Undim family: Difference between revisions
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The '''undim family''' tempers out {{monzo| 41 -20 -4 }}, equating the [[Pythagorean comma]] with a stack of four [[schisma]]s | The '''undim family''' of [[regular temperaments|temperaments]] [[tempering out|tempers out]] the [[undim comma]], {{monzo| 41 -20 -4 }}, equating the [[Pythagorean comma]] with a stack of four [[schisma]]s. This makes it a member of the [[schismic–Pythagorean equivalence continuum]], with {{nowrap| ''n'' {{=}} 4 }}. | ||
The second comma of the [[normal comma list]] defines which 7-limit family member we are looking at. Septimal undim (140 & | The name ''undim'' was given by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[diminished (temperament)|diminished]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>. | ||
The second comma of the [[normal comma list]] defines which 7-limit family member we are looking at. Septimal undim ({{nowrap| 140 & 152 }}) tempers out 5120/5103 (hemifamity). Unlit ({{nowrap| 152 & 316 }}) does 4375/4374 (ragisma) instead. Twilight ({{nowrap| 152 & 176 }}) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave. | |||
== Undim == | == Undim == | ||
Undim features a quarter-octave period, which acts as the [[1215/1024|ptolemaic augmented second (1215/1024)]]. That and five [[4/3|perfect fourths]] (i.e. a minor second, ~[[256/243]]) gives the interval class of 5. | |||
Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning. | Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]]. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: {{monzo| 41 -20 -4 }} | [[Comma list]]: {{monzo| 41 -20 -4 }} | ||
{{Mapping|legend=1| 4 0 41 | 0 1 -5 }} | {{Mapping|legend=1| 4 0 41 | 0 1 -5 }} | ||
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: mapping generators: ~1215/1024, ~3 | : mapping generators: ~1215/1024, ~3 | ||
[[Optimal tuning]] ([[POTE]]): ~1215/1024 = | [[Optimal tuning]] ([[POTE]]): ~1215/1024 = 300.0000, ~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 == | ||
Septimal undim tempers out the [[dimcomp comma]], mapping ~25/21 to the 1/4-octave period. It can be described as 12 & 140, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning. | Septimal undim tempers out the [[dimcomp comma]], mapping ~25/21 to the 1/4-octave period. It can be described as {{nowrap| 12 & 140 }}, and is the unique temperament that equates a syntonic~septimal comma with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma (interval region)|kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[292edo]] makes for an excellent tuning. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }} | {{Multival|legend=1| 4 -20 -44 -41 -81 -46 }} | ||
[[Optimal tuning]] ([[POTE]]): ~25/21 = 300. | [[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000, ~3/2 = 702.7362 | ||
{{Optimal ET sequence|legend=1| 140, 152, 292 }} | {{Optimal ET sequence|legend=1| 140, 152, 292 }} | ||
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{{Mapping|legend=1| 4 0 41 -160 | 0 1 -5 27 }} | {{Mapping|legend=1| 4 0 41 -160 | 0 1 -5 27 }} | ||
[[Optimal tuning]] ([[POTE]]): ~1215/1024 = | [[Optimal tuning]] ([[POTE]]): ~1215/1024 = 300.0000, ~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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Mapping: {{mapping| 4 0 41 -160 -113 | 0 1 -5 27 20 }} | Mapping: {{mapping| 4 0 41 -160 -113 | 0 1 -5 27 20 }} | ||
Optimal tuning (POTE): ~1215/1024 = | Optimal tuning (POTE): ~1215/1024 = 300.0000, ~3/2 = 702.5826 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 152, 468, 620 }} | ||
Badness: 0.070215 | Badness: 0.070215 | ||
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Mapping: {{mapping| 4 0 41 -160 -113 -334 | 0 1 -5 27 20 55 }} | Mapping: {{mapping| 4 0 41 -160 -113 -334 | 0 1 -5 27 20 55 }} | ||
Optimal tuning (POTE): ~1215/1024 = | Optimal tuning (POTE): ~1215/1024 = 300.0000, ~3/2 = 702.5741 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 152f, 316, 468, 620f, 1088bcdf }} | ||
Badness: 0.058390 | Badness: 0.058390 | ||
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: mapping generators: ~7168/6561, ~3 | : mapping generators: ~7168/6561, ~3 | ||
[[Optimal tuning]] ([[POTE]]): ~7168/6561 = | [[Optimal tuning]] ([[POTE]]): ~7168/6561 = 150.0000, ~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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Mapping: {{mapping| 8 0 82 -79 15 | 0 1 -5 8 1 }} | Mapping: {{mapping| 8 0 82 -79 15 | 0 1 -5 8 1 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.0000, ~3/2 = 702.5090 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 152, 328, 480, 1112bccddee, 1592bccdddeee }} | ||
Badness: 0.048007 | Badness: 0.048007 | ||
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Mapping: {{mapping| 8 0 82 -79 15 -186 | 0 1 -5 8 1 17 }} | Mapping: {{mapping| 8 0 82 -79 15 -186 | 0 1 -5 8 1 17 }} | ||
Optimal tuning (POTE): ~12/11 = | Optimal tuning (POTE): ~12/11 = 150.0000, ~3/2 = 702.4773 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 152f, 328, 480f, 808cdeff }} | ||
Badness: 0.041365 | Badness: 0.041365 | ||
Revision as of 08:31, 4 April 2025
The undim family of temperaments tempers out the undim comma, [41 -20 -4⟩, equating the Pythagorean comma with a stack of four schismas. This makes it a member of the schismic–Pythagorean equivalence continuum, with n = 4.
The name undim was given by Petr Pařízek in 2011 for it is some sort of opposite to diminished[1].
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
Undim features a quarter-octave period, which acts as the ptolemaic augmented second (1215/1024). That and five perfect fourths (i.e. a minor second, ~256/243) gives the interval class of 5.
Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious extension to prime 19 by observing that sharpening 1215/1024 by 1216/1215 results in 19/16, thus mapping 19/16 to 1\4. This interpretation is arguably much more harmonically plausible, owing to its simplicity and thereby greater tolerance to mistuning.
Subgroup: 2.3.5
Comma list: [41 -20 -4⟩
Mapping: [⟨4 0 41], ⟨0 1 -5]]
- mapping generators: ~1215/1024, ~3
Optimal tuning (POTE): ~1215/1024 = 300.0000, ~3/2 = 702.6054
Optimal ET sequence: 12, …, 104, 116, 128, 140, 152, 620, 772, 924c, 1076bc, 1228bc
Badness: 0.241703
Septimal undim
Septimal undim tempers out the dimcomp comma, mapping ~25/21 to the 1/4-octave period. It can be described as 12 & 140, and is the unique temperament that equates a syntonic~septimal comma with a stack of three marvel commas. A Pythagorean comma is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a kleisma, with three kleismas making a comma, so this temperament may be useful for modeling that. 292edo makes for an excellent tuning.
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 = 300.0000, ~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 = 300.000, ~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 = 300.000, ~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 = 300.0000, ~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 = 300.0000, ~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 = 300.0000, ~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 = 150.0000, ~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 = 150.0000, ~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 = 150.0000, ~3/2 = 702.4773
Optimal ET sequence: 152f, 328, 480f, 808cdeff
Badness: 0.041365