13th-octave temperaments: Difference between revisions

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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:
+== Aluminium ==
+: ''See also: [[Ragismic microtemperaments #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.
+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.
+[[Subgroup]]: 2.3.5
+[[Comma list]]: {{monzo| 92 -39 -13 }}
+[[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
+: Mapping generators: ~135/128, ~3
+[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
+{{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
+[[Badness]] (Sintel): 2.893
 == Tridecatonic ==
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 ''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.
-Subgroup: 2.3.5
+[[Subgroup]]: 2.3.5
-[[Comma]]: 2541865828329/2500000000000
+[[Comma list]]: 2541865828329/2500000000000
 [[Mapping]]: [{{Val| 13 21 31 }}, {{Val| 0 -1 -2 }}]
-[[POTE generator]]: ~3/2 = 701.0837
+: Mapping generators: ~531441/500000, ~3
+[[Optimal tuning]] ([[CTE]]): ~531441/500000 = 1\13, ~3/2 = 701.2353
 {{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }}
-[[Badness]]: 0.437494
+[[Badness]] (Sintel): 10.263
+== Triskaidekic ==
+: ''For extensions, see [[Orwellismic temperaments #Triskaidekic]].''
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 1220703125/1073741824
+{{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
+* [[CWE]]: ~128/125 = 92.308, ~3/2 = 698.068
+{{Optimal ET sequence|legend=1| 13, 26, 91cc }}
+[[Badness]] (Sintel): 52.583
+{{Navbox fractional-octave}}