Meantone family

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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 period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

Meantone

Main article: Meantone

Subgroup: 2.3.5

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨1 4 4]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2143

Minimax tuning:

Eigenmonzo basis: 2.5

Tuning ranges:

  • 5-odd-limit diamond monotone: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
  • 5-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955]
  • 5-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]

Optimal GPV sequence: 5, 7, 12, 19, 31, 50, 81, 131b

Badness: 0.007381

Scales: meantone5, meantone7, meantone12

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

Those all have a fifth as generator.

  • Injera adds [-7 8 0 -2⟩ with a half-octave period.
  • Mohajira adds [-23 11 0 2⟩ and splits the fifth in two.
  • Godzilla adds [-4 -1 0 2⟩ with an 8/7 generator, two of which give the fourth (4/3, an octave minus a fifth).
  • Mothra adds [-10 1 0 3⟩ with an 8/7 generator, three of which give the fifth.
  • Liese adds [-9 11 0 -3⟩ with a 10/7 generator, three of which give the twelfth (3/1, an octave plus a fifth).
  • Squares adds [-3 9 0 -4⟩ with a 9/7 generator, four of which give the eleventh (8/3, two octaves minus a fifth).
  • Jerome adds [3 7 0 -5⟩ and slices the fifth in five.

Temperaments discussed elsewhere include plutus.

Mohaha

See also: No-sevens 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 quartertones"; 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 11/10~12/11's, and that maps four 3/2's to 5/1. It has a heptatonic mos with three larger steps and four smaller ones, going sLsLsLs. Taking septimal meantone mapping of 7 leads to #Migration, flattone mapping of 7 leads to #Ptolemy, and dominant mapping of 7 leads to #Neutrominant.

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]

Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.8296

Optimal GPV sequence: 7, 17c, 24, 31, 69e, 100e, 131bee

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]

Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.8794

Optimal GPV sequence: 7, 17c, 24, 31, 55

Scales: mohaha7, mohaha10

Septimal meantone

Deutsch
Main article: Meantone
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The 7/4 of septimal meantone is the augmented sixth (C-A#), and other septimal intervals are 7/6, the augmented second (C-D#), 7/5, the augmented fourth (C-F#), and 21/16, the augmented third (C-E#). 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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9521

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
Eigenmonzo basis: 2.5

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
  • 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 7-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 700.000]
  • 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]

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

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

Badness: 0.013707

Scales: meantone5, meantone7, meantone12

Unidecimal meantone aka Huygens

See also: Meantone vs meanpop

Subgroup: 2.3.5.7.11

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

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.1676

Minimax tuning:

  • 11-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩
[[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⟩]
Eigenmonzo basis: 2.11/9

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]

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

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

Badness: 0.017027

Music

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.8552

Minimax tuning:

Eigenmonzo basis: 2.11/9

Optimal GPV sequence: 12f, 19e, 31

Badness: 0.018048

Meantonic

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 99/98, 105/104, 121/119

Mapping: [⟨1 0 -4 -13 -25 -20 -37], ⟨0 1 4 10 18 15 26]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.6486

Optimal GPV sequence: 12fg, 19eg, 31, 50e

Badness: 0.019037

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 99/98, 105/104, 121/119

Mapping: [⟨1 0 -4 -13 -25 -20 -37 -40], ⟨0 1 4 10 18 15 26 28]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.5551

Optimal GPV sequence: 12fghh, 19egh, 31, 50e

Badness: 0.017846

Meantoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 66/65, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -20 -7], ⟨0 1 4 10 18 15 7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.0360

Optimal GPV sequence: 12f, 19eg, 31g

Badness: 0.019433

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 57/56, 66/65, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -20 -7 -10], ⟨0 1 4 10 18 15 7 9]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2161

Optimal GPV sequence: 12f, 19egh, 31gh

Badness: 0.017437

Huygens

Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 99/98, 105/104, 120/119

Mapping: [⟨1 0 -4 -13 -25 -20 12], ⟨0 1 4 10 18 15 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9080

Optimal GPV sequence: 12f, 19e, 31

Badness: 0.019982

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119

Mapping: [⟨1 0 -4 -13 -25 -20 12 9], ⟨0 1 4 10 18 15 -5 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9308

Optimal GPV sequence: 12f, 19e, 31

Badness: 0.018047

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2582

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [8/13 0 0 1/26 0 -1/26⟩
Eigenmonzo basis: 2.13/7

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 697.674]

Optimal GPV sequence: 12, 31, 43, 74

Badness: 0.025899

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 120/119, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 -25 29 12], ⟨0 1 4 10 18 -16 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2996

Optimal GPV sequence: 12, 31, 43, 74g

Badness: 0.020889

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 -25 29 12 9], ⟨0 1 4 10 18 -16 -5 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.3271

