Scale diversity: 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:~~Sarzadoce~~|~~Sarzadoce~~]] and made on <tt>~~2013~~-12-~~15 21~~:01:15 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-01-10 13:17:45 UTC</tt>.<br>

: The original revision id was <tt>~~477653516~~</tt>.<br>

: The original revision id was <tt>481915814</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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Properties:

* Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.

* ~~EDO's~~ have a diversity of 1.

* EDOs have a diversity of 1.

* ~~Perfect Cyclic Difference Sets have a diversity of 2.~~

* Div(S) ≥ 0 since there are no intervals larger than an octave.

* By the [[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality|Cauchy-Schwarz Inequality]], it can be shown that Div(S) ~~<= 2.~~

* Similarly, it can be shown that 0 ~~<= Div(S) by noting that~~ there are no intervals larger than an octave.

=Definition:=

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<br />

Properties:<br />

<ul><li>Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.</li><li>~~EDO's~~ have a diversity of 1.</li><li>Perfect Cyclic Difference Sets have a diversity of 2.</li><li>By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality" rel="nofollow">Cauchy-Schwarz Inequality</a>, it can be shown that Div(S) ~~&lt;= 2.</li><li>Similarly, it can be shown that~~ 0 ~~&lt;= Div(S) by noting that~~ there are no intervals larger than an octave.</li></ul><br />

<ul><li>Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.</li><li>EDOs have a diversity of 1.</li><li>Div(S) ≥ 0 since there are no intervals larger than an octave.</li></ul><br />

<h1 id="toc0"><a name="Definition:"></a>Definition:</h1>

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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:Sarzadoce|Sarzadoce]] and made on <tt>2013-12-15 21:01:15 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-01-10 13:17:45 UTC</tt>.<br>
-: The original revision id was <tt>477653516</tt>.<br>
+: The original revision id was <tt>481915814</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 10: / Line 10: @@
 Properties:
 * Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.
-* EDO's have a diversity of 1.
+* EDOs have a diversity of 1.
-* Perfect Cyclic Difference Sets have a diversity of 2.
+* Div(S) ≥ 0 since there are no intervals larger than an octave.
-* By the [[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality|Cauchy-Schwarz Inequality]], it can be shown that Div(S) &lt;= 2.
-* Similarly, it can be shown that 0 &lt;= Div(S) by noting that there are no intervals larger than an octave.
 =Definition:=
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 &lt;br /&gt;
 Properties:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.&lt;/li&gt;&lt;li&gt;EDO's have a diversity of 1.&lt;/li&gt;&lt;li&gt;Perfect Cyclic Difference Sets have a diversity of 2.&lt;/li&gt;&lt;li&gt;By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality" rel="nofollow"&gt;Cauchy-Schwarz Inequality&lt;/a&gt;, it can be shown that Div(S) &amp;lt;= 2.&lt;/li&gt;&lt;li&gt;Similarly, it can be shown that 0 &amp;lt;= Div(S) by noting that there are no intervals larger than an octave.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.&lt;/li&gt;&lt;li&gt;EDOs have a diversity of 1.&lt;/li&gt;&lt;li&gt;Div(S) ≥ 0 since there are no intervals larger than an octave.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
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