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:

: This revision was by author [[User:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2011-08-15 11:39:39 UTC</tt>.

: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-08-15 15:55:52 UTC</tt>.

: The original revision id was <tt>~~246035683~~</tt>.

: The original revision id was <tt>246084097</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]].)

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]]).

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

On the other hand, primality may be desirable if you want, for example, a wholetone scale that is //not// absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, MOS scale of type [[1L 5s|1L+5s]]) or [[17edo]] (with whole tone scale 3 3 3 3 3 2, MOS Scale of type [[5L 1s|5L+1s]]). In general, making a chain of any interval of a prime //n//-EDO, thus treating the interval as the generator of a [[MOSScales|Moment of Symmetry]] scale, one can reach every tone in //n// steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo]]).

The larger //n// is, the less these points matter, since the difference between an //absolutely// uniform scale and an approximated, //nearly// uniform scale eventually become inaudible.

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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></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>).

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

On the other hand, primality may be desirable if you want, for example, a wholetone scale that is not absolutely uniform. In this case you might like <a class="wiki_link" href="/19edo">19edo</a> (with whole tone scale 3 3 3 3 3 4, MOS scale of type <a class="wiki_link" href="/1L%205s">1L+5s</a>) or <a class="wiki_link" href="/17edo">17edo</a> (with whole tone scale 3 3 3 3 3 2, MOS Scale of type <a class="wiki_link" href="/5L%201s">5L+1s</a>). In general, making a chain of any interval of a prime n-EDO, thus treating the interval as the generator of a <a class="wiki_link" href="/MOSScales">Moment of Symmetry</a> scale, one can reach every tone in n steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of <a class="wiki_link" href="/12EDO">12EDO</a> (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of <a class="wiki_link" href="/3edo">3edo</a>).

The larger n is, the less these points matter, since the difference between an absolutely uniform scale and an approximated, nearly uniform scale eventually become inaudible.

@@ 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>2011-08-15 11:39:39 UTC</tt>.<br>
+: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-08-15 15:55:52 UTC</tt>.<br>
-: The original revision id was <tt>246035683</tt>.<br>
+: The original revision id was <tt>246084097</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]].)
 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]]).
-On the other hand, primality may be desirable if you want, for example, a wholetone scale that is //not// absolutely uniform. (In this case you might like [[19edo]], for example.)
+On the other hand, primality may be desirable if you want, for example, a wholetone scale that is //not// absolutely uniform. In this case you might like [[19edo]] (with whole tone scale 3 3 3 3 3 4, MOS scale of type [[1L 5s|1L+5s]]) or [[17edo]] (with whole tone scale 3 3 3 3 3 2, MOS Scale of type [[5L 1s|5L+1s]]). In general, making a chain of any interval of a prime //n//-EDO, thus treating the interval as the generator of a [[MOSScales|Moment of Symmetry]] scale, one can reach every tone in //n// steps. For composite EDOs, this will only work with intervals that are co-prime to the EDO, for example 5 degrees of [[12EDO]] (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of [[3edo]]).
 The larger //n// is, the less these points matter, since the difference between an //absolutely// uniform scale and an approximated, //nearly// uniform scale eventually become inaudible.
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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;/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;).&lt;br /&gt;
 &lt;br /&gt;
-On the other hand, primality may be desirable if you want, for example, 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;
+On the other hand, primality may be desirable if you want, for example, 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; (with whole tone scale 3 3 3 3 3 4, MOS scale of type &lt;a class="wiki_link" href="/1L%205s"&gt;1L+5s&lt;/a&gt;) or &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt; (with whole tone scale 3 3 3 3 3 2, MOS Scale of type &lt;a class="wiki_link" href="/5L%201s"&gt;5L+1s&lt;/a&gt;). In general, making a chain of any interval of a prime &lt;em&gt;n&lt;/em&gt;-EDO, thus treating the interval as the generator of a &lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; scale, one can reach every tone in &lt;em&gt;n&lt;/em&gt; steps. For composite EDOs, this will only 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; (which generates the diatonic scale and a cycle of fifths that closes at 12 tones) but not 4 out of 12 (which generates a much smaller cycle of &lt;a class="wiki_link" href="/3edo"&gt;3edo&lt;/a&gt;).&lt;br /&gt;
 &lt;br /&gt;
 The larger &lt;em&gt;n&lt;/em&gt; is, the less these points matter, since the difference between an &lt;em&gt;absolutely&lt;/em&gt; uniform scale and an approximated, &lt;em&gt;nearly&lt;/em&gt; uniform scale eventually become inaudible.&lt;br /&gt;