13th-octave temperaments: Difference between revisions
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per discussion on discord: for the same logic as neon should belong in 10th-octave, aluminium similarly should be placed here. besides it adds to this page as opposed to just listing "discussed elsewhere". |
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{{Infobox fractional-octave|13}} | {{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. | 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. | ||
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 == | == Aluminium == | ||
: '' | : ''For extensions, see [[Ragismic microtemperaments #Aluminium]].'' | ||
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 | [[Subgroup]]: 2.3.5 | ||
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[[Comma list]]: {{monzo| 92 -39 -13 }} | [[Comma list]]: {{monzo| 92 -39 -13 }} | ||
{{Mapping|legend=1| 13 0 92 | 0 1 -3 }} | |||
: mapping generators: ~135/128, ~3 | |||
: | |||
[[Optimal tuning]] | [[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 }} | {{Optimal ET sequence|legend=1| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }} | ||
[[Badness]] (Sintel): 2. | [[Badness]] (Sintel): 2.89 | ||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||