Dicot family

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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:

  • 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, 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

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 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:

  • 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]]

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]

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]]

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

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

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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