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:~~Osmiorisbendi~~|~~Osmiorisbendi~~]] and made on <tt>2011-07-19 03:40:18 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-19 04:15:33 UTC</tt>.<br>

: The original revision id was <tt>~~241899344~~</tt>.<br>

: The original revision id was <tt>241901398</tt>.<br>

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

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

For these or similar reasons, some musicians seem not like the Prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]).

For these or similar reasons, some musicians seem not to like the Prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]).

On the other hand, primality may be a desirable feature if you happen to want, e.g., a wholetone scale that is //not// absolutely uniform. (In this case you might like [[19edo]], for example.)

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{todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here}. XXX

==The first "Prime EDOs"==

==The first Prime EDOs==

Multiples of an (prime?) EDO can inherit properties from that ~~(prime?)~~ EDO, in particular a tuning for certain intervals. A (multiple?) EDO~~, on~~ the ~~other hand~~, ~~is necessarily more complex~~.

Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below.

[[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],

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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, octave-repeating 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 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>).<br />

For these or similar reasons, some musicians seem not to 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>).<br />

<br />

On the other hand, primality may be a desirable feature if you happen to want, e.g., a wholetone scale that is <em>not</em> absolutely uniform. (In this case you might like <a class="wiki_link" href="/19edo">19edo</a>, for example.)<br />

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{todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here}. XXX<br />

<br />

<h2 id="toc1"><a name="Prime numbers in EDOs-The first ~~&quot;~~Prime EDOs~~&quot;~~"></a>The first ~~&quot;~~Prime EDOs~~&quot;~~</h2>

<h2 id="toc1"><a name="Prime numbers in EDOs-The first Prime EDOs"></a>The first Prime EDOs</h2>

Multiples of an (prime?) EDO can inherit properties from that ~~(prime?)~~ EDO, in particular a tuning for certain intervals. A (multiple?) EDO~~, on~~ the ~~other hand~~, ~~is necessarily more complex~~.<br />

Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below.<br />

<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:Osmiorisbendi|Osmiorisbendi]] and made on <tt>2011-07-19 03:40:18 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-07-19 04:15:33 UTC</tt>.<br>
-: The original revision id was <tt>241899344</tt>.<br>
+: The original revision id was <tt>241901398</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>
@@ Line 14: / Line 14: @@
 * N-EDO does not support any rank two temperament with period a fraction of an octave; all such temperaments are //linear// temperaments.
-For these or similar reasons, some musicians seem not like the Prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]).
+For these or similar reasons, some musicians seem not to like the Prime EDOs (e.g. the makers of [[http://www.armodue.com/risorse.htm|Armodue]]).
 On the other hand, primality may be a desirable feature if you happen to want, e.g., a wholetone scale that is //not// absolutely uniform. (In this case you might like [[19edo]], for example.)
@@ Line 23: / Line 23: @@
 {todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here}. XXX
-==The first "Prime EDOs"==
+==The first Prime EDOs==
-Multiples of an (prime?) EDO can inherit properties from that (prime?) EDO, in particular a tuning for certain intervals. A (multiple?) EDO, on the other hand, is necessarily more complex.
+Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below.
 [[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[11edo|11]], [[13edo|13]], [[17edo|17]],
@@ Line 47: / Line 47: @@
 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, octave-repeating 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 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;).&lt;br /&gt;
+For these or similar reasons, some musicians seem not to 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;).&lt;br /&gt;
 &lt;br /&gt;
 On the other hand, primality may be a desirable feature if you happen to want, e.g., a wholetone scale that is &lt;em&gt;not&lt;/em&gt; absolutely uniform. (In this case you might like &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, for example.)&lt;br /&gt;
@@ Line 56: / Line 56: @@
 {todo: add more useful things about prime numbers for musicians, composers, microtonalists, xenharmonicians, ekmelicians and theorists here}. XXX&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Prime numbers in EDOs-The first &amp;quot;Prime EDOs&amp;quot;"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;The first &amp;quot;Prime EDOs&amp;quot;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Prime numbers in EDOs-The first Prime EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;The first Prime EDOs&lt;/h2&gt;
-  Multiples of an (prime?) EDO can inherit properties from that (prime?) EDO, in particular a tuning for certain intervals. A (multiple?) EDO, on the other hand, is necessarily more complex.&lt;br /&gt;
+  Multiples of an EDO, including multiples of a prime EDO, can inherit properties from that EDO, in particular a tuning for certain intervals. A multiple however is by definition more complex; a prime EDO is always the least complex EDO divisible by that prime, and these are listed below.&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/11edo"&gt;11&lt;/a&gt;, &lt;a class="wiki_link" href="/13edo"&gt;13&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17&lt;/a&gt;,&lt;br /&gt;