Maximal evenness
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author hstraub and made on 2014-08-22 18:04:05 UTC.
- The original revision id was 519349914.
- 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:
Within every [[edo]] one can specify a "maximally even" (ME) scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo. Mathematically, ME scales of n notes in m edo are any [[mode]] of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the [[https://en.wikipedia.org/wiki/Floor_and_ceiling_functions|"floor"]] function rounds down to the nearest integer. The maximally even scale will be one: a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo). b. whose steps are distributed as evenly as possible. (a) and (b) above imply that the ME scale will be a [[MOSScales|moment of symmetry scale]]. The probably most popular heptatonic ME scale is the major scale of [[12edo]]: <span style="font-family: monospace; "> 2 2 1 2 2 2 1</span>, but also every [[http://en.wikipedia.org/wiki/Diatonic_scale|diatonic scale]] of 12edo is maximally even. Some more detailed examples follow. For instance, here are all the ME scales available in [[31edo]]: <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 2 .. 15 16</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 10 10 11</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 8 8 8 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 6 6 6 6 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 5 5 5 5 5 6</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 5 4 5 4 5 4 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 4 4 4 4 4 4 4 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 4 3 4 3 4 3 4 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 3 3 3 3 3 3 3 3 3 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 3 3 3 3 2 3 3 3 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 3 3 2 3 2 3 3 2 3 2 3 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 13 . 3 2 3 2 2 3 2 3 2 2 3 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 14 . 2 2 2 2 3 2 2 2 2 3 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 15 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 16 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 17 . 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 18 . 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 19 . 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 20 . 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 21 . 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 22 . 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 23 . 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 24 . 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 25 . 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 26 . 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 27 . 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 28 . 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 29 . 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 30 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2</span> And here are all the ME scales available in [[13edo]]: <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;">2 .. 6 7</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 4 4 5</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 3 3 3 4</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 2 3 2 3 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 2 2 2 2 2 3</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 2 2 2 2 2 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 2 2 1 2 2 1 2 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 2 1 2 1 2 1 2 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 2 1 1 2 1 1 2 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 1 1 1 1 2 1 1 1 1 1</span> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 1 1 1 1 1 1 1 1 1 1 1 2 </span> The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th). The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo. Maximally even sets tend to be familiar and musically relevant scale collections. Examples: * The maximally even heptatonic set of [[19edo]] is, like the one in 12edo, a diatonic scale. * The maximally even heptatonic sets of [[17edo]] and [[24edo]], in contrary, are Maqamic[7]. * The maximally even heptatonic set of [[22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara. * The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale. Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel Mandelbaum]]'s 1961 thesis [[http://www.anaphoria.com/mandelbaum.html|Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.
Original HTML content:
<html><head><title>Maximal evenness</title></head><body>Within every <a class="wiki_link" href="/edo">edo</a> one can specify a "maximally even" (ME) scale for every smaller edo. The maximally even scale is the closest the parent edo can get to representing the smaller edo. Mathematically, ME scales of n notes in m edo are any <a class="wiki_link" href="/mode">mode</a> of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow">"floor"</a> function rounds down to the nearest integer.<br /> <br /> The maximally even scale will be one:<br /> a. which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent edo).<br /> b. whose steps are distributed as evenly as possible.<br /> (a) and (b) above imply that the ME scale will be a <a class="wiki_link" href="/MOSScales">moment of symmetry scale</a>. <br /> <br /> The probably most popular heptatonic ME scale is the major scale of <a class="wiki_link" href="/12edo">12edo</a>: <span style="font-family: monospace; "> 2 2 1 2 2 2 1</span>, but also every <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Diatonic_scale" rel="nofollow">diatonic scale</a> of 12edo is maximally even. Some more detailed examples follow.<br /> <br /> For instance, here are all the ME scales available in <a class="wiki_link" href="/31edo">31edo</a>:<br /> <br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 2 .. 15 16</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 10 10 11</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 8 8 8 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 6 6 6 6 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 5 5 5 5 5 6</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 5 4 5 4 5 4 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 4 4 4 4 4 4 4 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 4 3 4 3 4 3 4 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 3 3 3 3 3 3 3 3 3 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 3 3 3 3 2 3 3 3 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 3 3 2 3 2 3 3 2 3 2 3 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 13 . 3 2 3 2 2 3 2 3 2 2 3 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 14 . 2 2 2 2 3 2 2 2 2 3 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 15 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 16 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 17 . 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 18 . 2 1 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 19 . 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 20 . 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 21 . 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 22 . 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 23 . 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 24 . 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 25 . 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 26 . 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 27 . 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 28 . 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 29 . 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 30 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2</span><br /> <br /> And here are all the ME scales available in <a class="wiki_link" href="/13edo">13edo</a>:<br /> <br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;">2 .. 6 7</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 3 .. 4 4 5</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 4 .. 3 3 3 4</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 5 .. 2 3 2 3 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 6 .. 2 2 2 2 2 3</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 7 .. 2 2 2 2 2 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 8 .. 2 2 1 2 2 1 2 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 9 .. 2 1 2 1 2 1 2 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 10 . 2 1 1 2 1 1 2 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 11 . 2 1 1 1 1 2 1 1 1 1 1</span><br /> <span style="font-family: Courier,monospace; font-size: 12px; line-height: normal;"> 12 . 1 1 1 1 1 1 1 1 1 1 1 2 </span><br /> <br /> The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31th of an octave instead of one 13th).<br /> <br /> The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo.<br /> <br /> Maximally even sets tend to be familiar and musically relevant scale collections. Examples:<br /> <br /> <ul><li>The maximally even heptatonic set of <a class="wiki_link" href="/19edo">19edo</a> is, like the one in 12edo, a diatonic scale.</li><li>The maximally even heptatonic sets of <a class="wiki_link" href="/17edo">17edo</a> and <a class="wiki_link" href="/24edo">24edo</a>, in contrary, are Maqamic[7].</li><li>The maximally even heptatonic set of <a class="wiki_link" href="/22edo">22edo</a> is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</li><li>The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.</li></ul><br /> Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in <a class="wiki_link" href="/Joel%20Mandelbaum">Joel Mandelbaum</a>'s 1961 thesis <a class="wiki_link_ext" href="http://www.anaphoria.com/mandelbaum.html" rel="nofollow">Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament</a>. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.</body></html>