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.
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.


== 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]]''
 
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 ==
: ''See also: [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[26th-octave temperaments]]''
 
''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
 
[[Comma list]]: 2541865828329/2500000000000
 
[[Mapping]]: [{{Val| 13 21 31 }}, {{Val| 0 -1 -2 }}]
 
: 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]] (Sintel): 10.263


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}

Latest revision as of 03:52, 21 July 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

← 12th-octave • 13th-octave • 14th-octave →

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 aluminium, tridecatonic, trideci, and triskaidekic.

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