Dicot family

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The 5-limit parent comma for the dicot family is 25/24, the classical chromatic semitone. The generator is a classical third (major and minor mean the same thing), and two such thirds give a fifth. In fact, (5/4)2 = (3/2)(25/24).

Possible tunings for dicot are 7edo, 10edo, 17edo, 24edo using the val ⟨24 38 55] (24c), and 31edo using the val ⟨31 49 71] (31c). In a sense, what dicot is all about is using neutral thirds and pretending that this is 5-limit, and like any temperament which seems to involve "pretending", dicot is close to the edge of what can sensibly be called a temperament at all. In other words, it is an exotemperament.

Dicot

Subgroup: 2.3.5

Comma list: 25/24

Mapping: [⟨1 1 2], ⟨0 2 1]]

mapping generators: ~2, ~5/4

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 348.594

Tuning ranges:

  • 5-odd-limit diamond monotone: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
  • 5-odd-limit diamond tradeoff: ~5/4 = [315.641, 386.314] (full comma to untempered)
  • 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]

Optimal ET sequence: 3, 4, 7, 17, 24c, 31c

Badness: 0.013028

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal dicot adds 36/35, sharp adds 28/27, and dichotic adds 64/63, all retaining the same period and generator.

Decimal adds 49/48, sidi adds 245/243, and jamesbond adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.

2.3.5.11 subgroup

Subgroup: 2.3.5.11

Comma list: 25/24, 45/44

Gencom: [2 5/4; 25/24 45/44]

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

Sval mapping: [⟨1 1 2 2], ⟨0 2 1 5]]

POL2 generator: ~5/4 = 346.734

Optimal ET sequence: 3e, 4e, 7, 24c, 31c, 38cc, 45cce

RMS error: 5.621 cents

Related temperaments: dicot, sharp, dichotic

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 25/24, 40/39, 45/44

Gencom: [2 5/4; 25/24 40/39 45/44]

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

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

POL2 generator: ~5/4 = 350.526

Optimal ET sequence: 3e, 7, 17, 24c

RMS error: 5.916 cents

Septimal dicot

Subgroup: 2.3.5.7

Comma list: 15/14, 25/24

Mapping: [⟨1 1 2 2], ⟨0 2 1 3]]

Wedgie: ⟨⟨ 2 1 3 -3 -1 4 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.381

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness: 0.019935

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 25/24

Mapping: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 342.125

Optimal ET sequence: 3de, 4e, 7

Badness: 0.019854

Eudicot

Subgroup: 2.3.5.7.11

Comma list: 15/14, 25/24, 33/32

Mapping: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.051

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness: 0.027114

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 15/14, 25/24, 33/32, 40/39

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 338.846

Optimal ET sequence: 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef

Badness: 0.023828

Flat

Subgroup: 2.3.5.7

Comma list: 21/20, 25/24

Mapping: [⟨1 1 2 3], ⟨0 2 1 -1]]

Wedgie: ⟨⟨ 2 1 -1 -3 -7 -5 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 331.916

Optimal ET sequence: 3, 4, 7d, 11cd, 18bcddd

Badness: 0.025381

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 337.532

Optimal ET sequence: 3, 4, 7d

Badness: 0.024988

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 14/13, 21/20, 25/24, 33/32

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 341.023

Optimal ET sequence: 3, 4, 7d

Badness: 0.023420

Sharp

Subgroup: 2.3.5.7

Comma list: 25/24, 28/27

Mapping: [⟨1 1 2 1], ⟨0 2 1 6]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 357.938

Wedgie: ⟨⟨ 2 1 6 -3 4 11 ]]

Optimal ET sequence: 3d, 7d, 10, 37cd, 47bccd, 57bccdd

Badness: 0.028942

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 28/27, 35/33

Mapping: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.106

Optimal ET sequence: 3de, 7d, 10, 17d, 27cde

Badness: 0.022366

Dichotic

Subgroup: 2.3.5.7

Comma list: 25/24, 64/63

Mapping: [⟨1 1 2 4], ⟨0 2 1 -4]]

Wedgie: ⟨⟨ 2 1 -4 -3 -12 -12 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.264

Optimal ET sequence: 3, 7, 10, 17, 27c, 37c, 64bccc

Badness: 0.037565

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 64/63

Mapping: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.262

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness: 0.030680

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 40/39, 45/44, 64/63

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.365

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness: 0.021674

Dichotomic

Subgroup: 2.3.5.7.11

Comma list: 22/21, 25/24, 33/32

Mapping: [⟨1 1 2 4 4], ⟨0 2 1 -4 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.073

Optimal ET sequence: 3, 7, 10e, 17e

Badness: 0.031719

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 25/24, 33/32, 40/39

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.313

Optimal ET sequence: 3, 7, 10e, 17e

Badness: 0.022741

Dichosis

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 64/63

Mapping: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.659

Optimal ET sequence: 3, 7e, 10

Badness: 0.041361

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 35/33, 40/39, 64/63

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.646

Optimal ET sequence: 3, 7e, 10

Badness: 0.027938

Decimal

Main article: Decimal

Subgroup: 2.3.5.7

Comma list: 25/24, 49/48

Mapping: [⟨2 0 3 4], ⟨0 2 1 1]]

mapping generators: ~7/5, ~7/4

Wedgie: ⟨⟨ 4 2 2 -6 -8 -1 ]]

Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 948.443 (~7/6 = 251.557)

Optimal ET sequence: 4, 10, 14c, 24c, 38ccd, 62cccdd

Badness: 0.028334

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 49/48

Mapping: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 946.507 (~7/6 = 253.493)

Optimal ET sequence: 10, 14c, 24c, 38ccd, 52cccde

Badness: 0.026712

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 45/44, 49/48, 91/90

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

Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 947.955 (~7/6 = 252.045)

Optimal ET sequence: 10, 14cf, 24cf

Badness: 0.021326

Decimated

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 49/48

Mapping: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 944.934 (~7/6 = 255.066)

Optimal ET sequence: 4, 10e, 14c

Badness: 0.031456

Decibel

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 49/48

Mapping: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

Optimal tuning (POTE): ~7/5 = 1\2, ~7/4 = 956.507 (~8/7 = 243.493)

Optimal ET sequence: 4, 6, 10

Badness: 0.032385

Sidi

Subgroup: 2.3.5.7

Comma list: 25/24, 245/243

Mapping: [⟨1 3 3 6], ⟨0 -4 -2 -9]]

mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 4 2 9 -12 3 15 ]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 427.208

Optimal ET sequence: 3d, 14c, 45cc, 59bcccd

Badness: 0.056586

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 99/98

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

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 427.273

Optimal ET sequence: 3de, 14c, 45cce, 59bcccdee

Badness: 0.032957

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