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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, 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 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 '''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 &amp; 152) tempers out 5120/5103 (hemifamity). Unlit (152 &amp; 316) does 4375/4374 (ragisma) instead. Twilight (152 &amp; 176) adds 6144/6125 (porwell) to the comma list and splits the period into two – 1/8 of an octave.  
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 }} = 2199023255552/2179240250625
[[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 = 1\4, ~3/2 = 702.6054
[[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 = 300.000, ~3/2 = 702.7362
[[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 = 1\4, ~3/2 = 702.5764
[[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 = 1\4, ~3/2 = 702.5826
Optimal tuning (POTE): ~1215/1024 = 300.0000, ~3/2 = 702.5826


{{Optimal ET sequence|legend=1| 152, 468, 620 }}
{{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 = 1\4, ~3/2 = 702.5741
Optimal tuning (POTE): ~1215/1024 = 300.0000, ~3/2 = 702.5741


{{Optimal ET sequence|legend=1| 152f, 316, 468, 620f, 1088bcdf }}
{{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 = 1\8, ~3/2 = 702.5090
[[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 = 1\8, ~3/2 = 702.5090
Optimal tuning (POTE): ~12/11 = 150.0000, ~3/2 = 702.5090


{{Optimal ET sequence|legend=1| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
{{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 = 1\8, ~3/2 = 702.4773
Optimal tuning (POTE): ~12/11 = 150.0000, ~3/2 = 702.4773


{{Optimal ET sequence|legend=1| 152f, 328, 480f, 808cdeff }}
{{Optimal ET sequence|legend=0| 152f, 328, 480f, 808cdeff }}


Badness: 0.041365
Badness: 0.041365
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