13th-octave temperaments: Difference between revisions
Jump to navigation
Jump to search
No edit summary Tags: Mobile edit Mobile web edit |
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". |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 2: | Line 2: | ||
{{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 | ||
| Line 12: | Line 16: | ||
[[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]]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}} | ||
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