Mapping: [⟨1 0 -4 6 13 -9], ⟨0 1 4 -2 -6 8]]

Vals: 5, 12, 17c, 46cde

POTE generator: ~3/2 = 704.905

Badness: 0.0273

Domineering

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 64/63

Mapping: [⟨1 0 -4 6 -6], ⟨0 1 4 -2 6]]

POTE generator: ~3/2 = 698.776

Vals: 5e, 7, 12, 19d, 43de

Badness: 0.0220

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 45/44, 52/49, 64/63

Mapping: [⟨1 0 -4 6 -6 10], ⟨0 1 4 -2 6 -4]]

POTE generator: ~3/2 = 695.762

Vals: 5ef, 7, 12, 19d, 31def

Badness: 0.0270

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -4 6 -6 10 12], ⟨0 1 4 -2 6 -4 -5]]

POTE generator: ~3/2 = 696.115

Vals: 5ef, 7, 12, 19d, 31def

Badness: 0.0245

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 -4 6 -6 10 12 9], ⟨0 1 4 -2 6 -4 -5 -3]]

POTE generator: ~3/2 = 696.217

Vals: 5ef, 7, 12, 19d, 31def

Badness: 0.0204

Dominatrix

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 64/63

Mapping: [⟨1 0 -4 6 -6 -1], ⟨0 1 4 -2 6 3]]

POTE generator: ~3/2 = 698.544

Vals: 5e, 7, 12f, 19df

Domination

Subgroup: 2.3.5.7.11

Comma list: 36/35, 64/63, 77/75

Mapping: [⟨1 0 -4 6 -14], ⟨0 1 4 -2 11]]

POTE generator: ~3/2 = 705.004

Vals: 5e, 12e, 17c, 46cd

Badness: 0.0366

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 64/63, 66/65

Mapping: [⟨1 0 -4 6 -14 -9], ⟨0 1 4 -2 11 8]]

POTE generator: ~3/2 = 705.496

Vals: 5e, 12e, 17c

Badness: 0.0274

Arnold

Subgroup: 2.3.5.7.11

Comma list: 22/21, 33/32, 36/35

Mapping: [⟨1 0 -4 6 5], ⟨0 1 4 -2 -1]]

POTE generator: ~3/2 = 698.491

Vals: 5, 7, 12e

Badness: 0.0261

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 27/26, 33/32, 36/35

Mapping: [⟨1 0 -4 6 5 -1], ⟨0 1 4 -2 3]]

POTE generator: ~3/2 = 696.743

Vals: 5, 7, 12ef, 19def

Badness: 0.0233

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 0 -4 6 5 -1 12], ⟨0 1 4 -2 3 -5]]

POTE generator: ~3/2 = 696.978

Vals: 5, 7, 12ef, 19def

Badness: 0.0245

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 0 -4 6 5 -1 12 9], ⟨0 1 4 -2 3 -5 -3]]

POTE generator: ~3/2 = 697.068

Vals: 5, 7, 12ef, 19def

Badness: 0.0211

Sharptone

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.

Subgroup: 2.3.5.7

Comma list: 21/20, 28/27

Mapping: [⟨1 0 -4 -2], ⟨0 1 4 3]]

Wedgie: ⟨⟨1 4 3 4 2 -4]]

POTE generator: ~3/2 = 700.140

Vals: 5, 7d, 12d

Badness: 0.0248

Meanertone

Subgroup: 2.3.5.7.11

Comma list: 21/20, 28/27, 33/32

Mapping: [⟨1 0 -4 -2 5], ⟨0 1 4 3 -1]]

POTE generator: ~3/2 = 696.615

Vals: 5, 7d, 12de

Badness: 0.0252

Plutus

Subgroup: 2.3.5.7

Comma list: 15/14, 81/80

Mapping: [⟨1 0 -4 -5], ⟨0 1 4 5]]

Wedgie: ⟨⟨1 4 5 4 5 0]]

