Undim family: Difference between revisions

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== Undim ==
== Undim ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 41 -20 -4 }} = 2199023255552/2179240250625
[[Comma list]]: {{monzo| 41 -20 -4 }} = 2199023255552/2179240250625


[[Mapping]]: [{{val| 4 0 41 }}, {{val| 0 1 -5 }}]
{{Mapping|legend=1| 4 0 41 | 0 1 -5 }}


Mapping generators: ~1215/1024, ~3
: mapping generators: ~1215/1024, ~3


[[POTE generator]]: ~3/2 = 702.6054
[[Optimal tuning]] ([[POTE]]): ~1215/1024 = 1\4, ~3/2 = 702.6054


{{Optimal ET sequence|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 }}
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== Septimal undim ==
== Septimal undim ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 390625/388962
[[Comma list]]: 5120/5103, 390625/388962


[[Mapping]]: [{{val| 4 0 41 81 }}, {{val| 0 1 -5 -11 }}]
{{Mapping|legend=1| 4 0 41 81 | 0 1 -5 -11 }}


{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }}
{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }}


[[POTE generator]]: ~3/2 = 702.7362
[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 702.7362


{{Optimal ET sequence|legend=1| 140, 152, 292 }}
{{Optimal ET sequence|legend=1| 140, 152, 292 }}
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Comma list: 1375/1372, 5120/5103, 5632/5625
Comma list: 1375/1372, 5120/5103, 5632/5625


Mapping: [{{val| 4 0 41 81 128 }}, {{val| 0 1 -5 -11 -18 }}]
Mapping: {{mapping| 4 0 41 81 128 | 0 1 -5 -11 -18 }}


POTE generator: ~3/2 = 702.6886
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 702.6886


{{Optimal ET sequence|legend=1| 140, 152, 292, 444d, 596d }}
{{Optimal ET sequence|legend=1| 140, 152, 292, 444d, 596d }}
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Comma list: 352/351, 625/624, 847/845, 1375/1372
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 }}]
Mapping: {{mapping| 4 0 41 81 128 148 | 0 1 -5 -11 -18 -21 }}


POTE generator: ~3/2 = 702.7363
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 702.7363


{{Optimal ET sequence|legend=1| 140, 152f, 292 }}
{{Optimal ET sequence|legend=1| 140, 152f, 292 }}
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== 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|legend=1| 4 0 41 -160 | 0 1 -5 27 }}


[[POTE generator]]: ~3/2 = 702.5764
[[Optimal tuning]] ([[POTE]]): ~1215/1024 = 1\4, ~3/2 = 702.5764


{{Optimal ET sequence|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }}
{{Optimal ET sequence|legend=1| 152, 316, 468, 620, 1088bcd, 1708bccdd }}
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Comma list: 3025/3024, 4375/4374, 5767168/5740875
Comma list: 3025/3024, 4375/4374, 5767168/5740875


Mapping: [{{val| 4 0 41 -160 -113 }}, {{val| 0 1 -5 27 20 }}
Mapping: {{mapping| 4 0 41 -160 -113 | 0 1 -5 27 20 }}


POTE generator: ~3/2 = 702.5826
Optimal tuning (POTE): ~1215/1024 = 1\4, ~3/2 = 702.5826


{{Optimal ET sequence|legend=1| 152, 468, 620 }}
{{Optimal ET sequence|legend=1| 152, 468, 620 }}
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Comma list: 1716/1715, 2080/2079, 3025/3024, 1835008/1828125
Comma list: 1716/1715, 2080/2079, 3025/3024, 1835008/1828125


Mapping: [{{val| 4 0 41 -160 -113 -334 }}, {{val| 0 1 -5 27 20 55 }}
Mapping: {{mapping| 4 0 41 -160 -113 -334 | 0 1 -5 27 20 55 }}


POTE generator: ~3/2 = 702.5741
Optimal tuning (POTE): ~1215/1024 = 1\4, ~3/2 = 702.5741


{{Optimal ET sequence|legend=1| 152f, 316, 468, 620f, 1088bcdf }}
{{Optimal ET sequence|legend=1| 152f, 316, 468, 620f, 1088bcdf }}
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== Twilight ==
== Twilight ==
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|legend=1| 8 0 82 -79 | 0 1 -5 8 }}


Mapping generators: ~7168/6561, ~3
: mapping generators: ~7168/6561, ~3


[[POTE generator]]: ~3/2 = 702.5090
[[Optimal tuning]] ([[POTE]]): ~7168/6561 = 1\8, ~3/2 = 702.5090


{{Optimal ET sequence|legend=1| 152, 328, 480, 1592bccddd }}
{{Optimal ET sequence|legend=1| 152, 328, 480, 1592bccddd }}
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Comma list: 6144/6125, 9801/9800, 19712/19683
Comma list: 6144/6125, 9801/9800, 19712/19683


Mapping: [{{val| 8 0 82 -79 15 }}, {{val| 0 1 -5 8 1 }}
Mapping: {{mapping| 8 0 82 -79 15 | 0 1 -5 8 1 }}


POTE generator: ~3/2 = 702.5090
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 702.5090


{{Optimal ET sequence|legend=1| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
{{Optimal ET sequence|legend=1| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
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Comma list: 1716/1715, 2080/2079, 3584/3575, 14641/14625
Comma list: 1716/1715, 2080/2079, 3584/3575, 14641/14625


Mapping: [{{val| 8 0 82 -79 15 -186 }}, {{val| 0 1 -5 8 1 17 }}
Mapping: {{mapping| 8 0 82 -79 15 -186 | 0 1 -5 8 1 17 }}


POTE generator: ~3/2 = 702.4773
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 702.4773


{{Optimal ET sequence|legend=1| 152f, 328, 480f, 808cdeff }}
{{Optimal ET sequence|legend=1| 152f, 328, 480f, 808cdeff }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Undim family| ]] <!-- main article -->
[[Category:Undim family| ]] <!-- main article -->
[[Category:Undim| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 08:50, 12 September 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

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

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