13th-octave temperaments: Difference between revisions

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== Aluminium ==

: ''~~See also:~~ [[Ragismic microtemperaments #Aluminium]]''

: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].''

Aluminium ~~is named after the 13th element, and~~ tempers out ~~the~~ {{monzo| 92 -39 -13 }} ~~comma which~~ sets [[135/128]] ~~interval to be equal~~ to 1/~~13th~~ of ~~the~~ octave.

Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit, and sets [[135/128]] to 1/13 of an [[octave]].

[[Subgroup]]: 2.3.5

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[[Comma list]]: {{monzo| 92 -39 -13 }}

~~[[Mapping]]:~~ {{~~mapping~~| 13 0 92 | 0 1 -3 }}

: mapping generators: ~135/128, ~3

: ~~Mapping generators~~: ~135/128, ~3

[[Optimal tuning]]s:

* [[WE]]: ~135/128 = 92.3068{{c}}, ~3/2 = 701.9791{{c}}

[[~~Optimal tuning~~]] ([[~~CTE~~]]): ~135/128 = ~~1\13~~, ~3/2 = 701.9897

: [[error map]]: {{val| -0.012 +0.012 +0.009 }}

* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 701.9897{{c}}

: error map: {{val| 0.000 +0.032 +0.033 }}

{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}

[[Badness]] (Sintel): 2.~~893~~

[[Badness]] (Sintel): 2.89

== Tridecatonic ==

: ''~~See also:~~ [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[~~26th-octave~~ temperaments]]''

: ''For extensions, see [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[Varunismic temperaments #Bosonic]].''

''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 ~~steps of 13edo~~.

Tridecatonic tempers out the [[devil's tridecalimma]], the comma which associates the [[10/9]] minor tone with [[13edo|2\13]].

[[Subgroup]]: 2.3.5

[[Comma list]]: ~~2541865828329/2500000000000~~

[[Comma list]]: {{monzo| -11 26 -13 }}

~~[[Mapping]]: [~~{{~~Val~~| 13 ~~21 31 }}, {{Val~~| 0 -1 -2 }}]

: mapping generators: ~531441/500000, ~3

: ~~Mapping generators~~: ~531441/500000, ~3

[[Optimal tuning]]s:

* [[WE]]: ~531441/500000 = 92.3166{{c}}, ~3/2 = 701.2353{{c}}

[[~~Optimal tuning~~]] ([[~~CTE~~]]): ~531441/500000 = ~~1\13~~, ~3/2 = 701.2353

: [[error map]]: {{val| +0.116 -0.688 +0.737 }}

* [[CWE]]: ~531441/500000 = 92.3077{{c}}, ~3/2 = 701.2353{{c}}

: error map: {{val| 0.000 -0.797 +0.617 }}

{{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }}

[[Badness]] (Sintel): 10.~~263~~

[[Badness]] (Sintel): 10.3

== Triskaidekic ==

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: Mapping generators: ~128/125, ~3

[[Optimal tuning]]s:

* [[WE]]: ~128/125 = 92.589, ~3/2 = 698.263

* [[WE]]: ~128/125 = 92.589{{c}}, ~3/2 = 698.263{{c}}

* [[CWE]]: ~128/125 = 92.308, ~3/2 = 698.068

: [[error map]]: {{val| +3.656 -0.036 -8.647 }}

* [[CWE]]: ~128/125 = 92.308{{c}}, ~3/2 = 698.068{{c}}

: error map: {{val| 0.000 -3.887 -17.083 }}

[[Badness]] (Sintel): 52.~~583~~

[[Badness]] (Sintel): 52.6

@@ Line 4: / Line 4: @@
 == Aluminium ==
-: ''See also: [[Ragismic microtemperaments #Aluminium]]''
+: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].''
-Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.
+Aluminium tempers out {{monzo| 92 -39 -13 }} in the 5-limit, and sets [[135/128]] to 1/13 of an [[octave]].
 [[Subgroup]]: 2.3.5
@@ Line 12: / Line 12: @@
 [[Comma list]]: {{monzo| 92 -39 -13 }}
-[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
+{{Mapping|legend=1| 13 0 92 | 0 1 -3 }}
+: mapping generators: ~135/128, ~3
-: Mapping generators: ~135/128, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~135/128 = 92.3068{{c}}, ~3/2 = 701.9791{{c}}
-[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
+: [[error map]]: {{val| -0.012 +0.012 +0.009 }}
+* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 701.9897{{c}}
+: error map: {{val| 0.000 +0.032 +0.033 }}
 {{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
-[[Badness]] (Sintel): 2.893
+[[Badness]] (Sintel): 2.89
 == Tridecatonic ==
-: ''See also: [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[26th-octave temperaments]]''
+: ''For extensions, see [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[Varunismic temperaments #Bosonic]].''
-''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.
+Tridecatonic tempers out the [[devil's tridecalimma]], the comma which associates the [[10/9]] minor tone with [[13edo|2\13]].
 [[Subgroup]]: 2.3.5
-[[Comma list]]: 2541865828329/2500000000000
+[[Comma list]]: {{monzo| -11 26 -13 }}
-[[Mapping]]: [{{Val| 13 21 31 }}, {{Val| 0 -1 -2 }}]
+{{Mapping|legend=1| 13 0 -11 | 0 1 2 }}
+: mapping generators: ~531441/500000, ~3
-: Mapping generators: ~531441/500000, ~3
+[[Optimal tuning]]s:
+* [[WE]]: ~531441/500000 = 92.3166{{c}}, ~3/2 = 701.2353{{c}}
-[[Optimal tuning]] ([[CTE]]): ~531441/500000 = 1\13, ~3/2 = 701.2353
+: [[error map]]: {{val| +0.116 -0.688 +0.737 }}
+* [[CWE]]: ~531441/500000 = 92.3077{{c}}, ~3/2 = 701.2353{{c}}
+: error map: {{val| 0.000 -0.797 +0.617 }}
 {{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }}
-[[Badness]] (Sintel): 10.263
+[[Badness]] (Sintel): 10.3
 == Triskaidekic ==
@@ Line 49: / Line 55: @@
 {{Mapping|legend=1| 13 0 30 | 0 1 0 }}
 : Mapping generators: ~128/125, ~3
 [[Optimal tuning]]s:
-* [[WE]]: ~128/125 = 92.589, ~3/2 = 698.263
+* [[WE]]: ~128/125 = 92.589{{c}}, ~3/2 = 698.263{{c}}
-* [[CWE]]: ~128/125 = 92.308, ~3/2 = 698.068
+: [[error map]]: {{val| +3.656 -0.036 -8.647 }}
+* [[CWE]]: ~128/125 = 92.308{{c}}, ~3/2 = 698.068{{c}}
+: error map: {{val| 0.000 -3.887 -17.083 }}
 {{Optimal ET sequence|legend=1| 13, 26, 91cc }}
-[[Badness]] (Sintel): 52.583
+[[Badness]] (Sintel): 52.6
 {{Navbox fractional-octave}}