POTE generator: ~3/2 = 682.895

Vals: 7, 37bcccdd, 44bccccdd

Badness: 0.0453

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 81/80

Mapping: [⟨1 0 -4 -5 -6], ⟨0 1 4 5 6]]

POTE generator: ~3/2 = 685.234

Vals: 5de, 7

Badness: 0.0325

Supermean

Subgroup: 2.3.5.7

Comma list: 81/80, 672/625

Mapping: [⟨1 0 -4 -21], ⟨0 1 4 15]]

POTE generator: ~3/2 = 704.889

Vals: 5d, 12d, 17c, 29c

Badness: 0.1342

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 132/125

Mapping: [⟨1 0 -4 -21 -14], ⟨0 1 4 15 11]]

POTE generator: ~3/2 = 705.096

Vals: 5de, 12de, 17c, 29c

Badness: 0.0633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 66/65, 81/80

Mapping: [⟨1 0 -4 -21 -14 -9], ⟨0 1 4 15 11 8]]

POTE generator: ~3/2 = 705.094

Vals: 5de, 12de, 17c, 29c

Godzilla

Main article: Semaphore and Godzilla

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.

Subgroup: 2.3.5.7

Comma list: 49/48, 81/80

Mapping: [⟨1 0 -4 2], ⟨0 2 8 1]]

Mapping generators: ~2, ~7/4

Wedgie: ⟨⟨2 8 1 8 -4 -20]]

POTE generator: ~8/7 = 252.635

valid range: [240.000, 257.143] (1\5 to 3\14)
nice range: [231.174, 266.871]
strict range: [240.000, 257.143]

Vals: 5, 14c, 19

Badness: 0.0267

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 81/80

Mapping: [⟨1 0 -4 2 -6], ⟨0 2 8 1 12]]

Mapping generators: ~2, ~7/4

POTE generator: ~8/7 = 254.027

Tuning ranges:

valid range: [252.632, 257.143] (4\19 to 3\14)
nice range: [231.174, 266.871]
strict range: [252.632, 257.143]

Vals: 14c, 19, 33cd, 52cd

Badness: 0.0290

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 81/80

Mapping: [⟨1 0 -4 2 -6 -5], ⟨0 2 8 1 12 11]]

Mapping generators: ~2, ~7/4

POTE generator: ~8/7 = 253.603

Tuning ranges:

valid range: 694.737 (4\19)
nice range: [621.581, 737.652]
strict range: 694.737

Vals: 14cf, 19, 33cdff, 52cdf

Badness: 0.0225

Semafour

Subgroup: 2.3.5.7.11

Comma list: 33/32, 49/48, 55/54

Mapping: [⟨1 0 -4 2 5], ⟨0 2 8 1 -2]]

Mapping generators: ~2, ~7/4

POTE generator: ~8/7 = 254.042

Vals: 14c, 19e, 33cdee

Badness: 0.0285

Varan

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 81/80

Mapping: [⟨1 0 -4 2 -10], ⟨0 2 8 1 17]]

Mapping generators: ~2, ~7/4

POTE generator: ~8/7 = 251.079

Vals: 19e, 24, 43de

Badness: 0.0396

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 81/80

Mapping: [⟨1 0 -4 2 -10 -5], ⟨0 2 8 1 17 11]]

Mapping generators: ~2, ~7/4

POTE generator: ~8/7 = 251.165

Vals: 19e, 24, 43de

Badness: 0.0257

Baragon

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 81/80

Mapping: [⟨1 0 -4 2 9], ⟨0 2 8 1 -7]]

Mapping generators: ~2, ~7/4

POTE generator: ~8/7 = 251.173

Vals: 19, 24, 43d

Badness: 0.0357

Injera

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

Origin of the name

Subgroup: 2.3.5.7

Comma list: 50/49, 81/80

Mapping: [⟨2 0 -8 -7], ⟨0 1 4 4]]

