Xenharmonic Wiki

Dicot family

Revision as of 13:47, 17 August 2025 by FloraC (talk | contribs) (- CTE & POTE tunings)
This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The dicot family of temperaments tempers out 25/24, the classical chromatic semitone. The head of this family, dicot, is generated by 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 sixths and pretending that these are 5-limit, and like any temperament which seems to involve a lot of "pretending", dicot is close to the edge of what can be sensibly 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:

  • WE: ~2 = 1206.283 ¢, ~6/5 = 350.420 ¢
error map: ⟨+6.283 +5.167 -23.328]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 351.086 ¢
error map: ⟨0.000 +0.216 -35.228]

Tuning ranges:

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

Badness (Sintel): 0.306

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, sharpie 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, sharpie, and dichotic.

Subgroup: 2.3.5.11

Comma list: 25/24, 45/44

Subgroup val 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:

  • WE: ~2 = 1206.750 ¢, ~6/5 = 348.684 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 348.954 ¢

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

Badness (Sintel): 0.370

2.3.5.11.13 subgroup

Subgroup: 2.3.5.11.13

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

Subgroup val 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:

  • WE: ~2 = 1202.433 ¢, ~6/5 = 351.237 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 350.978 ¢

Optimal ET sequence: 3e, 7, 17

Badness (Sintel): 0.536

Septimal dicot

Septimal dicot is the extension where 7/6 and 9/7 are also conflated into 5/4~6/5. Although 5/4~6/5 is a giant block already, 7/6 and 9/7 are often considered as thirds too. On that account one could argue for the canonicity of this extension, despite the relatively poor accuracy.

Subgroup: 2.3.5.7

Comma list: 15/14, 25/24

Mapping: [⟨1 1 2 2], ⟨0 2 1 3]]

Optimal tunings:

  • WE: ~2 = 1205.532 ¢, ~6/5 = 337.931 ¢
error map: ⟨+5.532 -20.561 -37.319 +56.032]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.561 ¢
error map: ⟨0.000 -24.834 -47.753 +46.856]

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.504

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:

  • WE: ~2 = 1203.346 ¢, ~6/5 = 343.078 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 343.260 ¢

Optimal ET sequence: 3de, 4e, 7

Badness (Sintel): 0.656

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:

  • WE: ~2 = 1205.828 ¢, ~6/5 = 337.683 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 336.909 ¢

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

Badness (Sintel): 0.896

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:

  • WE: ~2 = 1202.660 ¢, ~6/5 = 339.597 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 339.104 ¢

Optimal ET sequence: 3d, 4, 7

Badness (Sintel): 0.985

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]]

Optimal tunings:

  • WE: ~2 = 1220.466 ¢, ~6/5 = 337.577 ¢
error map: ⟨+20.466 -6.335 -7.804 -45.004]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 335.391 ¢
error map: ⟨0.000 -31.173 -50.922 -104.217]

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

Badness (Sintel): 0.642

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:

  • WE: ~2 = 1216.069 ¢, ~6/5 = 342.052 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 338.467 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.826

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:

  • WE: ~2 = 1211.546 ¢, ~6/5 = 344.304 ¢
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 341.373 ¢

Optimal ET sequence: 3, 4, 7d

Badness (Sintel): 0.968

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:

  • WE: ~2 = 1202.488 ¢, ~5/4 = 358.680 ¢
error map: ⟨+2.488 +17.893 -22.658 -14.258]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 358.495 ¢
error map: ⟨0.000 +15.035 -27.818 -17.854]

Optimal ET sequence: 3d, 7d, 10

Badness (Sintel): 0.732

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:

  • WE: ~2 = 1201.518 ¢, ~5/4 = 356.557 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.457 ¢

Optimal ET sequence: 3de, 7d, 10, 17d

Badness (Sintel): 0.739

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]]

Optimal tunings:

  • WE: ~2 = 1200.802 ¢, ~5/4 = 356.502 ¢
error map: ⟨+0.802 +11.851 -28.208 +8.374]
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 356.275 ¢
error map: ⟨0.000 +10.595 -30.039 +6.074]

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

Badness (Sintel): 0.951

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:

  • WE: ~2 = 1199.504 ¢, ~5/4 = 354.115 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.236 ¢

Optimal ET sequence: 7, 10, 17

Badness (Sintel): 1.01

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:

  • WE: ~2 = 1199.289 ¢, ~5/4 = 354.156 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.340 ¢

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

Badness (Sintel): 0.896

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:

  • WE: ~2 = 1203.949 ¢, ~5/4 = 355.239 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.024 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 1.05

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:

  • WE: ~2 = 1202.979 ¢, ~5/4 = 355.193 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 354.254 ¢

Optimal ET sequence: 3, 7, 10e

Badness (Sintel): 0.940

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:

  • WE: ~2 = 1197.526 ¢, ~5/4 = 359.915 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.745 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.37

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:

  • WE: ~2 = 1197.922 ¢, ~5/4 = 360.021 ¢
  • CWE: ~2 = 1200.000 ¢, ~5/4 = 360.722 ¢

Optimal ET sequence: 3, 7e, 10

Badness (Sintel): 1.15

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

Optimal tunings:

  • WE: ~7/5 = 603.286 ¢, ~7/4 = 953.637 ¢ (~7/6 = 252.935 ¢)
error map: ⟨+6.571 +5.318 -22.821 -2.047]
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.957 ¢ (~7/6 = 249.043 ¢)
error map: ⟨0.000 -0.041 -35.357 -17.869]

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

Badness (Sintel): 0.717

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:

  • WE: ~7/5 = 603.558 ¢, ~7/4 = 952.121 ¢ (~7/6 = 254.996 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 948.610 ¢ (~7/6 = 251.390 ¢)

Optimal ET sequence: 4e, 10, 14c, 24c

Badness (Sintel): 0.883

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:

  • WE: ~7/5 = 603.612 ¢, ~7/4 = 953.663 ¢ (~7/6 = 253.562 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 950.116 ¢ (~7/6 = 249.884 ¢)

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

Badness (Sintel): 0.881

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:

  • WE: ~7/5 = 604.535 ¢, ~7/4 = 952.076 ¢ (~7/6 = 256.994 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 946.108 ¢ (~7/6 = 253.892 ¢)

Optimal ET sequence: 4, 10e, 14c

Badness (Sintel): 1.04

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:

  • WE: ~7/5 = 599.404 ¢, ~7/4 = 955.557 ¢ (~8/7 = 243.251 ¢)
  • CWE: ~7/5 = 600.000 ¢, ~7/4 = 956.169 ¢ (~8/7 = 243.831 ¢)

Optimal ET sequence: 4, 6, 10

Badness (Sintel): 1.07

Sidi

Subgroup: 2.3.5.7

Comma list: 25/24, 245/243

Mapping: [⟨1 -1 1 -3], ⟨0 4 2 9]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1207.178 ¢, ~14/9 = 777.414 ¢
error map: ⟨+7.178 +0.523 -24.308 +6.367]
  • CWE: ~2 = 1200.000 ¢, ~14/9 = 773.872 ¢
error map: ⟨0.000 -6.464 -38.569 -3.973]

Optimal ET sequence: 3d, …, 11cd, 14c

Badness (Sintel): 1.43

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1207.200 ¢, ~11/7 = 777.363 ¢
  • CWE: ~2 = 1200.000 ¢, ~11/7 = 773.777 ¢

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

Badness (Sintel): 1.09

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