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 '''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 ==

Subgroup: 2.3.5

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.

[[Subgroup]]: 2.3.5

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

[[Comma list]]: {{monzo| 41 -20 -4 }}

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

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

: mapping generators: ~1215/1024, ~3

[[POTE ~~generator~~]]: ~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 }}

[[Badness]]: 0.241703

[[Badness]] (Smith): 0.241703

== Septimal undim ==

Subgroup: 2.3.5.7

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

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

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

[[Optimal tuning]]s:

* [[CTE]]: ~25/21 = 300.0000, ~3/2 = 702.7948

[[POTE ~~generator~~]]: ~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 }}

[[Badness]]: 0.062754

[[Badness]] (Smith): 0.062754

=== 11-limit ===

Line 38:

Line 53:

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 tunings:

* CTE: ~25/21 = 300.0000, ~3/2 = 702.7433

* POTE: ~25/21 = 300.0000, ~3/2 = 702.6886

Badness: 0.034837

Badness (Smith): 0.034837

=== 13-limit ===

Line 51:

Line 68:

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 tunings:

* CTE: ~25/21 = 300.0000, ~3/2 = 702.7792

* POTE: ~25/21 = 300.0000, ~3/2 = 702.7363

Badness: 0.028172

Badness (Smith): 0.028172

== Unlit ==

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2199023255552/2179240250625

~~[[Mapping]]: [~~{{~~val~~| 4 0 41 -160 ~~}}, {{val~~| 0 1 -5 27 }}

[[POTE ~~generator~~]]: ~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 }}

[[Badness]]: 0.268206

[[Badness]] (Smith): 0.268206

=== 11-limit ===

Line 77:

Line 100:

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 tunings:

* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5582

* POTE: ~1215/1024 = 300.0000, ~3/2 = 702.5826

Badness: 0.070215

Badness (Smith): 0.070215

=== 13-limit ===

Line 90:

Line 115:

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 tunings:

* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5562

* 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 (Smith): 0.058390

== Twilight ==

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 6144/6125, 31470387200/31381059609

~~[[Mapping]]: [~~{{~~val~~| 8 0 82 -79 ~~}}, {{val~~| 0 1 -5 8 }}

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

: mapping generators: ~7168/6561, ~3

[[POTE ~~generator~~]]: ~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 }}

Badness: 0.213094

Badness (Smith): 0.213094

=== 11-limit ===

Line 118:

Line 149:

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 tunings:

* CTE: ~12/11 = 150.0000, ~3/2 = 702.4692

* 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 (Smith): 0.048007

=== 13-limit ===

Line 131:

Line 164:

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 }}

Optimal tunings:

* CTE: ~12/11 = 150.0000, ~3/2 = 702.4168

* POTE: ~12/11 = 150.0000, ~3/2 = 702.4773

~~POTE generator: ~3/2~~ = ~~702.4773~~

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

Badness (Smith): 0.041365

~~Badness: 0.041365~~

== Notes ==

[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Undim family| ]]

[[Category:Undim| ]]

[[Category:Rank 2]]

@@ Line 1: / Line 1: @@
-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]].
+{{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 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 ==
-Subgroup: 2.3.5
+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.
+[[Subgroup]]: 2.3.5
-[[Comma list]]: {{monzo| 41 -20 -4 }} = 2199023255552/2179240250625
+[[Comma list]]: {{monzo| 41 -20 -4 }}
-[[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]]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 }}
-[[Badness]]: 0.241703
+[[Badness]] (Smith): 0.241703
 == Septimal undim ==
-Subgroup: 2.3.5.7
+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
 [[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 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~25/21 = 300.0000, ~3/2 = 702.7948
-[[POTE generator]]: ~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 }}
-[[Badness]]: 0.062754
+[[Badness]] (Smith): 0.062754
 === 11-limit ===
@@ Line 38: / Line 53: @@
 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 tunings:
+* CTE: ~25/21 = 300.0000, ~3/2 = 702.7433
+* POTE: ~25/21 = 300.0000, ~3/2 = 702.6886
-{{Optimal ET sequence|legend=1| 140, 152, 292, 444d, 596d }}
+{{Optimal ET sequence|legend=0| 140, 152, 292, 444d, 596d }}
-Badness: 0.034837
+Badness (Smith): 0.034837
 === 13-limit ===
@@ Line 51: / Line 68: @@
 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 tunings:
+* CTE: ~25/21 = 300.0000, ~3/2 = 702.7792
+* POTE: ~25/21 = 300.0000, ~3/2 = 702.7363
-{{Optimal ET sequence|legend=1| 140, 152f, 292 }}
+{{Optimal ET sequence|legend=0| 140, 152f, 292 }}
-Badness: 0.028172
+Badness (Smith): 0.028172
 == Unlit ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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]]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 }}
-[[Badness]]: 0.268206
+[[Badness]] (Smith): 0.268206
 === 11-limit ===
@@ Line 77: / Line 100: @@
 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 tunings:
+* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5582
+* 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 (Smith): 0.070215
 === 13-limit ===
@@ Line 90: / Line 115: @@
 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 tunings:
+* CTE: ~1215/1024 = 300.0000, ~3/2 = 702.5562
+* 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 (Smith): 0.058390
 == Twilight ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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]]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 }}
-Badness: 0.213094
+Badness (Smith): 0.213094
 === 11-limit ===
@@ Line 118: / Line 149: @@
 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 tunings:
+* CTE: ~12/11 = 150.0000, ~3/2 = 702.4692
+* 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 (Smith): 0.048007
 === 13-limit ===
@@ Line 131: / Line 164: @@
 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 }}
+Optimal tunings:
+* CTE: ~12/11 = 150.0000, ~3/2 = 702.4168
+* POTE: ~12/11 = 150.0000, ~3/2 = 702.4773
-POTE generator: ~3/2 = 702.4773
+{{Optimal ET sequence|legend=0| 152f, 328, 480f, 808cdeff }}
-{{Optimal ET sequence|legend=1| 152f, 328, 480f, 808cdeff }}
+Badness (Smith): 0.041365
-Badness: 0.041365
+== Notes ==
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Undim family| ]] <!-- main article -->
+[[Category:Undim| ]] <!-- key article -->
 [[Category:Rank 2]]