Optimal GPV sequence: 12, 31, 43, 74gh

Badness: 0.017611

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5155

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [14/25 -2/25 0 0 0 1/25⟩
Eigenmonzo basis: 2.13/9

Optimal GPV sequence: 12f, 31f, 43

Badness: 0.026421

Meridetonic

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 126/125, 273/272

Mapping: [⟨1 0 -4 -13 -25 -39 -56], ⟨0 1 4 10 18 27 38]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5076

Optimal GPV sequence: 12fg, 31fg, 43

Badness: 0.027706

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 126/125, 153/152, 273/272

Mapping: [⟨1 0 -4 -13 -25 -39 -56 -59], ⟨0 1 4 10 18 27 38 40]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.4848

Optimal GPV sequence: 12fghh, 31fgh, 43

Badness: 0.025315

Meridetoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 78/77, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -39 -7], ⟨0 1 4 10 18 27 7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.6098

Optimal GPV sequence: 12f, 31fg, 43g

Badness: 0.027518

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 57/56, 78/77, 81/80, 85/84, 99/98

Mapping: [⟨1 0 -4 -13 -25 -39 -7 -10], ⟨0 1 4 10 18 27 7 9]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.7012

Optimal GPV sequence: 12f, 19effgh, 31fgh, 43gh

Badness: 0.023613

Sauveuric

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 -25 -39 12], ⟨0 1 4 10 18 27 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5384

Optimal GPV sequence: 12f, 43

Badness: 0.023881

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 -25 -39 12 9], ⟨0 1 4 10 18 27 -5 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5550

Optimal GPV sequence: 12f, 43

Badness: 0.020540

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

Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 948.6109

Optimal GPV sequence: 19e, 43, 62

Badness: 0.031433

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 169/168, 221/220

Mapping: [⟨1 0 -4 -13 -25 -5 -22], ⟨0 2 8 20 36 11 33]]

Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 948.6173

Optimal GPV sequence: 19eg, 43, 62

Badness: 0.023380

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220

Mapping: [⟨1 0 -4 -13 -25 -5 -22 -25], ⟨0 2 8 20 36 11 33 37]]

Optimal tuning (CTE): ~2 = 1\1, ~19/11 = 948.6088

Optimal GPV sequence: 19egh, 43, 62

Badness: 0.018952

Semimeantone

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 847/845

Mapping: [⟨2 0 -8 -26 -50 -59], ⟨0 1 4 10 18 21]]

Mapping generators: ~55/39, ~3

Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 697.1678

Optimal GPV sequence: 12f, 38deefff, 50eff, 62, 136b

Badness: 0.040668

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 221/220, 289/288

Mapping: [⟨2 0 -8 -26 -50 -59 5], ⟨0 1 4 10 18 21 1]]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 697.1740

Optimal GPV sequence: 12f, 50eff, 62, 136bg

Badness: 0.031491

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220

Mapping: [⟨2 0 -8 -26 -50 -59 5 -1], ⟨0 1 4 10 18 21 1 3]]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 697.1871

Optimal GPV sequence: 12f, 50eff, 62

Badness: 0.024206

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.5311

Minimax tuning:

  • 11-odd-limit: ~3/2 = [0 0 1/4⟩
[[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⟩]
Eigenmonzo basis: 2.5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [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 GPV sequence: 12e, 19, 31, 81

Badness: 0.021543

Music

Tridecimal 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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3563

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [4/7 0 0 0 -1/28 1/28⟩
Eigenmonzo basis: 2.13/11

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]

Optimal GPV sequence: 19, 31, 50, 81

Badness: 0.020883

Meanpoppic

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 144/143, 273/272

Mapping: [⟨1 0 -4 -13 24 -20 -37], ⟨0 1 4 10 -13 15 26]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3508

Optimal GPV sequence: 19g, 31, 50, 81, 131bd

Badness: 0.019953

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272

Mapping: [⟨1 0 -4 -13 24 -20 -37 -40], ⟨0 1 4 10 -13 15 26 28]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3471

Optimal GPV sequence: 19gh, 31, 50, 81

Badness: 0.017791

Meanpoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 120/119, 126/125, 144/143

Mapping: [⟨1 0 -4 -13 24 -20 12], ⟨0 1 4 10 -13 15 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4388

Optimal GPV sequence: 19, 31

Badness: 0.022870

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125

Mapping: [⟨1 0 -4 -13 24 -20 12 9], ⟨0 1 4 10 -13 15 -5 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4838

Optimal GPV sequence: 12ef, 19, 31

Badness: 0.020488

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2827

Minimax tuning:

