Scale diversity: Difference between revisions

Revision as of 13:17, 10 January 2014

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This revision was by author genewardsmith and made on 2014-01-10 13:17:45 UTC.

The original revision id was 481915814.

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Original Wikitext content:

**Diversity** is a scale measurement which categorizes scales according to the "diversity" of available intervals. As a general rule of thumb, scales with many unique interval sizes will have a high diversity. Similarly, scales with many redundant intervals will be assigned a low diversity rating.

Properties:
* Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.
* EDOs have a diversity of 1.
* Div(S) ≥ 0 since there are no intervals larger than an octave.

=Definition:= 
[[math]]
\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)
[[math]]

[[math]]
X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))
[[math]]

__Where:__
S is a multiset.
N is the cardinality of S.
dia(S) is the [[Diamonds|diamond]] function.
sort(S) returns a tuple with all of the elements of S in non-decreasing order.
steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T

Original HTML content:

<html><head><title>Scale Diversity</title></head><body><strong>Diversity</strong> is a scale measurement which categorizes scales according to the &quot;diversity&quot; of available intervals. As a general rule of thumb, scales with many unique interval sizes will have a high diversity. Similarly, scales with many redundant intervals will be assigned a low diversity rating.<br />
<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>EDOs have a diversity of 1.</li><li>Div(S) ≥ 0 since there are no intervals larger than an octave.</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition:</h1>
 <!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)&lt;br/&gt;[[math]]
 --><script type="math/tex">\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))&lt;br/&gt;[[math]]
 --><script type="math/tex">X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))</script><!-- ws:end:WikiTextMathRule:1 --><br />
<br />
<u>Where:</u><br />
S is a multiset.<br />
N is the cardinality of S.<br />
dia(S) is the <a class="wiki_link" href="/Diamonds">diamond</a> function.<br />
sort(S) returns a tuple with all of the elements of S in non-decreasing order.<br />
steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T</body></html>

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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:=
@@ Line 34: / Line 32: @@
 &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;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Definition:&lt;/h1&gt;
   &lt;!-- ws:start:WikiTextMathRule:0:

Scale diversity: Difference between revisions

Revision as of 13:17, 10 January 2014

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