Undim family
The undim family tempers out [41 -20 -4⟩, equating the Pythagorean comma with a stack of four schismas, making it a member of the schismic-Pythagorean equivalence continuum. It features a quarter-octave period, which acts as the interval separating ~256/243 from ~5/4.
The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal undim (140 & 152) tempers out 5120/5103 (hemifamity). Unlit (152 & 316) does 4375/4374 (ragisma) instead. Twilight (152 & 176) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.
Undim
Subgroup: 2.3.5
Comma list: [41 -20 -4⟩ = 2199023255552/2179240250625
Mapping: [⟨4 0 41], ⟨0 1 -5]]
- mapping generators: ~1215/1024, ~3
Optimal tuning (POTE): ~1215/1024 = 1\4, ~3/2 = 702.6054
Optimal ET sequence: 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc
Badness: 0.241703
Septimal undim
Subgroup: 2.3.5.7
Comma list: 5120/5103, 390625/388962
Mapping: [⟨4 0 41 81], ⟨0 1 -5 -11]]
Wedgie: ⟨⟨4 -20 -44 -41 -81 -46]]
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 702.7362
Optimal ET sequence: 140, 152, 292
Badness: 0.062754
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 5120/5103, 5632/5625
Mapping: [⟨4 0 41 81 128], ⟨0 1 -5 -11 -18]]
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 702.6886
Optimal ET sequence: 140, 152, 292, 444d, 596d
Badness: 0.034837
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 625/624, 847/845, 1375/1372
Mapping: [⟨4 0 41 81 128 148], ⟨0 1 -5 -11 -18 -21]]
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 702.7363
Optimal ET sequence: 140, 152f, 292
Badness: 0.028172
Unlit
Subgroup: 2.3.5.7
Comma list: 4375/4374, 2199023255552/2179240250625
Mapping: [⟨4 0 41 -160], ⟨0 1 -5 27]]
Optimal tuning (POTE): ~1215/1024 = 1\4, ~3/2 = 702.5764
Optimal ET sequence: 152, 316, 468, 620, 1088bcd, 1708bccdd
Badness: 0.268206
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 5767168/5740875
Mapping: [⟨4 0 41 -160 -113], ⟨0 1 -5 27 20]]
Optimal tuning (POTE): ~1215/1024 = 1\4, ~3/2 = 702.5826
Optimal ET sequence: 152, 468, 620
Badness: 0.070215
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 1835008/1828125
Mapping: [⟨4 0 41 -160 -113 -334], ⟨0 1 -5 27 20 55]]
Optimal tuning (POTE): ~1215/1024 = 1\4, ~3/2 = 702.5741
Optimal ET sequence: 152f, 316, 468, 620f, 1088bcdf
Badness: 0.058390
Twilight
Subgroup: 2.3.5.7
Comma list: 6144/6125, 31470387200/31381059609
Mapping: [⟨8 0 82 -79], ⟨0 1 -5 8]]
- mapping generators: ~7168/6561, ~3
Optimal tuning (POTE): ~7168/6561 = 1\8, ~3/2 = 702.5090
Optimal ET sequence: 152, 328, 480, 1592bccddd
Badness: 0.213094
11-limit
Subgroup: 2.3.5.7.11
Comma list: 6144/6125, 9801/9800, 19712/19683
Mapping: [⟨8 0 82 -79 15], ⟨0 1 -5 8 1]]
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 702.5090
Optimal ET sequence: 152, 328, 480, 1112bccddee, 1592bccdddeee
Badness: 0.048007
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3584/3575, 14641/14625
Mapping: [⟨8 0 82 -79 15 -186], ⟨0 1 -5 8 1 17]]
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 702.4773
Optimal ET sequence: 152f, 328, 480f, 808cdeff
Badness: 0.041365