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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 354.664
error map: ⟨0.000 +7.374 -31.649]
  • POTE: ~2 = 1200.000, ~6/5 = 348.594
error map: ⟨0.000 -4.766 -37.719]

Tuning ranges:

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

Badness (Smith): 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.

Temperaments discussed elsewhere are:

The rest are considered below.

2.3.5.11 subgroup

The 2.3.5.11-subgroup extension is related to septimal dicot, sharp, and dichotic.

Subgroup: 2.3.5.11

Comma list: 25/24, 45/44

Sval mapping: [⟨1 1 2 2], ⟨0 2 1 5]]

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 352.287
  • POTE: ~2 = 1200.000, ~6/5 = 346.734

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

RMS error: 5.621 cents

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

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

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

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 352.420
  • POTE: ~2 = 1200.000, ~6/5 = 350.526

Optimal ET sequence: 3e, 7, 17, 24c

RMS error: 5.916 cents

Septimal dicot

Septimal dicot is the extension where 7/6 and 9/7 are added to the giant block of 5/4~6/5 third.

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 tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 342.257
error map: ⟨0.000 -17.441 -44.056 +57.946]
  • POTE: ~2 = 1200.000, ~6/5 = 336.381
error map: ⟨0.000 -29.193 -49.933 +40.316]

Optimal ET sequence: 3d, 4, 7

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 345.596
  • POTE: ~2 = 1200.000, ~6/5 = 342.125

Optimal ET sequence: 3de, 4e, 7

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 340.417
  • POTE: ~2 = 1200.000, ~6/5 = 336.051

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

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 340.835
  • POTE: ~2 = 1200.000, ~6/5 = 338.846

Optimal ET sequence: 3d, 4, 7

Badness (Smith): 0.023828

Flattie

This temperament used to be known as flat. Unlike septimal dicot where 7/6 is added to the neutral third, here 8/7 is added instead.

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 tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 346.438
error map: ⟨0.000 -9.080 -39.876 -115.264]
  • POTE: ~2 = 1200.000, ~6/5 = 331.916
error map: ⟨0.000 -38.123 -54.398 -100.742]

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

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 343.139
  • POTE: ~2 = 1200.000, ~6/5 = 337.532

Optimal ET sequence: 3, 4, 7d

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~6/5 = 343.655
  • POTE: ~2 = 1200.000, ~6/5 = 341.023

Optimal ET sequence: 3, 4, 7d

Badness (Smith): 0.023420

Sharpie

This temperament used to be known as sharp. This is where you find 7/6 at the major second and 7/4 at the major sixth.

Subgroup: 2.3.5.7

Comma list: 25/24, 28/27

Mapping: [⟨1 1 2 1], ⟨0 2 1 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 359.564
error map: ⟨0.000 +17.173 -26.750 -11.442]
  • POTE: ~2 = 1200.000, ~5/4 = 357.938
error map: ⟨0.000 +13.921 -28.376 -21.198]

Wedgie: ⟨⟨ 2 1 6 -3 4 11 ]]

Optimal ET sequence: 3d, 7d, 10

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 357.261
  • POTE: ~2 = 1200.000, ~5/4 = 356.106

Optimal ET sequence: 3de, 7d, 10, 17d

Badness (Smith): 0.022366

Dichotic

In dichotic, 7/4 is found at a stack of two perfect fourths.

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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 356.333
error map: ⟨0.000 +10.710 -29.981 +5.844]
  • POTE: ~2 = 1200.000, ~5/4 = 356.264
error map: ⟨0.000 +10.573 -30.050 +6.119]

Optimal ET sequence: 3, 7, 10, 17, 27c

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 354.183
  • POTE: ~2 = 1200.000, ~5/4 = 354.262

Optimal ET sequence: 7, 10, 17

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 354.247
  • POTE: ~2 = 1200.000, ~5/4 = 354.365

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

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 353.751
  • POTE: ~2 = 1200.000, ~5/4 = 354.073

Optimal ET sequence: 3, 7, 10e

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 353.850
  • POTE: ~2 = 1200.000, ~5/4 = 354.313

Optimal ET sequence: 3, 7, 10e

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 361.081
  • POTE: ~2 = 1200.000, ~5/4 = 360.659

Optimal ET sequence: 3, 7e, 10

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~5/4 = 361.061
  • POTE: ~2 = 1200.000, ~5/4 = 360.646

Optimal ET sequence: 3, 7e, 10

Badness (Smith): 0.027938

Decimal

Main article: Decimal
See also: Jubilismic clan

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 tunings:

  • CTE: ~7/5 = 600.000, ~7/4 = 955.608 (~8/7 = 244.392)
error map: ⟨0.000 +9.260 -30.706 -13.218]
  • POTE: ~7/5 = 600.000, ~7/4 = 948.443 (~7/6 = 251.557)
error map: ⟨0.000 -5.069 -37.871 -20.383]

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

Badness (Smith): 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 tunings:

  • CTE: ~7/5 = 600.000, ~7/4 = 952.812 (~8/7 = 247.188)
  • POTE: ~7/5 = 600.000, ~7/4 = 946.507 (~7/6 = 253.493)

Optimal ET sequence: 4e, 10, 14c, 24c

Badness (Smith): 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 tunings:

  • CTE: ~7/5 = 600.000, ~7/4 = 954.469 (~8/7 = 245.531)
  • POTE: ~7/5 = 600.000, ~7/4 = 947.955 (~7/6 = 252.045)

Optimal ET sequence: 4ef, 10, 14cf, 24cf

Badness (Smith): 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 tunings:

  • CTE: ~7/5 = 600.000, ~7/4 = 950.940 (~7/6 = 249.060)
  • POTE: ~7/5 = 600.000, ~7/4 = 944.934 (~7/6 = 255.066)

Optimal ET sequence: 4, 10e, 14c

Badness (Smith): 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 tunings:

  • CTE: ~7/5 = 600.000, ~7/4 = 955.608 (~8/7 = 244.392)
  • POTE: ~7/5 = 600.000, ~7/4 = 956.507 (~8/7 = 243.493)

Optimal ET sequence: 4, 6, 10

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 424.452
error map: ⟨0.000 +0.238 -35.217 +11.108]
  • POTE: ~2 = 1200.000, ~9/7 = 427.208
error map: ⟨0.000 -10.789 -40.731 -13.702]

Optimal ET sequence: 3d, …, 11cd, 14c

Badness (Smith): 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 tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 424.587
  • POTE: ~2 = 1200.000, ~9/7 = 427.273

Optimal ET sequence: 3de, …, 11cdee, 14c

Badness (Smith): 0.032957

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