Undim family

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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

POTE generator: ~3/2 = 702.6054

Optimal GPV sequence12, …, 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]]

POTE generator: ~3/2 = 702.7362

Optimal GPV sequence140, 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]]

POTE generator: ~3/2 = 702.6886

Optimal GPV 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]]

POTE generator: ~3/2 = 702.7363

Optimal GPV 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]

POTE generator: ~3/2 = 702.5764

Optimal GPV sequence152, 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]

POTE generator: ~3/2 = 702.5826

Optimal GPV 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]

POTE generator: ~3/2 = 702.5741

Optimal GPV 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

POTE generator: ~3/2 = 702.5090

Optimal GPV sequence152, 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]

POTE generator: ~3/2 = 702.5090

Optimal GPV 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]

POTE generator: ~3/2 = 702.4773

Optimal GPV sequence: 152f, 328, 480f, 808cdeff

Badness: 0.041365