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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. | | 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. |
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| == Aluminium ==
| | Temperaments discussed elsewhere are [[ragismic microtemperaments #Aluminium|aluminium]], [[octagar temperaments #Tridecatonic|tridecatonic]], [[ragismic microtemperaments #Trideci|trideci]], and [[orwellismic temperaments #Triskaidekic|triskaidekic]]. |
| : ''See also: [[Ragismic microtemperaments #Aluminium]]''
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| 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.
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| [[Subgroup]]: 2.3.5
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| [[Comma list]]: {{monzo| 92 -39 -13 }}
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| [[Mapping]]: {{mapping| 13 0 92 | 0 1 -3 }}
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| : Mapping generators: ~135/128, ~3
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| [[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897
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| {{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}
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| [[Badness]] (Sintel): 2.893
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| == Tridecatonic ==
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| : ''See also: [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[26th-octave temperaments]]''
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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.
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| [[Subgroup]]: 2.3.5
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| [[Comma list]]: 2541865828329/2500000000000
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| [[Mapping]]: [{{Val| 13 21 31 }}, {{Val| 0 -1 -2 }}]
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| : Mapping generators: ~531441/500000, ~3
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| [[Optimal tuning]] ([[CTE]]): ~531441/500000 = 1\13, ~3/2 = 701.2353
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| {{Optimal ET sequence|legend=1| 26, 39, 65, 286, 351, 416, 481, 546 }}
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| [[Badness]] (Sintel): 10.263
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| == Triskaidekic ==
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| : ''For extensions, see [[Orwellismic temperaments #Triskaidekic]].''
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| [[Subgroup]]: 2.3.5
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| [[Comma list]]: 1220703125/1073741824
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| {{Mapping|legend=1| 13 0 30 | 0 1 0 }}
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| : Mapping generators: ~128/125, ~3
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| [[Optimal tuning]]s:
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| * [[WE]]: ~128/125 = 92.589, ~3/2 = 698.263
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| * [[CWE]]: ~128/125 = 92.308, ~3/2 = 698.068
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| {{Optimal ET sequence|legend=1| 13, 26, 91cc }}
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| [[Badness]] (Sintel): 52.583
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| {{Navbox fractional-octave}} | | {{Navbox fractional-octave}} |