Prime number: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-~~06 16~~:22:16 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-18 19:38:16 UTC</tt>.<br>

: The original revision id was <tt>~~240252397~~</tt>.<br>

: The original revision id was <tt>241852539</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>

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<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">=Prime numbers in [[EDO]]s=

Whether a number n is prime or not has important consequences for the properties of the corresponding n-[[edo|EDO]], especially for lower values of n.

A //prime number// is an integer (whole number) greater than one which is divisible only by itself and one. There an an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether a number n is prime or not has important consequences for the properties of the corresponding n-[[edo|EDO]], especially for lower values of n.

* If the octave is divided into a prime number of equal parts, there is **no fully symmetric chord**, such as the diminished seventh chord in [[12edo]].

* There is also (besides the full scale of all notes of the edo) **no absolutely uniform scale**, like the wholetone scale in 12edo.

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<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>prime numbers</title></head><body><h1 id="toc0"><a name="Prime numbers in EDOs"></a>Prime numbers in <a class="wiki_link" href="/EDO">EDO</a>s</h1>

<br />

Whether a number n is prime or not has important consequences for the properties of the corresponding n-<a class="wiki_link" href="/edo">EDO</a>, especially for lower values of n.<br />

A <em>prime number</em> is an integer (whole number) greater than one which is divisible only by itself and one. There an an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether a number n is prime or not has important consequences for the properties of the corresponding n-<a class="wiki_link" href="/edo">EDO</a>, especially for lower values of n.<br />

<ul><li>If the octave is divided into a prime number of equal parts, there is <strong>no fully symmetric chord</strong>, such as the diminished seventh chord in <a class="wiki_link" href="/12edo">12edo</a>.</li><li>There is also (besides the full scale of all notes of the edo) <strong>no absolutely uniform scale</strong>, like the wholetone scale in 12edo.</li><li>Nor is there a thing like <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modes_of_limited_transposition" rel="nofollow">modes of limited transpostion</a>, as used by the composer Olivier Messiaen.</li><li>N-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are <em>linear</em> temperaments.</li></ul><br />

For these or similar reasons, some musicians seem not to like prime EDOs (e.g. the makers of <a class="wiki_link_ext" href="http://www.armodue.com/risorse.htm" rel="nofollow">Armodue</a>).<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-06 16:22:16 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-18 19:38:16 UTC</tt>.<br>
-: The original revision id was <tt>240252397</tt>.<br>
+: The original revision id was <tt>241852539</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>
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 <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">=Prime numbers in [[EDO]]s=
-Whether a number n is prime or not has important consequences for the properties of the corresponding n-[[edo|EDO]], especially for lower values of n.
+A //prime number// is an integer (whole number) greater than one which is divisible only by itself and one. There an an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether a number n is prime or not has important consequences for the properties of the corresponding n-[[edo|EDO]], especially for lower values of n.
 * If the octave is divided into a prime number of equal parts, there is **no fully symmetric chord**, such as the diminished seventh chord in [[12edo]].
 * There is also (besides the full scale of all notes of the edo) **no absolutely uniform scale**, like the wholetone scale in 12edo.
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 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;prime numbers&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Prime numbers in EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Prime numbers in &lt;a class="wiki_link" href="/EDO"&gt;EDO&lt;/a&gt;s&lt;/h1&gt;
   &lt;br /&gt;
-Whether a number n is prime or not has important consequences for the properties of the corresponding n-&lt;a class="wiki_link" href="/edo"&gt;EDO&lt;/a&gt;, especially for lower values of n.&lt;br /&gt;
+A &lt;em&gt;prime number&lt;/em&gt; is an integer (whole number) greater than one which is divisible only by itself and one. There an an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether a number n is prime or not has important consequences for the properties of the corresponding n-&lt;a class="wiki_link" href="/edo"&gt;EDO&lt;/a&gt;, especially for lower values of n.&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;If the octave is divided into a prime number of equal parts, there is &lt;strong&gt;no fully symmetric chord&lt;/strong&gt;, such as the diminished seventh chord in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;.&lt;/li&gt;&lt;li&gt;There is also (besides the full scale of all notes of the edo) &lt;strong&gt;no absolutely uniform scale&lt;/strong&gt;, like the wholetone scale in 12edo.&lt;/li&gt;&lt;li&gt;Nor is there a thing like &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Modes_of_limited_transposition" rel="nofollow"&gt;modes of limited transpostion&lt;/a&gt;, as used by the composer Olivier Messiaen.&lt;/li&gt;&lt;li&gt;N-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are &lt;em&gt;linear&lt;/em&gt; temperaments.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 For these or similar reasons, some musicians seem not to like prime EDOs (e.g. the makers of &lt;a class="wiki_link_ext" href="http://www.armodue.com/risorse.htm" rel="nofollow"&gt;Armodue&lt;/a&gt;).&lt;br /&gt;