Dicot family
The 5-limit parent comma for the dicot family is 25/24, the classical 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 classical third (major and minor mean the same thing), and that 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
- 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, 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 ]] adds 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.
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
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