19th-octave temperaments: Difference between revisions

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-[[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. 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]].
+For graywood, see [[Syntonic–kleismic equivalence continuum#Graywood]].
 == Meanmag ==
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 {{Mapping|legend=1| 19 30 44 0 | 0 0 0 1 }}
 : 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}}
+: [[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 tuning]] ([[POTE]]): ~25/24 = 1\19, ~8/7 = 238.396
+{{Optimal ET sequence|legend=1| 19, 57, 76, 171bbccdd }}
-{{Optimal ET sequence|legend=1| 19, 38, 57, 76, 95bc }}
+[[Badness]] (Sintel): 1.95
-[[Badness]]: 0.077023
 === 11-limit ===
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 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 ====
 Subgroup: 2.3.5.7.11.13
@@ Line 40: / Line 46: @@
 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 ==
@@ Line 52: / Line 60: @@
 {{Mapping|legend=1| 19 30 44 53 0 | 0 0 0 0 1 }}
 : 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 }}
-[[Badness]]: 0.036387
+[[Badness]] (Sintel): 1.20
 === 13-limit ===
@@ Line 68: / Line 77: @@
 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
-[[Category:Rank 2]]
+{{Navbox fractional-octave}}