11edo: Difference between revisions

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Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.

* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].

* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].

* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third."

* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".

* 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 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 [[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 is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

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.

[[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.]]

== Notation ==

@@ Line 14: / Line 14: @@
 Compared to 12edo, the intervals of 11edo are stretched:
-* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
+* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
-* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
+* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
-* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third."
+* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
-* 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.
-edo 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 [[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.
+edo 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.
 edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].
 edo 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.
-[[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.]]
 == Notation ==