Hemimean clan
The hemimean clan tempers out the no-threes hemimean comma 3136/3125.
Didacus
Subgroup: 2.5.7
Comma list: 3136/3125
Sval mapping: [⟨1 0 -3], ⟨0 2 5]]
Mapping generators: ~2, ~28/25
Gencom mapping: [⟨1 0 0 -3], ⟨0 0 2 5]]
Gencom: [2 28/25; 3136/3125]
POTE generator: ~28/25 = 193.772
Extensions
Full 7-limit extensions of didacus, in general, split the syntonic comma into two, each for 126/125~225/224, as 3136/3125 = (126/125)/(225/224). These include hemithirds, misty, semisept, sengagen, subpental, arch, and emka, considered below, as well as septimal meantone, passion, hexe, hemiwürschmidt, parakleismic, clyde, bischismic, and sycamore considered elsewhere.
A notable subgroup extension of didacus is roulette.
Hemithirds
Subgroup: 2.3.5.7
Comma list: 1029/1024, 3136/3125
Mapping: [⟨1 4 2 2], ⟨0 -15 2 5]]
Wedgie: ⟨⟨ 15 -2 -5 -38 -50 -6 ]]
POTE generator: ~28/25 = 193.244
- [[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [11/5 -1/10 0 1/10⟩, [5/2 -1/4 0 1/4⟩]
- Eigenmonzos: 2, 7/6
- [[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [82/35 -4/35 0 2/35⟩, [20/7 -2/7 0 1/7⟩]
- Eigenmonzos: 2, 7/6
Badness: 0.044284
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 3136/3125
Mapping: [⟨1 4 2 2 7], ⟨0 -15 2 5 -22]]
POTE generator: ~28/25 = 193.227
Minimax tuning:
- 11-odd-limit
- [[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]
- Eigenmonzos: 2, 11/7
Badness: 0.019003
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 385/384, 625/624
Mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]
POTE generator: ~28/25 = 193.166
Badness: 0.021738
Misty
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5120/5103
Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]
Wedgie: ⟨⟨ 3 -12 -30 -26 -56 -36 ]]
POTE generator: ~3/2 = 703.021
Badness: 0.036802
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4375/4356
Mapping: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]
POTE generator: ~3/2 = 703.210
Badness: 0.035932
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 364/363, 625/624
Mapping: [⟨3 0 26 56 77 92], ⟨0 1 -4 -10 -14 -17]]
POTE generator: ~3/2 = 703.303
Badness: 0.027820
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
Mapping: [⟨3 0 26 56 77 92 -2], ⟨0 1 -4 -10 -14 -17 3]]
POTE generator: ~3/2 = 703.233
Badness: 0.023908
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
Mapping: [⟨3 0 26 56 77 92 -2 8], ⟨0 1 -4 -10 -14 -17 3 1]]
POTE generator: ~3/2 = 703.253
Badness: 0.019878
Mystic
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 441/440, 896/891, 1001/1000
Mapping: [⟨3 0 26 56 77 -46], ⟨0 1 -4 -10 -14 12]]
POTE generator: ~3/2 = 703.311
Badness: 0.036872
Hemimist
Subgroup: 2.3.5.7.11
Comma list: 3136/3125, 5120/5103, 3025/3024
Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]
POTE generator: ~11/8 = 551.527
Badness: 0.047439
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 676/675, 847/845, 3025/3024
Mapping: [⟨3 0 26 56 8 23], ⟨0 2 -8 -20 1 -5]]
POTE generator: ~11/8 = 551.525
Badness: 0.022923
Semisept
Subgroup: 2.3.5
Comma list: 782757789696/762939453125
Mapping: [⟨1 12 6], ⟨0 -17 -6]]
POTE generator: ~192/125 = 735.146
