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19th-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|19}}
{{Technical data page}}
[[19edo]] has excellent 5-limit accuracy, but its quality of higher-limit approximation can be improved. This page accommodates a number of temperaments that are otherwise difficult to catalog because they belong to multiple families. Meanmag has the same 5-limit mapping as 19et with harmonics 7, 11, and 13 mapped to an independent generator. Undevigintone has the same 2.3.5.7.13 subgroup mapping as 19et with harmonic 11 mapped to an independent generator.  
{{Infobox fractional-octave|19}}
[[19edo]] has excellent [[5-limit]] accuracy, but its quality of higher-limit approximation can be improved. This page accommodates a number of temperaments that are otherwise difficult to catalog because they belong to multiple families.  


See also [[enneadecal]] and [[superenneadecal]].  
* Graywood has the same 3-limit mapping as 19et, but harmonic [[5/1|5]] is mapped to an independent generator.
* Meanmag has the same 5-limit mapping as 19et, but harmonic 7 is mapped to an independent generator.
* Undevigintone has the same 7-limit mapping as 19et, but harmonic 11 is mapped to an independent generator.  


== Graywood ==
For graywood, see [[Syntonic–kleismic equivalence continuum #Graywood]].
[[Subgroup]]: 2.3.5


[[Comma list]]: 1162261467/1073741824
See also [[ragismic microtemperaments #Enneadecal|enneadecal]] and [[sensamagic clan #Superenneadecal|superenneadecal]].


{{Mapping|legend=1|19 30 38|0 0 1}}
== Meanmag ==
 
[[Optimal tuning]] ([[CTE]]): ~2187/2048 = 1\19, ~5/4 = 386.314
 
[[Support]]ing [[ET]]s: {{EDOs|19, 38c, 57c, 76c, 95bcc }}
=== Meanmag ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 19 30 44 0 | 0 0 0 1 }}
{{Mapping|legend=1| 19 30 44 0 | 0 0 0 1 }}
: mapping generators: ~25/24, ~7
: mapping generators: ~25/24, ~7


{{Multival|legend=1| 0 0 19 0 30 44 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~25/24 = 63.2931{{c}}, ~7/4 = 963.6625{{c}}
[[Optimal tuning]] ([[POTE]]): ~25/24 = 1\19, ~8/7 = 238.396
: [[error map]]: {{val| +2.569 -3.162 -1.417 -0.026 }}
* [[CWE]]: ~25/24 = 63.1579{{c}}, ~7/4 = 963.4030{{c}}
: error map: {{val| 0.000 -7.218 -7.366 -5.423 }}


{{Optimal ET sequence|legend=1| 19, 38, 57, 76, 95bc }}
{{Optimal ET sequence|legend=1| 19, 57, 76, 171bbccdd }}


[[Badness]]: 0.077023
[[Badness]] (Sintel): 1.95


==== 11-limit ====
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Mapping: {{mapping| 19 30 44 0 119 | 0 0 0 1 -1 }}
Mapping: {{mapping| 19 30 44 0 119 | 0 0 0 1 -1 }}


Optimal tuning (POTE): ~25/24 = 1\19, ~8/7 = 233.486
Optimal tunings:
* WE: ~25/24 = 63.2535{{c}}, ~7/4 = 967.9769{{c}}
* CWE: ~25/24 = 63.1579{{c}}, ~7/4 = 966.6112{{c}}


{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=0| 19, 38, 57 }}


Badness: 0.066829
Badness (Sintel): 2.21


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 19 30 44 0 119 17 | 0 0 0 1 -1 1 }}
Mapping: {{mapping| 19 30 44 0 119 17 | 0 0 0 1 -1 1 }}


Optimal tuning (POTE): ~25/24 = 1\19, ~8/7 = 234.890
Optimal tunings:
* WE: ~25/24 = 63.2422{{c}}, ~7/4 = 966.3987{{c}}
* CWE: ~25/24 = 63.1579{{c}}, ~7/4 = 965.3984{{c}}


{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
{{Optimal ET sequence|legend=0| 19, 38, 57, 76 }}


Badness: 0.045844
Badness (Sintel): 1.89


== Undevigintone ==
== Undevigintone ==
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{{Mapping|legend=1| 19 30 44 53 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 19 30 44 53 0 | 0 0 0 0 1 }}
: mapping generators: ~28/27, ~11
: mapping generators: ~28/27, ~11


[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\19, ~11/8 = 538.047
[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 63.3591{{c}}, ~11/8 = 539.7611{{c}}
* [[CWE]]: ~28/27 = 63.1579{{c}}, ~11/8 = 540.6837{{c}}


{{Optimal ET sequence|legend=1| 19, 38d }}
{{Optimal ET sequence|legend=1| 19, 38d }}


[[Badness]]: 0.036387
[[Badness]] (Sintel): 1.20


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 19 30 44 53 0 70 | 0 0 0 0 1 0 }}
Mapping: {{mapping| 19 30 44 53 0 70 | 0 0 0 0 1 0 }}


Optimal tuning (POTE): ~28/27 = 1\19, ~11/8 = 537.061
Optimal tunings:
* WE: ~28/27 = 63.3741{{c}}, ~11/8 = 538.8996{{c}}
* CWE: ~28/27 = 63.1579{{c}}, ~11/8 = 539.4216{{c}}
 
{{Optimal ET sequence|legend=0| 19, 38df }}


{{Optimal ET sequence|legend=1| 19, 38df }}
Badness (Sintel): 0.948


Badness: 0.022933
{{Navbox fractional-octave}}
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