Macrotonal EDO: Difference between revisions

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2010-01-30 13:18:42 UTC</tt>.<br>

: The original revision id was <tt>116805319</tt>.<br>

* 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).

&lt;ul&gt;&lt;li&gt;1edo is a sparse scale indeed, since it consists of one pitch and its octave transpositions.&lt;/li&gt;&lt;li&gt;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).&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt; is the first 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 &amp;quot;fifth&amp;quot; and still call it a &amp;quot;fifth&amp;quot;. 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. Its fourth degree, at 960 cents, is 9 cents away from the seventh harmonic, 7:4, at 969 cents. Some Indonesian slendro scales come close to 5edo, as do some scales found in African tribal musics.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/6edo"&gt;6edo&lt;/a&gt; is equivalent to 12edo's &amp;quot;whole tone scale,&amp;quot; and does sound distinctly different from 12edo treated in the traditional way.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; 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 &amp;quot;fifth&amp;quot; and still call it a &amp;quot;fifth&amp;quot;. 7edo contains a pentatonic moment-of-symmetry scale: &lt;a class="wiki_link" href="/2L%203s"&gt;2L 3s&lt;/a&gt; (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).&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/8edo"&gt;8edo&lt;/a&gt; has no perfect fourths or fifths and sounds very xenharmonic. It is playable in &lt;a class="wiki_link" href="/24edo"&gt;24edo&lt;/a&gt; as a subset. It contains one MOS scale: &lt;a class="wiki_link" href="/3L%202s"&gt;3L 2s&lt;/a&gt; (2 1 2 1 2). It can be treated as two fully diminished seventh chords separated by a 150-cent &lt;a class="wiki_link" href="/neutral%20second"&gt;neutral tone&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt; contains a pentatonic MOS scale -- &lt;a class="wiki_link" href="/2L%203s"&gt;2L 3s&lt;/a&gt; (1 3 1 3 1) -- with a heptatonic extension -- &lt;a class="wiki_link" href="/2L%205s"&gt;2L 5s&lt;/a&gt; (1 1 2 1 1 2 1, sometimes called &amp;quot;mavila&amp;quot; or &amp;quot;antidiatonic&amp;quot;). 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.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/10edo"&gt;10edo&lt;/a&gt; contains a very close approximation of the 13th harmonic, and an MOS scale &lt;a class="wiki_link" href="/3L%204s"&gt;3L 4s&lt;/a&gt; (1 2 1 2 1 2 1). It can be treated as two 5edos separated by a 120 cent [large] semitone. 10edo is the only macrotonal edo which contains another xenharmonic macrotonal edo: it contains 5edo, in much the same way that 12edo contains 6edo, the &amp;quot;whole tone scale&amp;quot;. It is arguably the only macrotonal edo that can be said to contain &lt;a class="wiki_link" href="/tetrachord"&gt;tetrachords&lt;/a&gt;: its &amp;quot;perfect fourth,&amp;quot; at four degrees, can be divided 1-1-2, 1-2-1, or 2-1-1.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt; is a xenharmonic fifths-less tuning with a decent approximation of the 11th harmonic and the septimal supermajor third (also: a sharp but passable harmonic seventh). Since 11 is a prime number, 11edo offers several MOS scales: &lt;a class="wiki_link" href="/4L%203s"&gt;4L 3s&lt;/a&gt; (2 1 2 1 2 1 2), &lt;a class="wiki_link" href="/3L%202s"&gt;3L 2s&lt;/a&gt; (3 1 3 1 3), &lt;a class="wiki_link" href="/3L%205s"&gt;3L 5s&lt;/a&gt; (1 2 1 1 2 1 2 1), &lt;a class="wiki_link" href="/2L%203s"&gt;2L 3s&lt;/a&gt; (1 3 1 3 1), &lt;a class="wiki_link" href="/2L%205s"&gt;2L 5s&lt;/a&gt; (1 3 1 1 1 3 1) and others. It many ways, it is more complex than 12edo, even with one fewer tone. 11edo is a subset of that odd duck &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;