Prime number: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2012-01-26 03:59:33 UTC</tt>.

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2012-01-26 04:00:40 UTC</tt>.

: The original revision id was <tt>~~295468112~~</tt>.

: The original revision id was <tt>295468238</tt>.

: The revision comment was: <tt></tt>

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* Nor is there a thing like [[http://en.wikipedia.org/wiki/Modes_of_limited_transposition|modes of limited transpostion]], as used by the composer Olivier Messiaen.

* //n//-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are //linear// temperaments.

* Making a chain of any interval of the //n//-EDO, one can reach every tone in //n// steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]]~~, but not for 4 degrees~~.)

* Making a chain of any interval of the //n//-EDO, one can reach every tone in //n// steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]].)

For these or similar reasons, some musicians do not like the prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]) and others love them.

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

<ul><li>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 <a class="wiki_link" href="/12edo">12edo</a>.</li><li>There is also (besides the scale comprising all notes of the EDO) no absolutely uniform, octave-repeating scale, 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 linear temperaments.</li><li>Making a chain of any interval of the n-EDO, one can reach every tone in n steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of <a class="wiki_link" href="/12EDO">12EDO</a>~~, but not for 4 degrees~~.)</li></ul>

<ul><li>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 <a class="wiki_link" href="/12edo">12edo</a>.</li><li>There is also (besides the scale comprising all notes of the EDO) no absolutely uniform, octave-repeating scale, 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 linear temperaments.</li><li>Making a chain of any interval of the n-EDO, one can reach every tone in n steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of <a class="wiki_link" href="/12EDO">12EDO</a>.)</li></ul>

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

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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:hstraub|hstraub]] and made on <tt>2012-01-26 03:59:33 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2012-01-26 04:00:40 UTC</tt>.<br>
-: The original revision id was <tt>295468112</tt>.<br>
+: The original revision id was <tt>295468238</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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 * Nor is there a thing like [[http://en.wikipedia.org/wiki/Modes_of_limited_transposition|modes of limited transpostion]], as used by the composer Olivier Messiaen.
 * //n//-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are //linear// temperaments.
-* Making a chain of any interval of the //n//-EDO, one can reach every tone in //n// steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]], but not for 4 degrees.)
+* Making a chain of any interval of the //n//-EDO, one can reach every tone in //n// steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]].)
 For these or similar reasons, some musicians do not like the prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]) and others love them.
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   &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 are an infinite number of prime numbers, the first few of which are 2, 3, 5, 7, 11, 13 ... . Whether or not a number &lt;em&gt;n&lt;/em&gt; is prime has important consequences for the properties of the corresponding &lt;em&gt;n&lt;/em&gt;-&lt;a class="wiki_link" href="/edo"&gt;EDO&lt;/a&gt;, especially for lower values of &lt;em&gt;n&lt;/em&gt;.&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;If the octave is divided into a prime number of equal parts, there is &lt;em&gt;no fully symmetric chord&lt;/em&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 scale comprising all notes of the EDO) &lt;em&gt;no absolutely uniform, octave-repeating scale&lt;/em&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;&lt;em&gt;n&lt;/em&gt;-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;li&gt;Making a chain of any interval of the &lt;em&gt;n&lt;/em&gt;-EDO, one can reach every tone in &lt;em&gt;n&lt;/em&gt; steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of &lt;a class="wiki_link" href="/12EDO"&gt;12EDO&lt;/a&gt;, but not for 4 degrees.)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;If the octave is divided into a prime number of equal parts, there is &lt;em&gt;no fully symmetric chord&lt;/em&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 scale comprising all notes of the EDO) &lt;em&gt;no absolutely uniform, octave-repeating scale&lt;/em&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;&lt;em&gt;n&lt;/em&gt;-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;li&gt;Making a chain of any interval of the &lt;em&gt;n&lt;/em&gt;-EDO, one can reach every tone in &lt;em&gt;n&lt;/em&gt; steps. (For composite EDOs, this will work with intervals that are co-prime to the EDO, for example 5 degrees of &lt;a class="wiki_link" href="/12EDO"&gt;12EDO&lt;/a&gt;.)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 For these or similar reasons, some musicians do not like the 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;) and others love them.&lt;br /&gt;
 &lt;br /&gt;