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19th-octave temperaments: Difference between revisions

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{{Infobox fractional-octave|19}}
{{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. 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.  
[[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 [[harmonic]]s [[7/1|7]], [[11/1|11]], and [[13/1|13]] mapped to an independent generator. Undevigintone has the same [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping as 19et with harmonic 11 mapped to an independent generator.  


See also [[enneadecal]] and [[superenneadecal]].  
See also [[enneadecal]] and [[superenneadecal]].  


== Graywood ==
For graywood, see [[Syntonic–kleismic equivalence continuum#Graywood]].
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 1162261467/1073741824
 
{{Mapping|legend=1|19 30 38|0 0 1}}
 
[[Optimal tuning]] ([[CTE]]): ~2187/2048 = 1\19, ~5/4 = 386.314
 
[[Support]]ing [[ET]]s: {{EDOs|19, 38c, 57c, 76c, 95bcc }}


== Meanmag ==
== Meanmag ==
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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


[[Optimal tuning]] ([[POTE]]): ~25/24 = 1\19, ~8/7 = 238.396
[[Optimal tuning]]s:
* [[WE]]: ~25/24 = 63.2931{{c}}, ~7/4 = 963.6625{{c}}
: [[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 ===
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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}}
{{Optimal ET sequence|legend=1| 19, 38, 57, 76 }}
* CWE: ~25/24 = 63.1579{{c}}, ~7/4 = 965.3984{{c}}
 
Badness: 0.045844
 
== (5-limit) enneadecal ==
''For the 7-limit temperament, see [[Ragismic microtemperaments#Enneadecal]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 19073486328125/19042491875328


{{Mapping|legend=1|19 19 33|0 1 1}}
{{Optimal ET sequence|legend=0| 19, 38, 57, 76 }}


[[Optimal tuning]] ([[CTE]]): ~648/625 = 1\19, ~3/2 = 701.990
Badness (Sintel): 1.89
 
[[Support]]ing [[ET]]s: {{EDOs|171, 19, 152, 323, 494, 665, 190, 133, 836, 817, 475, 114, 361, 209 }}


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


Badness: 0.022933
Badness (Sintel): 0.948


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}
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