Meantone family: Difference between revisions

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]

Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb

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

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 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: ~3/2 = 696.495

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (12-note MOS, 2.3.5.7 POTE tuning)


#	Cents^[1]	Approximate ratios^[2]
0	0.00	1/1
1	696.5	3/2
2	193.0	9/8, 10/9
3	889.5	5/3
4	386.0	5/4
5	1082.5	15/8, 28/15
6	579.0	7/5
7	75.5	21/20, 25/24, 28/27
8	772.0	14/9, 25/16
9	268.5	7/6
10	965.0	7/4
11	461.4	21/16

↑ octave-reduced
↑ 2.3.5.7, odd limit ≤ 27

Technical data

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 10 4 13 12]]

Minimax tuning:

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] (19 to 12)
nice range: [694.786, 701.955]
strict range: [694.786, 700.000]

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

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness: 0.0137

Bimeantone

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

Period: 1\2

Optimal (POTE) generator: ~3/2 = 696.016

EDO generators: 22\38, 29\50

Scales (Scala files):

Technical data

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

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

Mapping generators: ~63/44, ~3

Optimal ET sequence: 12, 26de, 38d, 50

Badness: 0.0381

13-limit

Period: 1\2

Optimal (POTE) generator: ~3/2 = 695.836

EDO generators: 22\38, 29\50

Scales (Scala files):

Technical data

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

Optimal ET sequence: 12f, 26deff, 38df, 50

Badness: 0.0288

Unidecimal meantone aka Huygens

See also: Meantone vs meanpop

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.967

EDO generators: 18\31, 25\43

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

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] (31 to 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.

Optimal ET sequence: 12, 19e, 31, 105, 136b, 167be, 198be

Badness: 0.0170

Twinkle canon – 74 edo by Claudi Meneghin

Tridecimal meantone

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.642

EDO generators: 18\31

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Optimal ET sequence: 12f, 19e, 31

Badness: 0.0180

Grosstone

Period: 1\1

Optimal (POTE) generator: ~3/2 = 697.264

EDO generators: 18\31, 25\43

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Tuning ranges:

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

Optimal ET sequence: 12, 19ef, 31, 43, 74

Badness: 0.0259

Meridetone

Period: 1\1

Optimal (POTE) generator: ~3/2 = 697.529

EDO generators: 25\43

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Optimal ET sequence: 12f, 31f, 43

Badness: 0.0264

Hemimeantone

Period: 1\1

Optimal (POTE) generator: ~15/13 = 250.304

EDO generators: 9\43, 13\62

Scales (Scala files):

Technical data

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

Optimal ET sequence: 19e, 43, 62, 167bef

Badness: 0.0314

Meanpop

See also: Meantone vs meanpop

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.434

EDO generators: 11\19, 18\31, 29\50

Scales (Scala files):

Technical data

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

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

Mapping generator: ~2, ~3

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] (19 to 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.

Optimal ET sequence: 12e, 19, 31, 81

Badness: 0.0215

13-limit Meanpop

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.211

EDO generators: 11\19, 18\31, 29\50

Scales (Scala files):

Technical data

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

Tuning ranges:

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

Optimal ET sequence: 12ef, 19, 31, 50, 81, 131bd, 212bbddf

Badness: 0.0209

Meanplop

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.202

EDO generators: 11\19, 18\31

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Optimal ET sequence: 12e, 19, 31f, 50ff, 81fff

Badness: 0.0277

Meanenneadecal

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.250

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Optimal ET sequence: 7d, 12, 19, 31e, 50ee

Badness: 0.0214

13-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.146

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Optimal ET sequence: 12f, 19, 31e, 50ee

Badness: 0.0212

Vincenzo

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.060

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Badness: 0.0248

17-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.858

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0255

19-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.131

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0223

23-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.044

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0201

29-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.913

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0182

31-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.750

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0171

37-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.603

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0161

41-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.696

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0154

43-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.688

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0139

47-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 695.676

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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]]

Mapping generators: ~2, ~3

Badness: 0.0138

Meanundec

Period: 1\1

Optimal (POTE) generator: ~3/2 = 697.254

EDO generators: 7\12

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Optimal ET sequence: 7d, 12f, 19f, 31eff

Badness: 0.0242

Meanundeci

Period: 1\1

Optimal (POTE) generator: ~3/2 = 694.689

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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

POTE generator: ~3/2 = 694.689

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

Mapping generator: ~3

Optimal ET sequence: 7d, 12e, 19e

Badness: 0.0315

13-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 694.764

EDO generators: 7\12, 11\19

Scales (Scala files):

Technical data

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

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

Mapping generators: ~2, ~3

Optimal ET sequence: 7d, 12e, 19e

Badness: 0.0263

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.

