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Undim family: Difference between revisions

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Cleanup
+CTE tunings; +error maps; note the badness metric being used
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== 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.
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: 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 ==
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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 = 300.0000, ~3/2 = 702.7362
[[Optimal tuning]]s:
* [[CTE]]: ~25/21 = 300.0000, ~3/2 = 702.7948
: [[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 ===
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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 ===
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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 ==
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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 = 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 ===
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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 = 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 ===
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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 = 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 ==
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: 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 ===
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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 = 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 ===
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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 = 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 ==
Retrieved from "https://en.xen.wiki/w/Undim_family"