Undim family: Difference between revisions
Created page with "The '''undim family''' tempers out {{monzo| 41 -20 -4 }}, equating the Pythagorean comma with a stack of four schismas, making it a member of the schismic-Pythagorea..." |
+13-limit undim (pretty obvious extension) |
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| Line 12: | Line 12: | ||
Mapping generators: ~1215/1024, ~3 | Mapping generators: ~1215/1024, ~3 | ||
[[POTE generator]]: ~3/2 = 702. | [[POTE generator]]: ~3/2 = 702.6054 | ||
{{Val list|legend=1| 12, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }} | {{Val list|legend=1| 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc }} | ||
[[Badness]]: 0.241703 | [[Badness]]: 0.241703 | ||
| Line 27: | Line 27: | ||
{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }} | {{Multival|legend=1| 4 -20 -44 -41 -81 -46 }} | ||
[[POTE generator]]: ~3/2 = 702. | [[POTE generator]]: ~3/2 = 702.7362 | ||
{{Val list|legend=1| | {{Val list|legend=1| 140, 152, 292 }} | ||
[[Badness]]: 0.062754 | [[Badness]]: 0.062754 | ||
| Line 40: | Line 40: | ||
Mapping: [{{val| 4 0 41 81 128 }}, {{val| 0 1 -5 -11 -18 }}] | Mapping: [{{val| 4 0 41 81 128 }}, {{val| 0 1 -5 -11 -18 }}] | ||
POTE generator: ~3/2 = 702. | POTE generator: ~3/2 = 702.6886 | ||
Vals: {{Val list| | Vals: {{Val list| 140, 152, 292, 444d, 596d }} | ||
Badness: 0.034837 | Badness: 0.034837 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 352/351, 625/624, 847/845, 1375/1372 | |||
Mapping: [{{val| 4 0 41 81 128 148 }}, {{val| 0 1 -5 -11 -18 -21 }}] | |||
POTE generator: ~3/2 = 702.7363 | |||
Vals: {{Val list| 140, 152f, 292 }} | |||
Badness: 0.0282 | |||
== Unlit == | == Unlit == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
Comma list: 4375/4374, 2199023255552/2179240250625 | [[Comma list]]: 4375/4374, 2199023255552/2179240250625 | ||
Mapping: [{{val| 4 0 41 -160 }}, {{val| 0 1 -5 27 }} | [[Mapping]]: [{{val| 4 0 41 -160 }}, {{val| 0 1 -5 27 }} | ||
POTE generator: ~3/2 = 702.5764 | [[POTE generator]]: ~3/2 = 702.5764 | ||
{{Val list|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }} | {{Val list|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }} | ||
Badness: 0.268 | [[Badness]]: 0.268 | ||
=== 11-limit === | === 11-limit === | ||
| Line 88: | Line 101: | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
Comma list: 6144/6125, 31470387200/31381059609 | [[Comma list]]: 6144/6125, 31470387200/31381059609 | ||
Mapping: [{{val| 8 0 82 -79 }}, {{val| 0 1 -5 8 }} | [[Mapping]]: [{{val| 8 0 82 -79 }}, {{val| 0 1 -5 8 }} | ||
Mapping generators: ~7168/6561, ~3 | Mapping generators: ~7168/6561, ~3 | ||
POTE | [[POTE generator]]: ~3/2 = 702.5090 | ||
{{Val list|legend=1| 152, 328, 480, 1592bccddd }} | {{Val list|legend=1| 152, 328, 480, 1592bccddd }} | ||
| Line 107: | Line 120: | ||
Mapping: [{{val| 8 0 82 -79 15 }}, {{val| 0 1 -5 8 1 }} | Mapping: [{{val| 8 0 82 -79 15 }}, {{val| 0 1 -5 8 1 }} | ||
POTE | POTE generator: ~3/2 = 702.5090 | ||
Vals: {{Val list| 152, 328, 480, 1112bccddee, 1592bccdddeee }} | Vals: {{Val list| 152, 328, 480, 1112bccddee, 1592bccdddeee }} | ||
| Line 120: | Line 133: | ||
Mapping: [{{val| 8 0 82 -79 15 -186 }}, {{val| 0 1 -5 8 1 17 }} | Mapping: [{{val| 8 0 82 -79 15 -186 }}, {{val| 0 1 -5 8 1 17 }} | ||
POTE | POTE generator: ~3/2 = 702.4773 | ||
Vals: {{Val list| 152f, 328, 480f, 808cdeff }} | Vals: {{Val list| 152f, 328, 480f, 808cdeff }} | ||
Revision as of 20:15, 11 December 2021
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
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
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
Vals: Template:Val list
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
Vals: Template:Val list
Badness: 0.0282
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
Badness: 0.268
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
Vals: Template:Val list
Badness: 0.0702
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
Vals: Template:Val list
Badness: 0.0584
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
Badness: 0.213
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
Vals: Template:Val list
Badness: 0.0480
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
Vals: Template:Val list
Badness: 0.0414