Maximal evenness: Difference between revisions
Wikispaces>hstraub **Imported revision 480570568 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 481718428 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2014-01-09 18:12:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>481718428</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></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. | <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"]] is the floor function. | ||
The maximally even scale will be one: | The maximally even scale will be one: | ||
| Line 67: | Line 67: | ||
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.<br /> | <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> is the floor function.<br /> | ||
<br /> | <br /> | ||
The maximally even scale will be one:<br /> | The maximally even scale will be one:<br /> | ||