Mapping generators: ~7/5, ~3

POTE generator: ~3/2 = 694.375

valid range: [685.714, 700.000] (8\14 to 7\12)
nice range: [688.957, 701.955]
strict range: [688.957, 700.000]

Wedgie: ⟨⟨2 8 8 8 7 -4]]

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

Badness: 0.0311

Music

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

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 81/80

Mapping: [⟨2 0 -8 -7 -12], ⟨0 1 4 4 6]]

Mapping generators: ~7/5, ~3

POTE generator: ~3/2 = 692.840

Tuning ranges:

valid range: [685.714, 700.000] (8\14 to 7\12)
nice range: [682.458, 701.955]
strict range: [685.714, 700.000]

Vals: 12, 14c, 26, 90bce, 116bcce

Badness: 0.0231

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 81/80

Mapping: [⟨2 0 -8 -7 -12 -21], ⟨0 1 4 4 6 9]]

Mapping generators: ~7/5, ~3

POTE generator: ~3/2 = 692.673

Tuning ranges:

valid range: 692.308 (15\26)
nice range: [682.458, 701.955]
strict range: 692.308 (15\26)

Vals: 12f, 14cf, 26, 38e

Badness: 0.0216

Enjera

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 40/39, 45/44, 50/49

Mapping: [⟨2 0 -8 -7 -12 -2], ⟨0 1 4 4 6 3]]

Mapping generators: ~7/5, ~3

POTE generator: ~3/2 = 694.121

Vals: 12f, 14c, 26f, 38eff

Badness: 0.0265

Injerous

Subgroup: 2.3.5.7.11

Comma list: 33/32, 50/49, 55/54

Mapping: [⟨2 0 -8 -7 10], ⟨0 1 4 4 -1]]

Mapping generators: ~7/5, ~3

POTE generator: ~3/2 = 690.548

Vals: 12e, 14c, 26e, 40cee

Badness: 0.0386

Lahoh

Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 81/77

Mapping: [⟨2 0 -8 -7 7], ⟨0 1 4 4 0]]

Mapping generators: ~7/5, ~3

POTE generator: ~3/2 = 699.001

Vals: 12, 14ce

Badness: 0.0431

Mohaha

See also: Subgroup temperaments #Mohaha

Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can 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 2.3.5.11 subgroup). Within this paradigm, mohaha 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, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.

Subgroup: 2.3.5.11

Comma list: 81/80, 121/120

Sval mapping: [⟨1 1 0 2], ⟨0 2 8 5]]

Sval mapping generators: ~2, ~11/9

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]

Gencom: [2 11/9; 81/80 121/120]

POTE generator: ~11/9 = 348.0938

Vals: 7, 17c, 24, 31, 100e, 131bee

Badness: 0.0261

Scales: mohaha7, mohaha10

Mohoho

Subgroup: 2.3.5.11.13

Comma list: 66/65, 81/80, 121/120

Sval mapping: [⟨1 1 0 2 4], ⟨0 2 8 5 -1]]

Sval mapping generators: ~2, ~11/9

Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 8 0 5 -1]]

Gencom: [2 11/9; 66/65 81/80 121/120]

POTE generator: ~11/9 = 348.9155

Vals: 7, 17c, 24, 31, 55, 86ef, 141ceff

Badness: 0.0261

Scales: mohaha7, mohaha10

Mohajira

Main article: Mohajira

Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. 31edo makes for an excellent (7-limit) mohajira tuning, with generator 9/31.

