Meantone family

The 5-limit parent comma of the meantone family is the Didymus or syntonic comma, 81/80. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

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:

• 5-odd-limit: ~3/2 = [0 0 1/4⟩

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]

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.

• Septimal meantone adds [-13 10 0 -1⟩,
• Flattone adds [-17 9 0 1⟩,
• Dominant adds [6 -2 0 -1⟩,
• Sharptone adds [2 -3 0 1⟩,

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

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: Template:Val list

Scales: mohaha7, mohaha10

Septimal meantone

Deutsch

Main article: Meantone

English Wikipedia has an article on:

Septimal meantone temperament

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:

• 7- and 9-odd-limit: ~3/2 = [0 0 1/4⟩

[[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.

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: Template:Val list

Badness: 0.017027

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]]

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

Minimax tuning:

• 13- and 15-odd-limit: ~3/2 = [9/16 -1/8 0 0 1/16⟩

Eigenmonzo basis: 2.11/9

Optimal GPV sequence: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

Badness: 0.021543

Music

• Scott Joplin's "The Entertainer" tuned into meanpop^{[dead link]}
• Twinkle canon – 50 edo by Claudi Meneghin

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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). In general, most septimal subminor intervals are diminished and most septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. 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:

• 7-odd-limit: ~3/2 = [8/13 0 1/13 -1/13⟩

[[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

• 9-odd-limit: ~3/2 = [6/11 2/11 0 -1/11⟩

[[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.

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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]

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

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]

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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:

• 7- and 9-odd-limit: ~128/105 = [0 0 1/8⟩

[[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.

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:

• 11-odd-limit: ~11/9 = [0 0 1/8⟩

[[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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

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:

• 7- and 9-odd-limit: ~8/7 = [0 0 1/12⟩

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

Eigenmonzos (unchanged intervals): 2, 5

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: Template:Val list

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: Template:Val list

Badness: 0.023954

Music

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

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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.

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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

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:

• 7- and 9-odd-limit: ~9/7 = [1/2 0 -1/16⟩

[[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.

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

Badness: 0.056826

Jerome

Jerome is related to Hieronymus' tuning; the Hieronymus generator is 5^1/20, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.

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

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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]

Badness: 0.031130

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:

• 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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

Badness: 0.022933