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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]].  
{{Technical data page}}
The '''undim family''' of [[regular temperaments|temperaments]] [[tempering out|tempers out]] the [[undim comma]], {{monzo| 41 -20 -4 }}.  


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 [[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. Undim has a [[ploidacot]] of tetraploid monocot, and equates the [[Pythagorean comma]] with a stack of four [[schisma]]s, making it a member of the [[schismic–Pythagorean equivalence continuum]] with {{nowrap| ''n'' {{=}} 4 }}.  


== Undim ==
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>.  
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 }}
: mapping generators: ~1215/1024, ~3
: mapping generators: ~1215/1024, ~3


[[Optimal tuning]] ([[POTE]]): ~1215/1024 = 1\4, ~3/2 = 702.6054
[[Optimal tuning]]s:
* [[WE]]: ~1215/1024 = 299.9421{{c}}, ~3/2 = 702.4699{{c}}
: [[error map]]: {{val| -0.231 +0.283 +0.122 }}
* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6252{{c}}
: error map: {{val| 0.000 +0.670 +0.560 }}


{{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]] (Sintel): 5.67
 
=== Overview to extensions ===
The second comma of the [[normal forms #Normal forms for commas|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.
 
Note that all versions of undim (ones that do not already map 19 differently and more accurately) have an obvious place for prime 19 by observing that sharpening 1215/1024 by [[1216/1215]] results in [[19/16]], thus mapping 19/16 to [[4edo|1\4]].  


== 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 {{nowrap| 12 & 140 }}, and is the unique temperament that tempers together the syntonic and septimal commas and equates it with a stack of three [[marvel comma]]s. A [[Pythagorean comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an 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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{{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:
* [[WE]]: ~25/21 = 299.9451{{c}}, ~3/2 = 702.6076{{c}}
: [[error map]]: {{val| -0.220 +0.433 -0.505 +0.458 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7528{{c}}
: error map: {{val| 0.000 +0.798 -0.078 +0.894 }}


[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 702.7362
{{Optimal ET sequence|legend=1| 12, …, 128, 140, 152, 292 }}


{{Optimal ET sequence|legend=1| 140, 152, 292 }}
[[Badness]] (Sintel): 1.59
 
[[Badness]]: 0.062754


=== 11-limit ===
=== 11-limit ===
Line 42: 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 = 1\4, ~3/2 = 702.6886
Optimal tunings:
* WE: ~25/21 = 299.9389{{c}}, ~3/2 = 702.5455{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7038{{c}}


{{Optimal ET sequence|legend=1| 140, 152, 292, 444d, 596d }}
{{Optimal ET sequence|legend=0| 12, 128e, 140, 152, 292, 444d, 596d }}


Badness: 0.034837
Badness (Sintel): 1.15


=== 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 = 1\4, ~3/2 = 702.7363
Optimal tunings:
* WE: ~25/21 = 299.9476{{c}}, ~3/2 = 702.6135{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7480{{c}}


{{Optimal ET sequence|legend=1| 140, 152f, 292 }}
{{Optimal ET sequence|legend=0| 12f, 128ef, 140, 152f, 292 }}


Badness: 0.028172
Badness (Sintel): 1.16


== Unlit ==
== Unlit ==
Unlit tempers out 4375/4374, the [[ragisma]], and may be described as {{nowrap| 152 & 164 }}. This temperament is more faithful to the 5-limit optimal tuning, but comes with the cost of a much higher complexity. It was named by [[Flora Canou]] in 2021 for it is some sort of opposite of [[#Septimal undim|septimal undim]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 2199023255552/2179240250625
[[Comma list]]: 4375/4374, {{monzo| 41 -20 -4 }}


{{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]]s:
* [[WE]]: ~1215/1024 = 299.9374{{c}}, ~3/2 = 702.4299{{c}}
: [[error map]]: {{val| -0.250 +0.225 +0.223 +0.034 }}
* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5716{{c}}
: error map: {{val| 0.000 +0.617 +0.828 +0.607 }}


{{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]] (Sintel): 6.79


=== 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 = 1\4, ~3/2 = 702.5826
Optimal tunings:
* WE: ~1215/1024 = 299.9426{{c}}, ~3/2 = 702.4482{{c}}
* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5772{{c}}


{{Optimal ET sequence|legend=1| 152, 468, 620 }}
{{Optimal ET sequence|legend=0| 152, 468, 620 }}


Badness: 0.070215
Badness (Sintel): 2.32


=== 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 = 1\4, ~3/2 = 702.5741
Optimal tunings:
* WE: ~1215/1024 = 299.9452{{c}}, ~3/2 = 702.4459{{c}}
* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5703{{c}}


{{Optimal ET sequence|legend=1| 152f, 316, 468, 620f, 1088bcdf }}
{{Optimal ET sequence|legend=0| 152f, 316, 468, 620f, 1088bcdf }}


Badness: 0.058390
Badness (Sintel): 2.41


== Twilight ==
== Twilight ==
Twilight tempers out 6144/6125, the [[porwell comma]], and may be described as {{nowrap| 152 & 176 }}. It splits the period of undim into 1/8 of an octave, and has a [[ploidacot]] of octaploid monocot. It was named by [[Flora Canou]] in 2021 for it splits the period in halves to reconcile the different mappings of [[#Septimal undim|septimal undim]] and [[#Unlit|unlit]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 8 0 82 -79 | 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


[[Optimal tuning]] ([[POTE]]): ~7168/6561 = 1\8, ~3/2 = 702.5090
[[Optimal tuning]]s:
* [[WE]]: ~7168/6561 = 149.9598{{c}}, ~3/2 = 702.3209{{c}}
: [[error map]]: {{val| -0.321 +0.045 +0.395 +0.343 }}
* [[CWE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5023{{c}}
: error map: {{val| 0.000 +0.547 +1.175 +1.192 }}


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


Badness: 0.213094
Badness (Sintel): 5.39


=== 11-limit ===
=== 11-limit ===
Line 122: Line 154:
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 tunings:
* WE: ~12/11 = 149.9593{{c}}, ~3/2 = 702.3182{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5022{{c}}


{{Optimal ET sequence|legend=1| 152, 328, 480, 1112bccddee, 1592bccdddeee }}
{{Optimal ET sequence|legend=0| 152, 328, 480, 1112bccddee, 1592bccdddeee }}


Badness: 0.048007
Badness (Sintel): 1.59


=== 13-limit ===
=== 13-limit ===
Line 135: Line 169:
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 tunings:
* WE: ~12/11 = 149.9582{{c}}, ~3/2 = 702.2815{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4670{{c}}
 
{{Optimal ET sequence|legend=0| 152f, 328, 480f, 808cdeff }}


{{Optimal ET sequence|legend=1| 152f, 328, 480f, 808cdeff }}
Badness (Sintel): 1.71


Badness: 0.041365
== References ==


[[Category:Undim family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Undim family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Undim| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Undim_family"