Xenharmonic Wiki

Undim family: Difference between revisions

← Older edit
VisualWikitext
Review & expand
Naming of unlit and twilight
 
(2 intermediate revisions by the same user not shown)
Line 15: Line 15:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6754{{c}}
* [[WE]]: ~1215/1024 = 299.9421{{c}}, ~3/2 = 702.4699{{c}}
: [[error map]]: {{val| 0.0000 +0.7204 +0.3092 }}
: [[error map]]: {{val| -0.231 +0.283 +0.122 }}
* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6054{{c}}
* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.6252{{c}}
: error map: {{val| 0.0000 +0.6504 +0.6591 }}
: 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]] (Smith): 0.241703
[[Badness]] (Sintel): 5.67


=== Overview to extensions ===
=== Overview to extensions ===
Line 39: Line 39:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7948{{c}}
* [[WE]]: ~25/21 = 299.9451{{c}}, ~3/2 = 702.6076{{c}}
: [[error map]]: {{val| 0.0000 +0.8398 -0.2879 +0.4308 }}
: [[error map]]: {{val| -0.220 +0.433 -0.505 +0.458 }}
* [[POTE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7362{{c}}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7528{{c}}
: error map: {{val| 0.0000 +0.7812 +0.0051 +1.0754 }}
: error map: {{val| 0.000 +0.798 -0.078 +0.894 }}


{{Optimal ET sequence|legend=1| 140, 152, 292 }}
{{Optimal ET sequence|legend=1| 12, …, 128, 140, 152, 292 }}


[[Badness]] (Smith): 0.062754
[[Badness]] (Sintel): 1.59


=== 11-limit ===
=== 11-limit ===
Line 56: Line 56:


Optimal tunings:  
Optimal tunings:  
* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7433{{c}}
* WE: ~25/21 = 299.9389{{c}}, ~3/2 = 702.5455{{c}}
* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.6886{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7038{{c}}


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


Badness (Smith): 0.034837
Badness (Sintel): 1.15


=== 13-limit ===
=== 13-limit ===
Line 71: Line 71:


Optimal tunings:  
Optimal tunings:  
* CTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7792{{c}}
* WE: ~25/21 = 299.9476{{c}}, ~3/2 = 702.6135{{c}}
* POTE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7363{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 702.7480{{c}}


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


Badness (Smith): 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.  
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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5556{{c}}
* [[WE]]: ~1215/1024 = 299.9374{{c}}, ~3/2 = 702.4299{{c}}
: [[error map]]: {{val| 0.0000 +0.6006 +0.9081 +0.1761 }}
: [[error map]]: {{val| -0.250 +0.225 +0.223 +0.034 }}
* [[POTE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5764{{c}}
* [[CWE]]: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5716{{c}}
: error map: {{val| 0.0000 +0.6214 +0.8043 +0.7369 }}
: 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]] (Smith): 0.268206
[[Badness]] (Sintel): 6.79


=== 11-limit ===
=== 11-limit ===
Line 105: Line 105:


Optimal tunings:  
Optimal tunings:  
* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5582{{c}}
* WE: ~1215/1024 = 299.9426{{c}}, ~3/2 = 702.4482{{c}}
* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5826{{c}}
* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5772{{c}}


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


Badness (Smith): 0.070215
Badness (Sintel): 2.32


=== 13-limit ===
=== 13-limit ===
Line 120: Line 120:


Optimal tunings:  
Optimal tunings:  
* CTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5562{{c}}
* WE: ~1215/1024 = 299.9452{{c}}, ~3/2 = 702.4459{{c}}
* POTE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5741{{c}}
* CWE: ~1215/1024 = 300.0000{{c}}, ~3/2 = 702.5703{{c}}


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


Badness (Smith): 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.  
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
Line 138: Line 138:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.4765{{c}}
* [[WE]]: ~7168/6561 = 149.9598{{c}}, ~3/2 = 702.3209{{c}}
: [[error map]]: {{val| 0.0000 +0.5215 +1.3036 +0.9865 }}
: [[error map]]: {{val| -0.321 +0.045 +0.395 +0.343 }}
* [[POTE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}
* [[CWE]]: ~7168/6561 = 150.0000{{c}}, ~3/2 = 702.5023{{c}}
: error map: {{val| 0.0000 +0.5540 +1.1415 +1.2457 }}
: 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 (Smith): 0.213094
Badness (Sintel): 5.39


=== 11-limit ===
=== 11-limit ===
Line 155: Line 155:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4692{{c}}
* WE: ~12/11 = 149.9593{{c}}, ~3/2 = 702.3182{{c}}
* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5090{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.5022{{c}}


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


Badness (Smith): 0.048007
Badness (Sintel): 1.59


=== 13-limit ===
=== 13-limit ===
Line 170: Line 170:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4168{{c}}
* WE: ~12/11 = 149.9582{{c}}, ~3/2 = 702.2815{{c}}
* POTE: ~12/11 = 150.0000{{c}}, ~3/2 = 702.4773{{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=0| 152f, 328, 480f, 808cdeff }}


Badness (Smith): 0.041365
Badness (Sintel): 1.71


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Undim_family"