13th-octave temperaments: Difference between revisions

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A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.

~~Discussed elsewhere are:~~

Furthermore, one step of 13edo is very close to [[135/128]], a notable 5-limit interval, in fact it is a semiconvergent to the said representation. Hence, considered below is aluminium.

* ''[[~~Aluminium~~]]'', → [[~~Ragismic~~ microtemperaments#~~Aluminium~~|~~Ragismic microtemperaments~~]], ~~which is notable as~~ [[~~135/128]] is a [[semiconvergent~~]] ~~to 1\13 as well as a 5-limit interval~~.

Temperaments discussed elsewhere are [[octagar temperaments #Tridecatonic|tridecatonic]], [[ragismic microtemperaments #Trideci|trideci]], and [[orwellismic temperaments #Triskaidekic|triskaidekic]].

== ~~Tridecatonic~~ ==

== Aluminium ==

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

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

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

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

[[Subgroup]]: 2.3.5

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

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

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

: mapping generators: ~135/128, ~3

[[~~POTE generator~~]]: ~3/2 = 701.~~0837~~

[[Optimal tuning]]s:

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

: [[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| 26, 39, 65, ~~286~~, ~~351~~, ~~416~~, ~~481~~, ~~546~~ }}

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

[[Badness]]: 0.~~437494~~

[[Badness]] (Sintel): 2.89

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-{{Fractional-octave navigation|13}}
+{{Technical data page}}
+{{Infobox fractional-octave|13}}
+A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.
-Discussed elsewhere are:
+Furthermore, one step of 13edo is very close to [[135/128]], a notable 5-limit interval, in fact it is a semiconvergent to the said representation. Hence, considered below is aluminium.
-* ''[[Aluminium]]'', → [[Ragismic microtemperaments#Aluminium|Ragismic microtemperaments]], which is notable as [[135/128]] is a [[semiconvergent]] to 1\13 as well as a 5-limit interval.
+Temperaments discussed elsewhere are [[octagar temperaments #Tridecatonic|tridecatonic]], [[ragismic microtemperaments #Trideci|trideci]], and [[orwellismic temperaments #Triskaidekic|triskaidekic]].
-== Tridecatonic ==
+== Aluminium ==
-: ''See also: [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[26th-octave temperaments]]''
+: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].''
-''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.
+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
+[[Subgroup]]: 2.3.5
-[[Comma]]: 2541865828329/2500000000000
+[[Comma list]]: {{monzo| 92 -39 -13 }}
-[[Mapping]]: [{{Val| 13 21 31 }}, {{Val| 0 -1 -2 }}]
+{{Mapping|legend=1| 13 0 92 | 0 1 -3 }}
+: mapping generators: ~135/128, ~3
-[[POTE generator]]: ~3/2 = 701.0837
+[[Optimal tuning]]s:
+* [[WE]]: ~135/128 = 92.3068{{c}}, ~3/2 = 701.9791{{c}}
+: [[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| 26, 39, 65, 286, 351, 416, 481, 546 }}
+{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
-[[Badness]]: 0.437494
+[[Badness]] (Sintel): 2.89
 {{Navbox fractional-octave}}