Period: 1\1

Optimal (POTE) generator: ~3/2 = 693.779

EDO generators: 11\19, 15\26, 26\45, 37\64

Scales (Scala files): Flattone12

Interval table (12-note MOS, 2.3.5.7 POTE tuning)


#	Cents^[1]	Approximate ratios^[2]
0	0.00	1/1
1	693.8	3/2
2	187.6	9/8, 10/9
3	881.3	5/3
4	375.1	5/4, (16/13), (11/9)
5	1068.9	15/8, (24/13), (11/6)
6	562.7	(18/13), (11/8)
7	56.5
8	750.2	(20/13)
9	244.0	8/7
10	937.8	12/7
11	431.6	9/7

↑ octave-reduced
↑ 2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.

Technical data

Comma list: 81/80, 525/512

Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 -9 4 -17 -32]]

Minimax tuning:

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] (26 to 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.

Optimal ET sequence: 7, 19, 26, 45

Badness: 0.0386

11-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 693.126

EDO generators: 11\19, 15\26, 26\45, 37\64

Scales (Scala files): Flattone12

Technical data

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

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

Mapping generators: ~2, ~3

Tuning ranges:

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

Optimal ET sequence: 7, 19, 26, 45, 71bc, 116bcde

Badness: 0.0338

13-limit

Period: 1\1

Optimal (POTE) generator: ~3/2 = 693.058

EDO generators: 11\19, 15\26, 26\45, 37\64

Scales (Scala files): Flattone12

Technical data

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

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

Mapping generators: ~2, ~3

Tuning ranges:

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

Optimal ET sequence: 7, 19, 26, 45f, 71bcf, 116bcdef

Badness: 0.0223

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.

Period: 1\1

Optimal (POTE) generator: ~8/7 = 252.635

EDO generators: 4\19, 5\24, 9\43, 13\62

Scales (Scala files):

Interval table (9-note MOS, 2.3.5.7 POTE tuning)


#	Cents ^[1]	Approximate ratios^[2]
0	0.00	1/1
1	252.6	7/6, 8/7, (15/13)
2	505.3	4/3
3	757.9	14/9, (20/13)
4	1010.5	9/5, 16/9
5	63.2
6	315.8	6/5
7	568.4	7/5, (18/13)
8	821.1	8/5

↑ octave-reduced
↑ 2.3.5.7, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.

Technical data

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]]

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

Optimal ET sequence: 5, 14c, 19

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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 14cf, 19, 33cdff, 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|]

Optimal ET sequence: 14c, 19e, 33cdee

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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 19, 24, 43d

Badness: 0.0357

Music

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

Mohajira

Main article: Mohajira

Commas: 81/80, 6144/6125

Mohajira 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||

Optimal ET sequence: 7, 24, 31, 38, 55, 69

Badness: 0.0557

11-limit

Period: 1\1

Optimal (POTE) generator: ~11/9 = 348.477

EDO generators: 7\24, 9\31

Scales (Scala files):

Interval table (10-note MOS, 2.3.5.7.11 POTE tuning)


#	Cents^[1]	Approximate ratios^[2]
0	0.00	1/1
1	348.5	11/9
2	697.0	3/2
3	1045.4	11/6
4	193.9	9/8
5	542.4	11/8, 15/11
6	890.9	5/3
7	39.3
8	387.8	5/4
9	736.3	32/21

↑ octave-reduced
↑ 2.3.5.7.11, odd limit ≤ 27. JI readings in parentheses are outside the subgroup but are supported by the defining EDOs.

Technical data

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

Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]

Mapping generators: ~2, ~11/9

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

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 7, 24, 31, 38gh, 55

Badness: 0.0173

Dominant

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||

Optimal ET sequence: 5, 7, 12, 17c, 29cd

Badness: 0.0207

11-limit

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

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

nice range: [691.202, 715.587]

strict range: [700.000, 705.882]

POTE generator: ~3/2 = 703.254

Mapping generator: ~3

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

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 5, 7, 12ef, 19def

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||

Optimal ET sequence: 5, 7d, 12d

Badness: 0.0248

Meanertone

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

POTE generator: ~3/2 = 696.615

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

Optimal ET sequence: 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||

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 5de, 12de, 17c, 29c

Injera

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.

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||

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 2cd, 12, 14ce

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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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

Optimal ET sequence: 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

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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||

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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||

Optimal ET sequence: 5, 26, 31, 57, 88

Badness: 0.0371

11-limit

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

POTE generator: ~8/7 = 232.031

Mapping generator: ~8/7

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

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 31, 36, 67, 98

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

Optimal ET sequence: 14c, 17c, 31, 45, 76

Badness: 0.0460

Music:

By Chris Vaisvil

Square 8

11-limit

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

POTE generator: ~9/7 = 425.957

Mapping generator: ~9/7

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

Optimal ET sequence: 14c, 17c, 31, 45e, 76e

Badness: 0.0216

13-limit

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

POTE generator: ~9/7 = 425.550

Mapping generator: ~9/7

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

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 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|]

Optimal ET sequence: 14ce, 17ce, 31, 45, 76, 107b

Badness: 0.0568

Liese