  • 13- and 15-odd-limit: ~3/2 = [11/13 0 0 0 -1/13⟩
Eigenmonzo basis: 2.11

Optimal GPV sequence: 12e, 19, 31f

Badness: 0.027666

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 52/51, 65/64, 78/77, 81/80, 91/90

Mapping: [⟨1 0 -4 -13 24 10 12], ⟨0 1 4 10 -13 -4 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4069

Optimal GPV sequence: 12e, 19

Badness: 0.026836

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 39/38, 52/51, 65/64, 77/76, 81/80, 91/90

Mapping: [⟨1 0 -4 -13 24 10 12 9], ⟨0 1 4 10 -13 -4 -5 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4731

Optimal GPV sequence: 12e, 19

Badness: 0.023540

Meanploid

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 65/64, 78/77, 81/80, 85/84

Mapping: [⟨1 0 -4 -13 24 10 -7], ⟨0 1 4 10 -13 -4 7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.6614

Optimal GPV sequence: 12e, 19g, 31fg

Badness: 0.026094

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 57/56, 65/64, 76/75, 78/77, 81/80

Mapping: [⟨1 0 -4 -13 24 10 -7 -10], ⟨0 1 4 10 -13 -4 7 9]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.0160

Optimal GPV sequence: 12e, 19gh, 31fgh

Badness: 0.023104

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.1527

Tuning ranges:

  • 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
  • 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]

Optimal GPV sequence: 7d, 12, 19, 31e

Badness: 0.021423

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0983

Optimal GPV sequence: 7df, 12f, 19, 31e

Badness: 0.021182

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 81/80, 120/119

Mapping: [⟨1 0 -4 -13 -6 -20 12], ⟨0 1 4 10 6 15 -5]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2161

Optimal GPV sequence: 12f, 19, 31e

Badness: 0.022980

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119

Mapping: [⟨1 0 -4 -13 -6 -20 12 9], ⟨0 1 4 10 6 15 -5 -3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2774

Optimal GPV sequence: 12f, 19, 31e

Badness: 0.020293

Meanenneadecoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 34/33, 45/44, 51/50, 56/55, 78/77

Mapping: [⟨1 0 -4 -13 -6 -20 -7], ⟨0 1 4 10 6 15 7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4501

Optimal GPV sequence: 7dfg, 12f, 19g

Badness: 0.020171

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 78/77

Mapping: [⟨1 0 -4 -13 -6 -20 -7 -10], ⟨0 1 4 10 6 15 7 9]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.121

Optimal GPV sequence: 7dfgh, 12f, 19gh

Badness: 0.018045

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.7897

Optimal GPV sequence: 7d, 12, 19

Badness: 0.024763

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0114

Optimal GPV sequence: 12, 19

Badness: 0.025535

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.1196

Optimal GPV sequence: 12, 19

Badness: 0.022302

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0585

Optimal GPV sequence: 12, 19

Badness: 0.020139

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.9824

Optimal GPV sequence: 12, 19

Badness: 0.018168

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.7980

Optimal GPV sequence: 12, 19

Badness: 0.017069

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.6746

Optimal GPV sequence: 12, 19

Badness: 0.016129

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.7239

Optimal GPV sequence: 12, 19

Badness: 0.015356

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.7160

Optimal GPV sequence: 12, 19

Badness: 0.013906

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.6849

Optimal GPV sequence: 12, 19

Badness: 0.013818

Vincenzoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 34/33, 45/44, 51/50, 56/55, 65/64

Mapping: [⟨1 0 -4 -13 -6 10 -7], ⟨0 1 4 10 6 -4 7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4125

Optimal GPV sequence: 7dg, 12, 19g

Badness: 0.022099

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 34/33, 45/44, 51/50, 56/55, 57/55, 65/64

Mapping: [⟨1 0 -4 -13 -6 10 -7 -10], ⟨0 1 4 10 6 -4 7 9]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9500

Optimal GPV sequence: 7dgh, 12, 19gh

Badness: 0.019904

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.6202

Optimal GPV sequence: 7d, 12f, 19f

Badness: 0.024243

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 27/26, 34/33, 40/39, 45/44, 56/55

Mapping: [⟨1 0 -4 -13 -6 -1 -7], ⟨0 1 4 10 6 3 7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2789

Optimal GPV sequence: 7dg, 12f

Badness: 0.021400

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 27/26, 34/33, 40/39, 45/44, 56/55, 57/55

Mapping: [⟨1 0 -4 -13 -6 -1 -7 -10], ⟨0 1 4 10 6 3 7 9]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.8486

Optimal GPV sequence: 7dgh, 12f

Badness: 0.018996

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.7022

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

Badness: 0.031539

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2408

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

Badness: 0.026288

Migration

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

Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.5324

Optimal GPV sequence: 7d, 24d, 31, 100de, 131bdee, 162bdee

Badness: 0.025516

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]]

Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.5444

Optimal GPV sequence: 7d, 24d, 31

Badness: 0.028071

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

Optimal tuning (CTE): ~63/44 = 1\2, ~3/2 = 696.5199

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

Badness: 0.038122

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]]

Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 696.3410

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

Badness: 0.028817

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 189/187, 221/220

Mapping: [⟨2 0 -8 -26 -31 -40 5], ⟨0 1 4 10 12 15 1]]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 696.3526

Optimal GPV sequence: 12f, 38df, 50

Badness: 0.022666

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220

Mapping: [⟨2 0 -8 -26 -31 -40 5 -1], ⟨0 1 4 10 12 15 1 3]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3837

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

Badness: 0.017785

Trimean

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 1344/1331

Mapping: [⟨1 2 4 7 5], ⟨0 -3 -12 -30 -11]]

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.7074

Optimal GPV sequence: 43, 50, 93

Badness: 0.050729

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 144/143, 364/363

Mapping: [⟨1 2 4 7 5 3], ⟨0 -3 -12 -30 -11 5]]

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.7121

Optimal GPV sequence: 43, 50, 93

Badness: 0.035445

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 126/125, 144/143, 189/187, 221/220

Mapping: [⟨1 2 4 7 5 3 8], ⟨0 -3 -12 -30 -11 5 -28]]

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.7047

Optimal GPV sequence: 43, 50, 93

Badness: 0.025221

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

Minimax tuning:

[[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 (unchanged intervals): 2, 7/5
[[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 (unchanged intervals): 2, 9/7

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
  • 7-odd-limit diamond tradeoff: ~3/2 = [692.353, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
  • 7-odd-limit diamond monotone and tradeoff: ~3/2 = [692.353, 694.737]
  • 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]

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

Optimal GPV sequence: 7, 19, 26, 45

Badness: 0.038553

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:

  • 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
  • 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]

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

Badness: 0.033839

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:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]

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

Badness: 0.022260

Scales: flattone12

Ptolemy

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

Optimal GPV sequence: 7, 31dd, 38d, 45e, 83bcddee

Badness: 0.058785

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

Optimal GPV sequence: 7, 31ddf, 38df, 45ef, 83bcddeeff

Badness: 0.034316

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

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
  • 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 715.587]
  • 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 715.587]

Optimal GPV sequence: 5, 7, 12, 41cd, 53cdd, 65ccddd

Badness: 0.020690

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:

  • 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
  • 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [700.000, 705.882]

POTE generator: ~3/2 = 703.254

Optimal GPV sequence: 5, 12, 17c, 29cde

Badness: 0.024180

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:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 705.882

Optimal GPV sequence: 12f, 17c, 29cdef

Badness: 0.024108

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]]

POTE generator: ~3/2 = 704.905

Optimal GPV sequence: 5, 12, 17c, 46cde

Badness: 0.027295

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

Optimal GPV sequence: 5e, 7, 12, 19d, 43de

Badness: 0.021978

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

Optimal GPV sequence: 5ef, 7, 12, 19d, 31def

Badness: 0.027039

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

Optimal GPV sequence: 5ef, 7, 12, 19d, 31def

Badness: 0.024539

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

Optimal GPV sequence: 5ef, 7, 12, 19d, 31def

Badness: 0.020398

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

Optimal GPV sequence: 5e, 7, 12f, 19df

Badness: 0.018289

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

Optimal GPV sequence: 5e, 12e, 17c, 46cd

Badness: 0.036562

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

Optimal GPV sequence: 5e, 12e, 17c

Badness: 0.027435

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

Optimal GPV sequence: 5, 7, 12e

Badness: 0.026141

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 -1 3]]

POTE generator: ~3/2 = 696.743

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

Badness: 0.023300

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 -1 3 -5]]

POTE generator: ~3/2 = 696.978

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

Badness: 0.024535

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 -1 3 -5 -3]]

POTE generator: ~3/2 = 697.068

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

Badness: 0.021098

Neutrominant

Deutsch

Main article: Neutrominant

The neutrominant temperament (formerly maqamic temperament) has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. 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: 36/35, 64/63, 121/120

Mapping: [⟨1 1 0 4 2], ⟨0 2 8 -4 5]]

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 350.934

Optimal GPV sequence: 7, 17c, 24d, 41cd

Badness: 0.040240

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 64/63, 66/65, 121/120

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

Mapping generators: ~2, ~11/9

POTE generator: ~11/9 = 350.816

Optimal GPV sequence: 7, 17c, 24d, 41cd

Badness: 0.027214

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

Optimal GPV sequence: 5, 7d, 12d

Badness: 0.024848

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

Optimal GPV sequence: 5, 7d, 12de

Badness: 0.025167

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

Optimal GPV sequence: 5d, 12d, 17c, 29c

Badness: 0.134204

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

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

Badness: 0.063262

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

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

Badness: 0.040324

Godzilla

Deutsch

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

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~7/6 = [240.000, 257.143] (1\5 to 3\14)
  • 7- and 9-odd-limit diamond tradeoff: ~7/6 = [231.174, 266.871]
  • 7- and 9-odd-limit diamond monotone and tradeoff: ~7/6 = [240.000, 257.143]