7-limit

Subgroup: 2.3.5.7

Comma list: 81/80, 6144/6125

Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]

Mapping generators: ~2, ~128/105

Wedgie: ⟨⟨2 8 -11 8 -23 -48]]

POTE generator: ~128/105 = 348.415

7- and 9-odd-limit: 1/4 comma

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

Eigenmonzos: 2, 5

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

Vals: 7, 24, 31

Badness: 0.0557

Scales: mohaha7, mohaha10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 176/175

Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 348.477

Minimax tuning:

11-odd-limit: 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

Vals: 7, 24, 31

Badness: 0.0261

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 105/104, 121/120

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 348.558

Vals: 7, 24, 31

Badness: 0.0234

Scales: mohaha7, mohaha10

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 348.736

Vals: 7, 24, 31, 86ef

Badness: 0.0206

Scales: mohaha7, mohaha10

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 348.810

Vals: 7, 24, 31, 55, 86efh

Badness: 0.0173

Scales: mohaha7, mohaha10

Mohamaq

7-limit

Subgroup: 2.3.5.7

Comma list: 81/80, 392/375

Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]

Mapping generators: ~2, ~25/21

POTE generator: ~25/21 = 350.586

Vals: 17c, 24, 65c, 89cd

Badness: 0.0777

Scales: mohaha7, mohaha10

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 243/242

Mapping: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 350.565

Vals: 17c, 24, 65c, 89cd

Badness: 0.0362

Scales: mohaha7, mohaha10

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 243/242

Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 350.745

Vals: 17c, 24, 41c, 65c

Badness: 0.0287

Scales: mohaha7, mohaha10

Migration

Migration takes #Septimal meantone mapping of 7 and #Mohaha mapping of 11.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 126/125

Mapping: [⟨1 1 0 -3 2], ⟨0 2 8 20 5]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 348.182

Vals: 7d, 24d, 31, 100de, 131bdee, 162bdee

Badness: 0.0255

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 121/120, 126/125

Mapping: [⟨1 1 0 -3 2 4], ⟨0 2 8 20 5 -1]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 348.490

Vals: 7d, 24d, 31, 55d

Badness: 0.0281

Ptolemy

Ptolemy takes #Flattone mapping of 7 and #Mohaha mapping of 11.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 525/512

Mapping: [⟨1 1 0 8 2], ⟨0 2 8 -18 5]]

POTE generator: ~11/9 = 346.922

Vals: 7, 31dd, 38d, 45e, 83bcddee

Badness: 0.0588

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 81/80, 105/104, 121/120

Mapping: [⟨1 1 0 8 2 6], ⟨0 2 8 -18 5 -8]]

POTE generator: ~11/9 = 346.910

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

Badness: 0.0343

Maqamic

Main article: Maqamic

Maqamic takes #Dominant mapping of 7 and #Mohaha mapping of 11, so it is 36/35 that vanishes instead of 176/175 as in mohajira. 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.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 36/35, 121/120

Mapping: [⟨1 1 0 4 2], ⟨0 2 8 -4 5]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 350.934

Vals: 7, 17c, 24d, 41cd

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 36/35, 121/120, 144/143

Mapping: [⟨1 1 0 4 2 4], ⟨0 2 8 -4 5 -1]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 350.816

Vals: 7, 17c, 24d, 41cd

Orphic

Subgroup: 2.3.5.7

Comma list: 81/80, 5898240/5764801

Mapping: [⟨2 1 -4 4], ⟨0 4 16 3]]

Mapping generators: ~2401/1728, ~343/288

Wedgie: ⟨⟨8 32 6 32 -13 -76]]

POTE generator: ~7/6 = 275.794

Vals: 26, 48c, 74, 174bd, 248bd

Badness: 0.2588

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 73728/73205

Mapping: [⟨2 1 -4 4 8], ⟨0 4 16 3 -2]]

Mapping generators: ~363/256, ~77/64

POTE generator: ~7/6 = 275.762

Vals: 26, 48c, 74, 248bd, 322bd

Badness: 0.1015

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 144/143, 2200/2197

Mapping: [⟨2 1 -4 4 8 2], ⟨0 4 16 3 -2 10]]

Mapping generators: ~55/39, ~63/52

POTE generator: ~7/6 = 275.774

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

Badness: 0.0535

Mothra

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.

