Xenharmonic Wiki

Undim family: Difference between revisions

← Older edit
VisualWikitext
Cleanup
Tags: Mobile edit Mobile web edit
 
(4 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Technical data page}}
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 '''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 }}.  


Line 6: Line 7:


== 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.  
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]]) give 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.
Line 18: Line 19:
: mapping generators: ~1215/1024, ~3
: mapping generators: ~1215/1024, ~3


[[Optimal tuning]] ([[POTE]]): ~1215/1024 = 300.0000, ~3/2 = 702.6054
[[Optimal tuning]]s:
* [[CTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.6754
: [[error map]]: {{val| 0.0000 +0.7204 +0.3092 }}
* [[POTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.6054
: error map: {{val| 0.0000 +0.6504 +0.6591 }}


{{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 }}


[[Badness]]: 0.241703
[[Badness]] (Smith): 0.241703


== Septimal undim ==
== Septimal undim ==
Line 33: Line 38:
{{Mapping|legend=1| 4 0 41 81 | 0 1 -5 -11 }}
{{Mapping|legend=1| 4 0 41 81 | 0 1 -5 -11 }}


{{Multival|legend=1| 4 -20 -44 -41 -81 -46 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~25/21 = 300.0000, ~3/2 = 702.7948
[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.0000, ~3/2 = 702.7362
: [[error map]]: {{val| 0.0000 +0.8398 -0.2879 +0.4308 }}
* [[POTE]]: ~25/21 = 300.0000, ~3/2 = 702.7362
: error map: {{val| 0.0000 +0.7812 +0.0051 +1.0754 }}


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


[[Badness]]: 0.062754
[[Badness]] (Smith): 0.062754


=== 11-limit ===
=== 11-limit ===
Line 48: Line 55:
Mapping: {{mapping| 4 0 41 81 128 | 0 1 -5 -11 -18 }}
Mapping: {{mapping| 4 0 41 81 128 | 0 1 -5 -11 -18 }}


Optimal tuning (POTE): ~25/21 = 300.000, ~3/2 = 702.6886
Optimal tunings:
* CTE: ~25/21 = 300.0000, ~3/2 = 702.7433
* POTE: ~25/21 = 300.0000, ~3/2 = 702.6886


{{Optimal ET sequence|legend=0| 140, 152, 292, 444d, 596d }}
{{Optimal ET sequence|legend=0| 140, 152, 292, 444d, 596d }}


Badness: 0.034837
Badness (Smith): 0.034837


=== 13-limit ===
=== 13-limit ===
Line 61: Line 70:
Mapping: {{mapping| 4 0 41 81 128 148 | 0 1 -5 -11 -18 -21 }}
Mapping: {{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 tunings:
* CTE: ~25/21 = 300.0000, ~3/2 = 702.7792
* POTE: ~25/21 = 300.0000, ~3/2 = 702.7363


{{Optimal ET sequence|legend=0| 140, 152f, 292 }}
{{Optimal ET sequence|legend=0| 140, 152f, 292 }}


Badness: 0.028172
Badness (Smith): 0.028172


== Unlit ==
== Unlit ==
Line 74: Line 85:
{{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 = 300.0000, ~3/2 = 702.5764
[[Optimal tuning]]s:
* [[CTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.5556
: [[error map]]: {{val| 0.0000 +0.6006 +0.9081 +0.1761 }}
* [[POTE]]: ~1215/1024 = 300.0000, ~3/2 = 702.5764
: error map: {{val| 0.0000 +0.6214 +0.8043 +0.7369 }}


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


[[Badness]]: 0.268206
[[Badness]] (Smith): 0.268206


=== 11-limit ===
=== 11-limit ===
Line 87: Line 102:
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 = 300.0000, ~3/2 = 702.5826
Optimal tunings:
* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5582
* POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5826


{{Optimal ET sequence|legend=0| 152, 468, 620 }}
{{Optimal ET sequence|legend=0| 152, 468, 620 }}


Badness: 0.070215
Badness (Smith): 0.070215


=== 13-limit ===
=== 13-limit ===
Line 100: Line 117:
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 = 300.0000, ~3/2 = 702.5741
Optimal tunings:
* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5562
* POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5741


{{Optimal ET sequence|legend=0| 152f, 316, 468, 620f, 1088bcdf }}
{{Optimal ET sequence|legend=0| 152f, 316, 468, 620f, 1088bcdf }}


Badness: 0.058390
Badness (Smith): 0.058390


== Twilight ==
== Twilight ==
Line 115: Line 134:
: mapping generators: ~7168/6561, ~3
: mapping generators: ~7168/6561, ~3


[[Optimal tuning]] ([[POTE]]): ~7168/6561 = 150.0000, ~3/2 = 702.5090
[[Optimal tuning]]s:
* [[CTE]]: ~7168/6561 = 150.0000, ~3/2 = 702.4765
: [[error map]]: {{val| 0.0000 +0.5215 +1.3036 +0.9865 }}
* [[POTE]]: ~7168/6561 = 150.0000, ~3/2 = 702.5090
: error map: {{val| 0.0000 +0.5540 +1.1415 +1.2457 }}


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


Badness: 0.213094
Badness (Smith): 0.213094


=== 11-limit ===
=== 11-limit ===
Line 128: Line 151:
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 = 150.0000, ~3/2 = 702.5090
Optimal tunings:
* CTE: ~12/11 = 150.0000, ~3/2 = 702.4692
* POTE: ~12/11 = 150.0000, ~3/2 = 702.5090


{{Optimal ET sequence|legend=0| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
{{Optimal ET sequence|legend=0| 152, 328, 480, 1112bccddee, 1592bccdddeee }}


Badness: 0.048007
Badness (Smith): 0.048007


=== 13-limit ===
=== 13-limit ===
Line 141: Line 166:
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 = 150.0000, ~3/2 = 702.4773
Optimal tunings:
* CTE: ~12/11 = 150.0000, ~3/2 = 702.4168
* POTE: ~12/11 = 150.0000, ~3/2 = 702.4773


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


Badness: 0.041365
Badness (Smith): 0.041365


== Notes ==
== Notes ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Undim family| ]] <!-- main article -->
[[Category:Undim family| ]] <!-- main article -->
[[Category:Undim| ]] <!-- key article -->
[[Category:Undim| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Undim_family"