23edo: Difference between revisions

Line 20:

23-EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.

23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them ([[5/3]], [[7/3]], [[11/3]], [[7/5]], [[11/7]], [[11/5]]) and combinations of them ([[15/8]], [[21/16]], [[33/32]], [[35/32]], [[77/64]], [[55/32]]) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See [[Harmony of 23edo~~|here~~]] for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].

23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them ([[5/3]], [[7/3]], [[11/3]], [[7/5]], [[11/5]], [[11/7]]) and combinations of them ([[15/8]], [[21/16]], [[33/32]], [[35/32]], [[55/32]], [[77/64]]) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See ''[[Harmony of 23edo]]'' for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].

As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.

@@ Line 20: / Line 20: @@
 -EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.
--EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them ([[5/3]], [[7/3]], [[11/3]], [[7/5]], [[11/7]], [[11/5]]) and combinations of them ([[15/8]], [[21/16]], [[33/32]], [[35/32]], [[77/64]], [[55/32]]) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See [[Harmony of 23edo|here]] for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].
+-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them ([[5/3]], [[7/3]], [[11/3]], [[7/5]], [[11/5]], [[11/7]]) and combinations of them ([[15/8]], [[21/16]], [[33/32]], [[35/32]], [[55/32]], [[77/64]]) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See ''[[Harmony of 23edo]]'' for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].
 As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.