Subgroup: 2.3.5.7

Comma list: 81/80, 1029/1024

Mapping: [⟨1 1 0 3], ⟨0 3 12 -1]]

Mapping generators: ~2, ~8/7

POTE generator: ~8/7 = 232.193

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

Wedgie: ⟨⟨3 12 -1 12 -10 -36]]

7- and 9-odd-limit: 1/4 comma

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

Eigenmonzos: 2, 5

Vals: 5, 26, 31

Badness: 0.0371

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 385/384

Mapping: [⟨1 1 0 3 5], ⟨0 3 12 -1 -8]]

Mapping generators: ~2, ~8/7

POTE generator: ~8/7 = 232.031

Vals: 5, 26, 31, 88, 150be, 181bee

Badness: 0.0256

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 144/143

Mapping: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14]]

Mapping generators: ~2, ~8/7

POTE generator: ~8/7 = 231.811

Vals: 5, 26, 31, 57, 88

Badness: 0.0240

Cynder

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1029/1024

Mapping: [⟨1 1 0 3 0], ⟨0 3 12 -1 18]]

Mapping generators: ~2, ~8/7

POTE generator: ~8/7 = 231.317

Vals: 5e, 26, 57e, 83bce

Badness: 0.0557

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 81/80, 640/637

Mapping: [⟨1 1 0 3 0 1], ⟨0 3 12 -1 18 14]]

Mapping generators: ~2, ~8/7

POTE generator: ~8/7 = 232.293

Vals: 5e, 26, 57e, 83bce

Badness: 0.0341

Mosura

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 540/539

Mapping: [⟨1 1 0 3 -1], ⟨0 3 12 -1 23]]

Mapping generators: ~2, ~8/7

POTE generator: ~8/7 = 232.419

Vals: 31, 129, 160be, 191bce, 222bce, 253bcee

Badness: 0.0313

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 144/143, 176/175, 196/195

Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]

Mapping generators: ~2, ~8/7

POTE generator: ~8/7 = 232.640

Vals: 31, 36, 67, 98

Badness: 0.0369

Music

Prelude for solo piano in mothra16, brat 4 tuning by Chris Vaisvil

Squares

Squares splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. 31edo, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.

Subgroup: 2.3.5.7

Comma list: 81/80, 2401/2400

Mapping: [⟨1 3 8 6], ⟨0 -4 -16 -9]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 425.942

7- and 9-odd-limit: 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

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

Vals: 14c, 17c, 31

Badness: 0.0460

Scales: skwares8, skwares11, skwares14

Music

Square 8 by Chris Vaisvil

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 121/120

Mapping: [⟨1 3 8 6 7], ⟨0 -4 -16 -9 -10]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 425.957

Vals: 14c, 17c, 31

Badness: 0.0216

Scales: skwares8, skwares11, skwares14

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 121/120

Mapping: [⟨1 3 8 6 7 3], ⟨0 -4 -16 -9 -10 2]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 425.550

Vals: 14c, 17c, 31, 79cf

Badness: 0.0255

Scales: skwares8, skwares11, skwares14

Agora

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 121/120

Mapping: [⟨1 3 8 6 7 14], ⟨0 -4 -16 -9 -10 -29]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 426.276

Vals: 14cf, 31, 45ef, 76e

Badness: 0.0245

Scales: skwares8, skwares11, skwares14

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 3 8 6 7 14 8], ⟨0 -4 -16 -9 -10 -29 -11]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 426.187

Vals: 14cf, 31, 76e

Scales: skwares8, skwares11, skwares14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 3 8 6 7 14 8 11], ⟨0 -4 -16 -9 -10 -29 -11 -19]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 426.225

Vals: 14cf, 31, 76e

Scales: skwares8, skwares11, skwares14

Cuboctahedra

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 1375/1372

Mapping: [⟨1 3 8 6 -4], ⟨0 -4 -16 -9 21]]

Mapping generators: ~2, ~9/7

POTE generator: ~9/7 = 425.993

Vals: 14ce, 17ce, 31, 107b

Badness: 0.0568

Scales: skwares8, skwares11, skwares14

Liese