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]].'' | : ''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]]. | 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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[[Badness]] (Sintel): 2.89 | [[Badness]] (Sintel): 2.89 | ||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||
Latest revision as of 21:37, 6 August 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.
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 tridecatonic, trideci, and triskaidekic.
Aluminium
- For extensions, see Ragismic microtemperaments #Aluminium.
Aluminium tempers out [92 -39 -13⟩ in the 5-limit, and sets 135/128 to 1/13 of an octave.
Subgroup: 2.3.5
Comma list: [92 -39 -13⟩
Mapping: [⟨13 0 92], ⟨0 1 -3]]
- mapping generators: ~135/128, ~3
- WE: ~135/128 = 92.3068 ¢, ~3/2 = 701.9791 ¢
- error map: ⟨-0.012 +0.012 +0.009]
- CWE: ~135/128 = 92.3077 ¢, ~3/2 = 701.9897 ¢
- error map: ⟨0.000 +0.032 +0.033]
Optimal ET sequence: 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc
Badness (Sintel): 2.89