Optimal GPV sequence: 5, 14c, 19

Badness: 0.026747

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:

  • 11-odd-limit diamond monotone: ~7/6 = [252.632, 257.143] (4\19 to 3\14)
  • 11-odd-limit diamond tradeoff: ~7/6 = [231.174, 266.871]
  • 11-odd-limit diamond monotone and tradeoff: ~7/6 = [252.632, 257.143]

Optimal GPV sequence: 14c, 19, 33cd, 52cd

Badness: 0.028947

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:

  • 13- and 15-odd-limit diamond monotone: ~7/6 = 252.632 (4\19)
  • 13- and 15-odd-limit diamond tradeoff: ~7/6 = [231.174, 289.210]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~7/6 = 252.632

Optimal GPV sequence: 14cf, 19, 33cdff, 52cdff

Badness: 0.022503

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

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

Badness: 0.028510

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

Optimal GPV sequence: 19e, 24, 43de

Badness: 0.039647

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

Optimal GPV sequence: 19e, 24, 43de

Badness: 0.025676

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

Optimal GPV sequence: 5, 14ce, 19, 24, 43d

Badness: 0.035673

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 81/80, 91/90

Mapping: [⟨1 0 -4 2 9 -5], ⟨0 2 8 1 -7 11]]

Mapping generators: ~2, ~7/4

POTE generator: ~8/7 = 251.198

Optimal GPV sequence: 5, 14cef, 19, 24, 43d

Badness: 0.026703

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.

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

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
Eigenmonzos (unchanged intervals): 2, 5

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
  • 7-odd-limit diamond tradeoff: ~128/105 = [347.393, 350.978]
  • 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]
  • 7-odd-limit diamond monotone and tradeoff: ~128/105 = [347.393, 350.000]
  • 9-odd-limit diamond monotone and tradeoff: ~128/105 = [347.368, 350.000]

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

Optimal GPV sequence: 7, 24, 31

Badness: 0.055714

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:

[[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 (unchanged intervals): 2, 5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
  • 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]
  • 11-odd-limit diamond monotone and tradeoff: ~11/9 = [348.387, 350.000]

Optimal GPV sequence: 7, 24, 31

Badness: 0.026064

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

Optimal GPV sequence: 7, 24, 31

Badness: 0.023388

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

Optimal GPV sequence: 7, 24, 31, 86ef

Badness: 0.020576

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

Optimal GPV sequence: 7, 24, 31, 55, 86efh

Badness: 0.017302

Scales: mohaha7, mohaha10

Mohamaq

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

Optimal GPV sequence: 7d, 17c, 24, 65cc, 89ccd

Badness: 0.077734

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

Optimal GPV sequence: 7d, 17c, 24, 65cc, 89ccd

Badness: 0.036207

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

Optimal GPV sequence: 7d, 17c, 24, 41c, 65cc

Badness: 0.028738

Scales: mohaha7, mohaha10

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 subgroup, 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

Wedgie: ⟨⟨3 12 -1 12 -10 -36]]

POTE generator: ~8/7 = 232.193

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

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3 0 -1/12 0⟩]
Eigenmonzos (unchanged intervals): 2, 5

Optimal GPV sequence: 5, 26, 31

Badness: 0.037146

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

Optimal GPV sequence: 5, 26, 31, 88, 150be, 181bee

Badness: 0.025642

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

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

Badness: 0.023954

Music

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

Optimal GPV sequence: 5e, 26, 57e, 83bce

Badness: 0.055706

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 = 231.293

Optimal GPV sequence: 5e, 26, 57e, 83bce

Badness: 0.034124

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

Optimal GPV sequence: 31, 129, 160be, 191bce, 222bce, 253bcee

Badness: 0.031334

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

Optimal GPV sequence: 31, 36, 67, 98

Badness: 0.036857

Liese

Deutsch

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.

Subgroup: 2.3.5.7

Comma list: 81/80, 686/675

Mapping: [⟨1 0 -4 -3], ⟨0 3 12 11]]

Mapping generators: ~2, ~10/7

Wedgie: ⟨⟨3 12 11 12 9 -8]]

POTE generator: ~10/7 = 632.406

Minimax tuning:

  • 7- and 9-odd-limit: ~10/7 = [1/3 0 1/12⟩
[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]
Eigenmonzos (unchanged intervals): 2, 5

Algebraic generator: Radix, the real root of x5 - 2x4 + 2x3 - 2x2 + 2x - 2, also a root of x6 - x5 - 2. The recurrence converges.

