Undim family: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 14: Line 14:
[[POTE generator]]: ~3/2 = 702.6054
[[POTE generator]]: ~3/2 = 702.6054


{{Val list|legend=1| 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}
{{Optimal ET sequence|legend=1| 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }}


[[Badness]]: 0.241703
[[Badness]]: 0.241703
Line 29: Line 29:
[[POTE generator]]: ~3/2 = 702.7362
[[POTE generator]]: ~3/2 = 702.7362


{{Val list|legend=1| 140, 152, 292 }}
{{Optimal ET sequence|legend=1| 140, 152, 292 }}


[[Badness]]: 0.062754
[[Badness]]: 0.062754
Line 42: Line 42:
POTE generator: ~3/2 = 702.6886
POTE generator: ~3/2 = 702.6886


Optimal GPV sequence: {{Val list| 140, 152, 292, 444d, 596d }}
{{Optimal ET sequence|legend=1| 140, 152, 292, 444d, 596d }}


Badness: 0.034837
Badness: 0.034837
Line 55: Line 55:
POTE generator: ~3/2 = 702.7363
POTE generator: ~3/2 = 702.7363


Optimal GPV sequence: {{Val list| 140, 152f, 292 }}
{{Optimal ET sequence|legend=1| 140, 152f, 292 }}


Badness: 0.028172
Badness: 0.028172
Line 68: Line 68:
[[POTE generator]]: ~3/2 = 702.5764
[[POTE generator]]: ~3/2 = 702.5764


{{Val list|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }}
{{Optimal ET sequence|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }}


[[Badness]]: 0.268206
[[Badness]]: 0.268206
Line 81: Line 81:
POTE generator: ~3/2 = 702.5826
POTE generator: ~3/2 = 702.5826


Optimal GPV sequence: {{Val list| 152, 468, 620 }}
{{Optimal ET sequence|legend=1| 152, 468, 620 }}


Badness: 0.070215
Badness: 0.070215
Line 94: Line 94:
POTE generator: ~3/2 = 702.5741
POTE generator: ~3/2 = 702.5741


Optimal GPV sequence: {{Val list| 152f, 316, 468, 620f, 1088bcdf }}
{{Optimal ET sequence|legend=1| 152f, 316, 468, 620f, 1088bcdf }}


Badness: 0.058390
Badness: 0.058390
Line 109: Line 109:
[[POTE generator]]: ~3/2 = 702.5090
[[POTE generator]]: ~3/2 = 702.5090


{{Val list|legend=1| 152, 328, 480, 1592bccddd }}
{{Optimal ET sequence|legend=1| 152, 328, 480, 1592bccddd }}


Badness: 0.213094
Badness: 0.213094
Line 122: Line 122:
POTE generator: ~3/2 = 702.5090
POTE generator: ~3/2 = 702.5090


Optimal GPV sequence: {{Val list| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
{{Optimal ET sequence|legend=1| 152, 328, 480, 1112bccddee, 1592bccdddeee }}


Badness: 0.048007
Badness: 0.048007
Line 135: Line 135:
POTE generator: ~3/2 = 702.4773
POTE generator: ~3/2 = 702.4773


Optimal GPV sequence: {{Val list| 152f, 328, 480f, 808cdeff }}
{{Optimal ET sequence|legend=1| 152f, 328, 480f, 808cdeff }}


Badness: 0.041365
Badness: 0.041365

Revision as of 16:18, 7 May 2023

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 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 ]]

POTE generator: ~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]]

POTE generator: ~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]]

POTE generator: ~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]

POTE generator: ~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]

POTE generator: ~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]

POTE generator: ~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

POTE generator: ~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]

POTE generator: ~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]

POTE generator: ~3/2 = 702.4773

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

Badness: 0.041365

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