Badness: 0.630576
7-limit
Subgroup: 2.3.5.7
Comma list: 3136/3125, 1728/1715
Mapping: [⟨1 12 6 12], ⟨0 -17 -6 -15]]
Wedgie: ⟨⟨ 17 6 15 -30 -24 18 ]]
POTE generator: ~75/49 = 735.155
Badness: 0.050472
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 1331/1323
Mapping: [⟨1 12 6 12 20], ⟨0 -17 -6 -15 -27]]
POTE generator: ~55/36 = 735.125
Badness: 0.022476
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 540/539, 1375/1372
Mapping: [⟨1 12 6 12 20 -11], ⟨0 -17 -6 -15 -27 24]]
POTE generator: ~55/36 = 735.126
Badness: 0.025204
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
Mapping: [⟨1 12 6 12 20 -11 -10], ⟨0 -17 -6 -15 -27 24 23]]
POTE generator: ~26/17 = 735.125
Badness: 0.019919
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
Mapping: [⟨1 12 6 12 20 -11 -10 -8], ⟨0 -17 -6 -15 -27 24 23 20]]
POTE generator: ~26/17 = 735.116
Badness: 0.016301
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
Mapping: [{{val| 1 12 6 12 20 -11 -10 -8 18|, ⟨0 -17 -6 -15 -27 24 23 20 -22]]
POTE generator: ~26/17 = 735.106
Badness: 0.014957
Semishly
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 176/175, 196/195, 275/273
Mapping: [⟨1 12 6 12 20 8], ⟨0 -17 -6 -15 -27 -7]]
POTE generator: ~13/10 = 464.980
Badness: 0.028408
Sengagen
Subgroup: 2.3.5.7
Comma list: 3136/3125, 420175/419904
Mapping: [⟨1 1 2 2], ⟨0 29 16 40]]
Wedgie: ⟨⟨ 29 16 40 -42 -18 48 ]]
POTE generator: ~686/675 = 24.217
Badness: 0.057978
Subpental
Subgroup: 2.3.5.7
Comma list: 3136/3125, 19683/19600
Mapping: [⟨1 6 8 17], ⟨0 -14 -18 -45]]
Wedgie: ⟨⟨ 14 18 45 -4 32 54 ]]
POTE generator: ~56/45 = 378.467
Badness: 0.054303
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 8019/8000
Mapping: [⟨1 6 8 17 -6], ⟨0 -14 -18 -45 30]]
POTE generator: ~56/45 = 378.440
Badness: 0.045352
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 3136/3125
Mapping: [⟨1 6 8 17 -6 16], ⟨0 -14 -18 -45 30 -39]]
POTE generator: ~56/45 = 378.437
Badness: 0.023940
Arch
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5250987/5242880
Mapping: [⟨1 2 2 2], ⟨0 -18 14 35]]
Wedgie: ⟨⟨ 18 -14 -35 -64 -106 -42 ]]
POTE generator: ~64/63 = 27.668
Badness: 0.094345
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4000/3993
Mapping: [⟨1 2 2 2 3], ⟨0 -18 14 35 20]]
POTE generator: ~64/63 = 27.663
Badness: 0.036541
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 3136/3125
Mapping: [⟨1 2 2 2 3 4], ⟨0 -18 14 35 20 -13]]
POTE generator: ~64/63 = 27.660
Badness: 0.019504
Emka
Subgroup: 2.3.5
Comma list: [-50 -8 27⟩
Mapping: [⟨1 14 6], ⟨0 -27 -8]]
POTE generator: ~9765625/7077888 = 551.784
Badness: 0.544752
7-limit
Subgroup: 2.3.5.7
Comma list: 3136/3125, 84035/82944
Mapping: [⟨1 14 6 12], ⟨0 -27 -8 -20]]
POTE generator: ~48/35 = 551.782
Badness: 0.144338
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 2401/2376, 3136/3125
Mapping: [⟨1 14 6 12 3], ⟨0 -27 -8 -20 1]]
POTE generator: ~11/8 = 551.765
Vals: Template:Val list
Badness: 0.054744
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 364/363, 385/384, 625/624
Mapping: [⟨1 14 6 12 3 6], ⟨0 -27 -8 -20 1 -5]]
POTE generator: ~11/8 = 551.758
Vals: Template:Val list
Badness: 0.029741