Optimal GPV sequence: 17c, 19, 55, 74d

Badness: 0.046706

Liesel

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 540/539

Mapping: [⟨1 0 -4 -3 4], ⟨0 3 12 11 -1]]

POTE generator: ~10/7 = 633.073

Optimal GPV sequence: 17c, 19, 36, 91cee

Badness: 0.040721

13-limit

Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 81/80, 91/90

Mapping: [⟨1 0 -4 -3 4 0], ⟨0 3 12 11 -1 7]]

POTE generator: ~10/7 = 633.042

Optimal GPV sequence: 17c, 19, 36, 91ceef

Badness: 0.027304

Elisa

Subgroup: 2.3.5.7.11

Comma list: 77/75, 81/80, 99/98

Mapping: [⟨1 0 -4 -3 -5], ⟨0 3 12 11 16]]

POTE generator: ~10/7 = 633.061

Optimal GPV sequence: 17c, 19e, 36e

Badness: 0.041592

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 77/75, 81/80, 99/98

Mapping: [⟨1 0 -4 -3 -5 0], ⟨0 3 12 11 16 7]]

POTE generator: ~10/7 = 632.991

Optimal GPV sequence: 17c, 19e, 36e

Badness: 0.026922

Lisa

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 343/330

Mapping: [⟨1 0 -4 -3 -6], ⟨0 3 12 11 18]]

POTE generator: ~10/7 = 631.370

Optimal GPV sequence: 17cee, 19

Badness: 0.054829

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 81/80, 91/88, 147/143

Mapping: [⟨1 0 -4 -3 -6 0], ⟨0 3 12 11 18 7]]

POTE generator: ~10/7 = 631.221

Optimal GPV sequence: 17cee, 19

Badness: 0.036144

Lithium

Lithium is named after the 3rd element for being period-3, and also for lithium's molar mass of 6.9 g/mol since 69edo supports it.

Subgroup: 2.3.5.7

Comma list: 81/80, 3125/3087

Mapping: [⟨3 0 -12 -20], ⟨0 1 4 6]]

Mapping generators: ~56/45, ~3

Optimal tuning (CTE): ~56/45 = 1\3, ~3/2 = 695.827

Optimal GPV sequence: 12, 33cd, 45, 57

Badness: 0.0692

Squares

Main article: 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

Wedgie: ⟨⟨4 16 9 16 3 -24]]

POTE generator: ~9/7 = 425.942

Minimax tuning:

[[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]
Eigenmonzos (unchanged intervals): 2, 5

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

Optimal GPV sequence: 14c, 17c, 31

Badness: 0.045993

Scales: skwares8, skwares11, skwares14

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]]

POTE generator: ~9/7 = 425.957

Optimal GPV sequence: 14c, 17c, 31

Badness: 0.021636

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]]

POTE generator: ~9/7 = 425.550

Optimal GPV sequence: 14c, 17c, 31, 79cf

Badness: 0.025514

Squad

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 91/90, 99/98

Mapping: [⟨1 3 8 6 7 9], ⟨0 -4 -16 -9 -10 -15]]

POTE generator: ~9/7 = 425.7516

Optimal GPV sequence: 14cf, 17c, 31f

Badness: 0.026877

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]]

POTE generator: ~9/7 = 426.276

Optimal GPV sequence: 14cf, 31, 45ef, 76e

Badness: 0.024522

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]]

POTE generator: ~9/7 = 426.187

Optimal GPV sequence: 14cf, 31, 76e

Badness: 0.022573

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]]

POTE generator: ~9/7 = 426.225

Optimal GPV sequence: 14cf, 31, 76e

Badness: 0.018839

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]]

POTE generator: ~9/7 = 425.993

Optimal GPV sequence: 14ce, 17ce, 31, 107b, 138b, 169be, 200be

Badness: 0.056826

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.

