11edo: Difference between revisions
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== Theory == | == Theory == | ||
{{Harmonics in equal|11|intervals=odd}} | {{Harmonics in equal|11|intervals=odd}} | ||
Compared to 12edo, the intervals of 11edo are stretched: | Compared to 12edo, the intervals of 11edo are stretched: | ||
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* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4. | * The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4. | ||
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3. | * The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3. | ||
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation. | |||
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords. | 11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords. | ||
11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo. | 11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo. | ||
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]". | |||
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]] | [[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]] | ||
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! Solfege | ! Solfege | ||
! Approximate Ratios* | ! Approximate Ratios* | ||
! colspan="2" | [[Ups and | ! colspan="2" | [[Ups and downs notation|Up/down notation]] <br> with major wider <br> than minor | ||
! colspan="2" | Up/down notation <br> with major narrower <br> than minor | ! colspan="2" | Up/down notation <br> with major narrower <br> than minor | ||
! [[Smitonic]]<br>(3rd-gen)<br>notation | ! [[Smitonic]]<br>(3rd-gen)<br>notation | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
=== Uniform maps === | === Uniform maps === | ||
{{Uniform map| | {{Uniform map|edo=11}} | ||
=== Commas === | === Commas === | ||
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{{Main|List of 11edo MOS scales}} | {{Main|List of 11edo MOS scales}} | ||
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included. | Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included. | ||
== Instruments == | == Instruments == | ||
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== See also == | == See also == | ||
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. | * [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. | ||
== Notes == | == Notes == | ||
<references group=note/> | <references group=note/> | ||
[[Category:Listen]] | [[Category:Listen]] | ||
{{Todo|add rank 2 temperaments table}} | |||