Dicot family

The 5-limit parent comma for the dicot family is 25/24, the chromatic semitone. Its monzo is [-3 -1 2⟩, and flipping that yields ⟨⟨2 1 -3]] for the wedgie. This tells us the generator is a third (major and minor mean the same thing), and that two thirds gives a fifth. In fact, (5/4)^2 = 3/2 * 25/24. Possible tunings for dicot are 7edo, 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's 5-limit, and like any temperament which seems to involve pretending, dicot is at the edge of what can sensibly be called a temperament at all. In other words, it is an exotemperament.

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal dicot, with wedgie ⟨⟨2 1 3 -3 -1 4]] adds 36/35, sharp with wedgie ⟨⟨2 1 6 -3 4 11]] adds 28/27, and dichotic with wedgie ⟨⟨2 1 -4 -3 -12 -12]] ads 64/63, all retaining the same period and generator. Decimal with wedgie ⟨⟨4 2 2 -6 -8 -1]] adds 49/48, sidi with wedgie ⟨⟨4 2 9 -3 6 15]] adds 245/243, and jamesbond with wedgie ⟨⟨0 0 7 0 11 16]] 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.

Dicot

Subgroup: 2.3.5

Comma list: 25/24

Mapping: [⟨1 1 2], ⟨0 2 1]]

POTE generator: ~5/4 = 348.594

Tuning ranges:

• valid range: [300.000, 400.000] (1\4 to 1\3)

Badness: 0.013028

7-limit

Comma list: 15/14, 25/24

POTE generator: ~5/4 = 336.381

Mapping: [⟨1 1 2 2], ⟨0 2 1 3]]

Wedgie: ⟨⟨2 1 3 -3 -1 4]]

Badness: 0.019935

11-limit

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

POTE generator: ~5/4 = 342.125

Mapping: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

Badness: 0.019854

Eudicot

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

POTE generator: ~5/4 = 336.051

Mapping: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

Badness: 0.027114

13-limit

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

POTE generator: ~5/4 = 338.846

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

Badness: 0.023828

Flat

Comma list: 21/20, 25/24

POTE generator: ~5/4 = 331.916

Map: [⟨1 1 2 3], ⟨0 2 1 -1]]

Wedgie: ⟨⟨2 1 -1 -3 -7 -5]]

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

Badness: 0.025381

11-limit

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

POTE generator: ~5/4 = 337.532

Map: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

Vals: 3, 4, 7d

Badness: 0.024988

13-limit

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

POTE generator: ~5/4 = 341.023

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

Vals: 3, 4, 7d

Badness: 0.023420

Sharp

Comma list: 25/24, 28/27

POTE generator: ~5/4 = 357.938

Map: [⟨1 1 2 1], ⟨0 2 1 6]]

Wedgie: ⟨⟨2 1 6 -3 4 11]]

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

Badness: 0.028942

11-limit

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

POTE generator: ~5/4 = 356.106

Map: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

Vals: 3de, 7d, 10, 17d, 27cde

Badness: 0.022366

Decimal

Comma list: 25/24, 49/48

POTE generator: ~7/6 = 251.557

Map: [⟨2 0 3 4], ⟨0 2 1 1]]

Wedgie: ⟨⟨4 2 2 -6 -8 -1]]

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

Badness: 0.028334

11-limit

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

POTE generator: ~7/6 = 253.493

Map: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

Vals: 10, 14c, 24c, 38ccd, 52cccde

Badness: 0.026712

Decimated

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

POTE generator: ~7/6 = 255.066

Map: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

Vals: 4, 10e, 14c

Badness: 0.031456

Decibel

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

POTE generator: ~8/7 = 243.493

Map: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

Vals: 4, 6, 10

Badness: 0.032385

Dichotic

Comma list: 25/24, 64/63

POTE generator: ~5/4 = 356.264

Map: [⟨1 1 2 4], ⟨0 2 1 -4]]

Wedgie: ⟨⟨2 1 -4 -3 -12 -12]]

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

Badness: 0.037565

11-limit

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

POTE generator: ~5/4 = 354.262

Map: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

Vals: 7, 10, 17, 27ce, 44cce

Badness: 0.030680

Dichosis

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

POTE generator: ~5/4 = 360.659

Map: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

Vals: 3, 7e, 10

Badness: 0.041361

Jamesbond

Comma list: 25/24, 81/80

POTE generator: ~8/7 = 258.139

Map: [⟨7 11 16 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨0 0 7 0 11 16]]

Optimal ET sequence: 7, 14c

Badness: 0.041714

11-limit

Comma list: 25/24, 33/32, 45/44

POTE generator: ~8/7 = 258.910

Map: [⟨7 11 16 0 24], ⟨0 0 0 1 0]]

Vals: 7, 14c

Badness: 0.023524

13-limit

Comma list: 25/24, 27/26, 33/32, 40/39

POTE generator: ~8/7 = 250.764

Map: [⟨7 11 16 0 24 26], ⟨0 0 0 1 0 0]]

Vals: 7, 14c

Badness: 0.023003

Septimal

Comma list: 25/24, 33/32, 45/44, 65/63

POTE generator: ~8/7 = 247.445

Map: [⟨7 11 16 0 24 6], ⟨0 0 0 1 0 1]]

Vals: 7, 14cf

Badness: 0.022569

Sidi

Comma list: 25/24, 245/243

POTE generator: ~9/7 = 427.208

Map: [⟨1 3 3 6], ⟨0 -4 -2 -9]]

Wedgie: ⟨⟨4 2 9 -12 3 15]]

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

Badness: 0.056586

11-limit

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

POTE generator: ~9/7 = 427.273

Map: [⟨1 3 3 6 7], ⟨0 -4 -2 -9 -10]]

Vals: 3de, 14c, 17, 45cce, 59bcccdee

Badness: 0.032957

Quad

Comma list: 9/8, 25/24

POTE generator: ~8/7 = 324.482

Map: [⟨4 6 9 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨0 0 4 0 6 9]]

Optimal ET sequence: 4

Badness: 0.045911