Maximal evenness: Difference between revisions
Wikispaces>genewardsmith **Imported revision 481718428 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 481797458 - Original comment: included another popular example** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2014-01-10 03:53:50 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>481797458</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>included another popular example</tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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 [[https://en.wikipedia.org/wiki/Floor_and_ceiling_functions|"floor"]] | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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: | 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). | 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. | 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]]. | (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]]: | For instance, here are all the ME scales available in [[31edo]]: | ||
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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.</pre></div> | 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.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Maximal evenness</title></head><body>Within every <a class="wiki_link" href="/edo">edo</a> one can specify a &quot;maximally even&quot; (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 <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow">&quot;floor&quot;</a> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Maximal evenness</title></head><body>Within every <a class="wiki_link" href="/edo">edo</a> one can specify a &quot;maximally even&quot; (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 <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow">&quot;floor&quot;</a> function rounds down to the nearest integer.<br /> | ||
<br /> | <br /> | ||
The maximally even scale will be one:<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 /> | 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 /> | 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 /> | (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 /> | <br /> | ||
For instance, here are all the ME scales available in <a class="wiki_link" href="/31edo">31edo</a>:<br /> | For instance, here are all the ME scales available in <a class="wiki_link" href="/31edo">31edo</a>:<br /> | ||