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.

[[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]].

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: mapping generators: ~25/24, ~7

[[Optimal tuning]] ([[~~POTE~~]]): ~25/24 = 63.1579{{c}}, ~8/7 = ~~238~~.~~396~~{{c}}

[[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 }}

[[Badness]] (~~Smith~~): 0.~~077023~~

[[Badness]] (Sintel): 1.95

=== 11-limit ===

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

Optimal ~~tuning (POTE)~~: ~25/24 = 63.1579{{c}}, ~8/7 = ~~233~~.~~486~~{{c}}

Optimal tunings:

* WE: ~25/24 = 63.2535{{c}}, ~7/4 = 967.9769{{c}}

* CWE: ~25/24 = 63.1579{{c}}, ~7/4 = 966.6112{{c}}

Badness (~~Smith~~): 0.~~066829~~

Badness (Sintel): 2.21

==== 13-limit ====

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

Optimal ~~tuning (POTE)~~: ~25/24 = 63.1579{{c}}, ~8/7 = ~~234~~.~~890~~{{c}}

Optimal tunings:

* WE: ~25/24 = 63.2422{{c}}, ~7/4 = 966.3987{{c}}

* CWE: ~25/24 = 63.1579{{c}}, ~7/4 = 965.3984{{c}}

Badness (~~Smith~~): 0.~~045844~~

Badness (Sintel): 1.89

== Undevigintone ==

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: mapping generators: ~28/27, ~11

[[Optimal tuning]] ([[~~POTE~~]]): ~28/27 = 63.1579{{c}}, ~11/8 = ~~538~~.~~047~~{{c}}

[[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}}

[[Badness]] (~~Smith~~): 0.~~036387~~

[[Badness]] (Sintel): 1.20

=== 13-limit ===

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Mapping: {{mapping| 19 30 44 53 0 70 | 0 0 0 0 1 0 }}

Optimal ~~tuning (POTE)~~: ~28/27 = 63.1579{{c}}, ~11/8 = ~~537~~.~~061~~{{c}}

Optimal tunings:

* WE: ~28/27 = 63.3741{{c}}, ~11/8 = 538.8996{{c}}

* CWE: ~28/27 = 63.1579{{c}}, ~11/8 = 539.4216{{c}}

Badness (~~Smith~~): 0.~~022933~~

Badness (Sintel): 0.948

@@ Line 1: / Line 1: @@
 {{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]].
@@ Line 12: / Line 12: @@
 : mapping generators: ~25/24, ~7
-[[Optimal tuning]] ([[POTE]]): ~25/24 = 63.1579{{c}}, ~8/7 = 238.396{{c}}
+[[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]] (Smith): 0.077023
+[[Badness]] (Sintel): 1.95
 === 11-limit ===
@@ Line 25: / Line 29: @@
 Mapping: {{mapping| 19 30 44 0 119 | 0 0 0 1 -1 }}
-Optimal tuning (POTE): ~25/24 = 63.1579{{c}}, ~8/7 = 233.486{{c}}
+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=0| 19, 38, 57, 76 }}
+{{Optimal ET sequence|legend=0| 19, 38, 57 }}
-Badness (Smith): 0.066829
+Badness (Sintel): 2.21
 ==== 13-limit ====
@@ Line 38: / Line 44: @@
 Mapping: {{mapping| 19 30 44 0 119 17 | 0 0 0 1 -1 1 }}
-Optimal tuning (POTE): ~25/24 = 63.1579{{c}}, ~8/7 = 234.890{{c}}
+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=0| 19, 38, 57, 76 }}
-Badness (Smith): 0.045844
+Badness (Sintel): 1.89
 == Undevigintone ==
@@ Line 52: / Line 60: @@
 : mapping generators: ~28/27, ~11
-[[Optimal tuning]] ([[POTE]]): ~28/27 = 63.1579{{c}}, ~11/8 = 538.047{{c}}
+[[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]] (Smith): 0.036387
+[[Badness]] (Sintel): 1.20
 === 13-limit ===
@@ Line 65: / Line 75: @@
 Mapping: {{mapping| 19 30 44 53 0 70 | 0 0 0 0 1 0 }}
-Optimal tuning (POTE): ~28/27 = 63.1579{{c}}, ~11/8 = 537.061{{c}}
+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 }}
-Badness (Smith): 0.022933
+Badness (Sintel): 0.948
 {{Navbox fractional-octave}}