23edo: Difference between revisions
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'''23 equal divisions of the octave''' ('''23edo''') is a musical system which divides the [[octave]] into 23 [[equal]] parts of approximately 52.2 [[cent]]s. | |||
== Theory == | == Theory == | ||
{{Harmonics in equal|23}} | {{Harmonics in equal|23}} | ||
23edo has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit 46edo, and may be regarded as a basis for analyzing the harmony of 23edo so far, as approximations to just intervals goes. 23edo is the 9th [[prime edo]], following [[19edo]] and coming before [[29edo]]. | |||
23edo was proposed by ethnomusicologist [[Wikipedia: 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. | |||
23edo 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 23edo 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 | As with [[9edo]], [[16edo]], and [[25edo]], one way to treat 23edo is as a tuning of the [[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|antidiatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23edo), which extends to 9 notes [[7L 2s|superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23edo using the [[Armodue]] system, but just like notating 17edo with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23edo, the "Armodue 6th" is sharper than it is in 16edo, just like the diatonic 5th in 17edo is sharper than in 12edo. In other words, 2b is lower in pitch than 1#, just like how in 17edo Eb is lower than D#. | ||
However, one can also map 3/2 to 14 degrees of | However, one can also map 3/2 to 14 degrees of 23edo without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo]] and [[18edo]] and produces [[mos scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[[Wikipedia: Paul Rapoport %28music critic%29|Rapoport]]/Wilson 13edo "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23edo a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23edo as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note mos scale. | ||
== Selected just intervals == | == Selected just intervals == | ||
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== Notation == | == Notation == | ||
23edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 | 23edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 is not E. Chord names are different because C - E - G is not P1 - M3 - P5. | ||
The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 23edo "on the fly". | The second approach preserves the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 23edo "on the fly". | ||
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== Commas == | == Commas == | ||
23edo [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 23 36 53 65 80 85 }}.) Also note the discussion above, where there are some commas mentioned that are not in the standard comma list (e.g., 28/27). | |||
{| class="commatable wikitable center-all left-3 right-4 left-6" | {| class="commatable wikitable center-all left-3 right-4 left-6" | ||
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|} | |} | ||
== | == Scales == | ||
Important | Important [[mos]]ses include: | ||
* mavila 2L5s 4334333 (13\23, 1\1) | * mavila 2L5s 4334333 (13\23, 1\1) | ||
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* [[semiquartal]] 5L4s 332323232 (5\23, 1\1) | * [[semiquartal]] 5L4s 332323232 (5\23, 1\1) | ||
The chart below shows some of the | The chart below shows some of the mos modes of [[mavila]] available in 23edo, mainly Pentatonic (5-note), antidiatonic (7-note), 9- and 16-note mosses. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note mosses: | ||
[[File:23edoMavilaMOS.jpg|alt=23edoMavilaMOS.jpg|23edoMavilaMOS.jpg]] | [[File:23edoMavilaMOS.jpg|alt=23edoMavilaMOS.jpg|23edoMavilaMOS.jpg]] | ||
=== 23 tone | === 23-tone mos scales === | ||
{| class="wikitable" | {| class="wikitable" | ||
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|} | |} | ||
===Kosmorsky's Sephiroth modes=== | === Kosmorsky's Sephiroth modes === | ||
Kosmorsky has argued that the most significant modes of 23 edo are those of the 2 2 2 3 2 2 3 2 2 3 scale ([[3L 7s|3L 7s fair mosh]]); This is derived from extending the ~1/3 comma tempered 13th Harmonic, two of which add up to the 21st harmonic and three add up to the 17th harmonic almost perfectly. Interestingly, the chord 8:13:21:34 is a fragment of the fibonacci sequence. | |||
Notated in ascending (standard) form. I have named these 10 modes according to the Sephiroth as follows: | Notated in ascending (standard) form. I have named these 10 modes according to the Sephiroth as follows: | ||
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3 2 2 2 3 2 2 3 2 2 - Hod | 3 2 2 2 3 2 2 3 2 2 - Hod | ||
==Books== | == Books == | ||
[[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG]] | [[File:Libro_Icositrifónico.PNG|alt=Libro_Icositrifónico.PNG|302x365px|Libro_Icositrifónico.PNG]] | ||
==Instruments== | == Instruments == | ||
[[File:Icositriphonic_Guitar.PNG|alt=Icositriphonic_Guitar.PNG|601x305px|Icositriphonic_Guitar.PNG]] | [[File:Icositriphonic_Guitar.PNG|alt=Icositriphonic_Guitar.PNG|601x305px|Icositriphonic_Guitar.PNG]] | ||