Macrotonal EDO
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Andrew_Heathwaite and made on 2009-11-03 01:27:49 UTC.
- The original revision id was 99714691.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[edo]]: "equal division of the octave". A tuning system which derives its pitches by equally dividing an octave. [[macrotonal]]: literally, "larger than a half step". A "macrotonal" scale would be one where all step sizes exceed 100 cents. Paradoxically, "macrotonal" is a subset of "microtonal," according to the loose definition of microtonal meaning "tuning systems other than 12-tone equal temperament". "Macrotonal Edo," then, refers to a tuning system which cuts the octave into fewer than 12 equal parts. * 1edo is a sparse scale indeed, since it consists of one pitch and its octave transpositions. * 2, 3, and 4edo are all represented in 12edo (since 12 is divisible by 2, 3, and 4). In 12edo, they function as dyads and chords, not as scales (tritone, augmented triad, and fully diminished seventh chord, respectively). * [[5edo]] is the first truly xenharmonic edo. Its third degree, at 720 cents (and its inversion at 480 cents) seems to represent an upper limit for how wide you can make a "fifth" and still call it a "fifth". The single step (240 cents) confounds an interval naming system based on 7-tone scales, and may sound like a second or third depending on timbre and context. Some Indonesian slendro scales come close to 5edo, as do some scales found in African tribal musics. * [[6edo]] is equivalent to 12edo's "whole tone scale," and does sound distinctly different from 12edo treated in the traditional way. * [[7edo]] distinguishes a second, a third, a fourth, a fifth, a sixth, and a seventh, and thus passes as a complete heptatonic scale. It only distinguishes one neutral version of each interval class (as opposed to the major and minor seconds, thirds, sixths and sevenths of some more complex systems). Its fourth degree, at 686 cents (and its inversion at 514 cents), seems to represent a lower limit for how narrow you can make a "fifth" and still call it a "fifth". 7edo contains a pentatonic moment-of-symmetry scale: 2L+3s (1 2 1 2 1) (the same MOS class as traditional chain-of-fifths pentatonics such as is possible in Pythagorean, meantone, 12edo, and some higher edos). * [[8edo]] has no perfect fourths or fifths and sounds very xenharmonic. It is playable in [[24edo]] as a subset. It contains one MOS scale: 3L+2s (2 1 2 1 2). It can be treated as two fully diminished seventh chords separated by a 150-cent [[neutral second|neutral tone]]. * [[9edo]] contains a pentatonic MOS scale -- 2L+3s (1 3 1 3 1) -- with a heptatonic extension -- 2L+5s (1 1 2 1 1 2 1). Indonesian pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way. 9edo contains three augmented triads separated by a 133 cent neutral tone. * [[10edo]] contains a very close approximation of the 13th harmonic, and an MOS scale 3L+4s (1 2 1 2 1 2 1) (see also [[17edo neutral scale]]). It can be treated as two 5edos separated by a 120 cent semitone. * [[11edo]] is a xenharmonic fifths-less tuning with a decent approximation of the 11th harmonic and the septimal supermajor third. Due to its primeness, 11edo offers several MOS scales: 4L+3s (2 1 2 1 2 1 2), 3L+2s (3 1 3 1 3), 3L+5s (1 2 1 1 2 1 2 1), 2L+3s (1 3 1 3 1), 2L+5s (1 3 1 1 1 3 1) and others. 11edo is also a subset of that odd duck [[22edo]]. I ([[user:Andrew_Heathwaite|1257229669]]) offer macrotonal edos as one possible "starting point" for exploring non-12 tunings. Each offers its own set of unique constraints. Some seem to offer less variety than 12edo does (so are, in a way, "simpler" -- eg. 5edo, 7edo), and some seem to offer more variety (eg. 11edo). As a set, they offer abundant variety and could keep a student happily confused for a good while, perhaps a lifetime. [[file:macrotonal_edos.pdf]]
Original HTML content:
<html><head><title>macrotonal edos</title></head><body><a class="wiki_link" href="/edo">edo</a>: "equal division of the octave". A tuning system which derives its pitches by equally dividing an octave.<br />
<br />
<a class="wiki_link" href="/macrotonal">macrotonal</a>: literally, "larger than a half step". A "macrotonal" scale would be one where all step sizes exceed 100 cents. Paradoxically, "macrotonal" is a subset of "microtonal," according to the loose definition of microtonal meaning "tuning systems other than 12-tone equal temperament".<br />
<br />
"Macrotonal Edo," then, refers to a tuning system which cuts the octave into fewer than 12 equal parts.<br />
<br />
