13th-octave temperaments: Difference between revisions

Line 3:

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.

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

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.

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

== Aluminium ==

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

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

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

: mapping generators: ~135/128, ~3

[[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| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}

[[Badness]] (Sintel): 2.89

@@ Line 3: / Line 3: @@
 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.
-Temperaments discussed elsewhere are [[ragismic microtemperaments #Aluminium|aluminium]], [[octagar temperaments #Tridecatonic|tridecatonic]], [[ragismic microtemperaments #Trideci|trideci]], and [[orwellismic temperaments #Triskaidekic|triskaidekic]].
+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.
+Temperaments discussed elsewhere are [[octagar temperaments #Tridecatonic|tridecatonic]], [[ragismic microtemperaments #Trideci|trideci]], and [[orwellismic temperaments #Triskaidekic|triskaidekic]].
+== Aluminium ==
+: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].''
+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
+[[Comma list]]: {{monzo| 92 -39 -13 }}
+{{Mapping|legend=1| 13 0 92 | 0 1 -3 }}
+: mapping generators: ~135/128, ~3
+[[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| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
+[[Badness]] (Sintel): 2.89
 {{Navbox fractional-octave}}