19th-octave temperaments: Difference between revisions

(One intermediate revision by one other user not shown)

Line 1:

[[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]].

For graywood, see [[Syntonic–kleismic equivalence continuum#Graywood]].

== Meanmag ==

Line 12:

Line 14:

: 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 ===

Line 25:

Line 31:

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 ====

Line 38:

Line 46:

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 ==

Line 52:

Line 62:

: 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 ===

Line 65:

Line 77:

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]].
+For graywood, see [[Syntonic–kleismic equivalence continuum#Graywood]].
 == Meanmag ==
@@ Line 12: / Line 14: @@
 : 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 31: @@
 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 46: @@
 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 62: @@
 : 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 77: @@
 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}}