<ul><li>1edo is a sparse scale indeed, since it consists of one pitch and its octave transpositions.</li><li>2, 3, and 4edo are all represented in 12edo (since 12 is divisible by 2, 3, and 4). In 12edo, they function as dyads and chords, not as scales (tritone, augmented triad, and fully diminished seventh chord, respectively).</li><li><a class="wiki_link" href="/5edo">5edo</a> is the first truly xenharmonic edo. Its third degree, at 720 cents (and its inversion at 480 cents) seems to represent an upper limit for how wide you can make a "fifth" and still call it a "fifth". The single step (240 cents) confounds an interval naming system based on 7-tone scales, and may sound like a second or third depending on timbre and context. Some Indonesian slendro scales come close to 5edo, as do some scales found in African tribal musics.</li><li><a class="wiki_link" href="/6edo">6edo</a> is equivalent to 12edo's "whole tone scale," and does sound distinctly different from 12edo treated in the traditional way.</li><li><a class="wiki_link" href="/7edo">7edo</a> distinguishes a second, a third, a fourth, a fifth, a sixth, and a seventh, and thus passes as a complete heptatonic scale. It only distinguishes one neutral version of each interval class (as opposed to the major and minor seconds, thirds, sixths and sevenths of some more complex systems). Its fourth degree, at 686 cents (and its inversion at 514 cents), seems to represent a lower limit for how narrow you can make a "fifth" and still call it a "fifth". 7edo contains a pentatonic moment-of-symmetry scale: 2L+3s (1 2 1 2 1) (the same MOS class as traditional chain-of-fifths pentatonics such as is possible in Pythagorean, meantone, 12edo, and some higher edos).</li><li><a class="wiki_link" href="/8edo">8edo</a> has no perfect fourths or fifths and sounds very xenharmonic. It is playable in <a class="wiki_link" href="/24edo">24edo</a> as a subset. It contains one MOS scale: 3L+2s (2 1 2 1 2). It can be treated as two fully diminished seventh chords separated by a 150-cent <a class="wiki_link" href="/neutral%20second">neutral tone</a>.</li><li><a class="wiki_link" href="/9edo">9edo</a> contains a pentatonic MOS scale -- 2L+3s (1 3 1 3 1) -- with a heptatonic extension -- 2L+5s (1 1 2 1 1 2 1). Indonesian pelog scales sometimes use five-tone subsets of a seven-tone superset in a similar way. 9edo contains three augmented triads separated by a 133 cent neutral tone.</li><li><a class="wiki_link" href="/10edo">10edo</a> contains a very close approximation of the 13th harmonic, and an MOS scale 3L+4s (1 2 1 2 1 2 1) (see also <a class="wiki_link" href="/17edo%20neutral%20scale">17edo neutral scale</a>). It can be treated as two 5edos separated by a 120 cent semitone.</li><li><a class="wiki_link" href="/11edo">11edo</a> is a xenharmonic fifths-less tuning with a decent approximation of the 11th harmonic and the septimal supermajor third. Due to its primeness, 11edo offers several MOS scales: 4L+3s (2 1 2 1 2 1 2), 3L+2s (3 1 3 1 3), 3L+5s (1 2 1 1 2 1 2 1), 2L+3s (1 3 1 3 1), 2L+5s (1 3 1 1 1 3 1) and others. 11edo is also a subset of that odd duck <a class="wiki_link" href="/22edo">22edo</a>.</li></ul><br />
I (<!-- ws:start:WikiTextUserlinkRule:00:[[user:Andrew_Heathwaite|1257229669]] --><span class="membersnap">- <a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;"><img src="http://www.wikispaces.com/user/pic/Andrew_Heathwaite-lg.jpg" width="16" height="16" alt="Andrew_Heathwaite" class="userPicture" /></a> <a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;">Andrew_Heathwaite</a> <small>Nov 3, 2009</small></span><!-- ws:end:WikiTextUserlinkRule:00 -->) offer macrotonal edos as one possible "starting point" for exploring non-12 tunings. Each offers its own set of unique constraints. Some seem to offer less variety than 12edo does (so are, in a way, "simpler" -- eg. 5edo, 7edo), and some seem to offer more variety (eg. 11edo). As a set, they offer abundant variety and could keep a student happily confused for a good while, perhaps a lifetime.<br />
<br />
<!-- ws:start:WikiTextFileRule:20:<img src="http://www.wikispaces.com/site/embedthumbnail/file/macrotonal_edos.pdf?h=52&w=320" class="WikiFile" id="wikitext@@file@@macrotonal_edos.pdf" title="File: macrotonal_edos.pdf" width="320" height="52" /> --><div class="objectEmbed"><a href="/file/view/macrotonal_edos.pdf/99649243/macrotonal_edos.pdf" onclick="ws.common.trackFileLink('/file/view/macrotonal_edos.pdf/99649243/macrotonal_edos.pdf');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="macrotonal_edos.pdf" /></a><div><a href="/file/view/macrotonal_edos.pdf/99649243/macrotonal_edos.pdf" onclick="ws.common.trackFileLink('/file/view/macrotonal_edos.pdf/99649243/macrotonal_edos.pdf');" class="filename" title="macrotonal_edos.pdf">macrotonal_edos.pdf</a><br /><ul><li><a href="/file/detail/macrotonal_edos.pdf">Details</a></li><li><a href="/file/view/macrotonal_edos.pdf/99649243/macrotonal_edos.pdf">Download</a></li><li style="color: #666">268 KB</li></ul></div></div><!-- ws:end:WikiTextFileRule:20 --></body></html>