Subgroup: 2.3.5.7

Comma list: 81/80, 17280/16807

Mapping: [⟨1 1 0 2], ⟨0 5 20 7]]

Mapping generators: ~2, ~54/49

Wedgie: ⟨⟨5 20 7 20 -3 -40]]

POTE generator: ~54/49 = 139.343

Optimal GPV sequence: 17c, 26, 43, 69, 112bd

Badness: 0.108656

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 864/847

Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]

Mapping generators: ~2, ~12/11

POTE generator: ~12/11 = 139.428

Optimal GPV sequence: 17c, 26, 43, 69

Badness: 0.047914

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 144/143

Mapping: [⟨1 1 0 2 3 3], ⟨0 5 20 7 4 6]]

Mapping generators: ~2, ~12/11

POTE generator: ~12/11 = 139.387

Optimal GPV sequence: 17c, 26, 43, 69

Badness: 0.029285

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 144/143, 189/187

Mapping: [⟨1 1 0 2 3 3 2], ⟨0 5 20 7 4 6 18]]

Mapping generators: ~2, ~12/11

POTE generator: ~12/11 = 139.362

Optimal GPV sequence: 17cg, 26, 43, 69

Badness: 0.020878

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143

Mapping: [⟨1 1 0 2 3 3 2 1], ⟨0 5 20 7 4 6 18 28]]

Mapping generators: ~2, ~12/11

POTE generator: ~12/11 = 139.313

Optimal GPV sequence: 17cgh, 26, 43, 69

Badness: 0.018229

Meantritone

The meantritone temperament tempers out the mirkwai comma (16875/16807) and trimyna comma (50421/50000) in the 7-limit. In this temperament, three septimal tritones equals ~30/11 (an octave plus 15/11-wide super-fourth) and five of them equals ~16/3 (double-compound fourth). The name "meantritone" is a portmanteau of meantone and tritone, the latter is a generator of this temperament.

Subgroup: 2.3.5.7

Comma list: 81/80, 16875/16807

Mapping: [⟨1 4 12 12], ⟨0 -5 -20 -19]]

Wedgie: ⟨⟨5 20 19 20 16 -12]]

POTE generator: ~7/5 = 580.766

Optimal GPV sequence: 2cd, 29cd, 31

Badness: 0.082239

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 2541/2500

Mapping: [⟨1 4 12 12 17], ⟨0 -5 -20 -19 -28]]

POTE generator: ~7/5 = 580.647

Optimal GPV sequence: 2cde, 29cde, 31

Badness: 0.042869

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

Wedgie: ⟨⟨2 8 8 8 7 -4]]

POTE generator: ~3/2 = 694.375

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
  • 7-odd-limit diamond tradeoff: ~3/2 = [688.957, 701.955]
  • 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
  • 7-odd-limit diamond monotone and tradeoff: ~3/2 = [688.957, 700.000]
  • 9-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]

Optimal GPV sequence: 12, 26, 38, 102bcd, 140bccd, 178bbccdd

Badness: 0.031130

Music

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:

  • 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
  • 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
  • 11-odd-limit diamond monotone and tradeoff: ~3/2 = [685.714, 700.000]

Optimal GPV sequence: 12, 14c, 26, 90bce, 116bcce

Badness: 0.023124

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:

  • 13- and 15-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
  • 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]
  • 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = 692.308

Optimal GPV sequence: 12f, 14cf, 26, 38e

Badness: 0.021565

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 50/49, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5], ⟨0 1 4 4 6 9 1]]

POTE generator: ~3/2 = 692.487

Optimal GPV sequence: 12f, 14cf, 26

Badness: 0.018358

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84

Mapping: [⟨2 0 -8 -7 -12 -21 5 -1], ⟨0 1 4 4 6 9 1 3]]

POTE generator: ~3/2 = 692.299

Optimal GPV sequence: 12f, 14cf, 26

Badness: 0.015118

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

Optimal GPV sequence: 12f, 14c, 26f, 38eff

Badness: 0.026542

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

Optimal GPV sequence: 12e, 14c, 26e, 40cee

Badness: 0.038577

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

Optimal GPV sequence: 2cd, 10cd, 12

Badness: 0.043062

Teff

Main article: Teff

Teff (found by Mason Green) is to injera what mohajira is to meantone; it splits the generator in half in order to accommodate higher limit intervals, creating a half-octave quarter-tone temperament.

Subgroup: 2.3.5.7.11

Comma list: 50/49, 81/80, 864/847

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

Mapping generators: ~7/5, ~16/11

POTE generator: ~11/8 = 552.5303

Optimal GPV sequence: 24d, 26, 50d

Badness: 0.070689

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 81/80, 144/143

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

POTE generator: ~11/8 = 552.5324

Optimal GPV sequence: 24d, 26, 50d

Badness: 0.040047

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6], ⟨0 2 8 8 -1 5 2]]

POTE generator: ~11/8 = 552.6558

Optimal GPV sequence: 24d, 26

Badness: 0.029499

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143

Mapping: [⟨2 1 -4 -3 8 2 6 2], ⟨0 2 8 8 -1 5 2 6]]

POTE generator: ~11/8 = 552.6382

Optimal GPV sequence: 24d, 26

Badness: 0.023133

Pombe

Pombe (named after the African millet beer) is a variant of #Teff by Kaiveran Lugheidh that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.

Subgroup: 2.3.5.7

Comma list: 81/80, 300125/294912

Mapping: [⟨2 1 -4 11], ⟨0 2 8 -5]]

Mapping generators: ~735/512, ~35/24

Wedgie: ⟨⟨4 16 -10 16 -27 -68]]

POTE generator: ~48/35 = 552.2206

Optimal GPV sequence: 24, 26, 50, 126bcd, 176bcdd, 226bbcdd

Badness: 0.116104

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 245/242, 385/384

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

POTE generator: ~11/8 = 552.0929

Optimal GPV sequence: 24, 26, 50

Badness: 0.052099

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 245/242

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

POTE generator: ~11/8 = 552.1498

Optimal GPV sequence: 24, 26, 50

Badness: 0.031039

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 144/143, 245/242, 273/272

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

POTE generator: ~11/8 = 552.1579

Optimal GPV sequence: 24, 26, 50

Badness: 0.021260

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209

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

POTE generator: ~11/8 = 552.1196

Optimal GPV sequence: 24, 26, 50

Badness: 0.016548

Orphic

Subgroup: 2.3.5.7

Comma list: 81/80, 5898240/5764801

Mapping: [⟨2 5 12 7], ⟨0 -4 -16 -3]]

Mapping generators: ~2401/1728, ~7/6

Wedgie: ⟨⟨8 32 6 32 -13 -76]]

POTE generator: ~7/6 = 275.794

Optimal GPV sequence: 26, 48c, 74, 174bd, 248bbd

Badness: 0.258825

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 5 12 7 6], ⟨0 -4 -16 -3 2]]

Mapping generators: ~363/256, ~7/6

POTE generator: ~7/6 = 275.762

Optimal GPV sequence: 26, 48c, 74, 248bbd, 322bbdd

Badness: 0.101499

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 5 12 7 6 12], ⟨0 -4 -16 -3 2 -10]]

Mapping generators: ~55/39, ~7/6

POTE generator: ~7/6 = 275.774

Optimal GPV sequence: 26, 48c, 74, 174bd, 248bbd, 322bbdd

Badness: 0.053482

Cloudtone

The cloudtone temperament (5&50) tempers out the cloudy comma, 16807/16384 and the syntonic comma, 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 81/80, 16807/16384

Mapping: [⟨5 0 -20 14], ⟨0 1 4 0]]

Mapping generators: ~8/7, ~3

Wedgie: ⟨⟨5 20 0 20 -14 -56]]

POTE generator: ~3/2 = 695.720

Optimal GPV sequence: 5, 45, 50

Badness: 0.102256

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 2401/2376

Mapping: [⟨5 0 -20 14 41], ⟨0 1 4 0 -3]]

Mapping generators: ~8/7, ~3

POTE generator: ~3/2 = 696.536

Optimal GPV sequence: 5, 45, 50, 155bdd, 205bddd

Badness: 0.070378

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 2401/2376

Mapping: [⟨5 0 -20 14 41 -21], ⟨0 1 4 0 -3 5]]

Mapping generators: ~8/7, ~3

POTE generator: ~3/2 = 696.162

Optimal GPV sequence: 5, 45f, 50

Badness: 0.048829

Meanmag

Subgroup: 2.3.5.7

Comma list: 81/80, 3125/3072

Mapping: [⟨19 30 44 0], ⟨0 0 0 1]]

Mapping generators: ~25/24, ~7

Wedgie: ⟨⟨0 0 19 0 30 44]]

POTE generator: ~8/7 = 238.396

Optimal GPV sequence: 19, 38, 57, 76, 95bc

Badness: 0.077023

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 625/616

Mapping: [⟨19 30 44 0 119], ⟨0 0 0 1 -1]]

Mapping generators: ~25/24, ~7

POTE generator: ~8/7 = 233.486

Optimal GPV sequence: 19, 38, 57, 76

Badness: 0.066829

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 625/616

Mapping: [⟨19 30 44 0 119 17], ⟨0 0 0 1 -1 1]]

Mapping generators: ~25/24, ~7

POTE generator: ~8/7 = 234.890

Optimal GPV sequence: 19, 38, 57, 76

Badness: 0.045844

Undevigintone

Subgroup: 2.3.5.7.11

Comma list: 49/48, 81/80, 126/125

Mapping: [⟨19 30 44 53 0], ⟨0 0 0 0 1]]

Mapping generators: ~21/20, ~11

POTE generator: ~11/8 = 538.047

Optimal GPV sequence: 19, 38d

Badness: 0.036387

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 81/80, 126/125

Mapping: [⟨19 30 44 53 0 70], ⟨0 0 0 0 1 0]]

Mapping generators: ~21/20, ~11

POTE generator: ~11/8 = 537.061

Optimal GPV sequence: 19, 38df

Badness: 0.022933

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