Scale diversity: Difference between revisions

Revision as of 13:40, 10 January 2014

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This revision was by author genewardsmith and made on 2014-01-10 13:40:43 UTC.

The original revision id was 481921922.

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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

The scale diversity measure was proposed by [[Ryan Avella]].

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<br />
<br />
The scale diversity measure was proposed by <a class="wiki_link" href="/Ryan%20Avella">Ryan Avella</a>.</body></html>

@@ 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>2014-01-10 13:39:41 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-01-10 13:40:43 UTC</tt>.<br>
-: The original revision id was <tt>481921730</tt>.<br>
+: The original revision id was <tt>481921922</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 29: / Line 29: @@
 steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T
-The scale diversity measure was proposed by [[Ryan Avela]].</pre></div>
+The scale diversity measure was proposed by [[Ryan Avella]].</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Scale Diversity&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;Diversity&lt;/strong&gt; is a scale measurement which categorizes scales according to the &amp;quot;diversity&amp;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.&lt;br /&gt;
@@ Line 53: / Line 53: @@
 steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T&lt;br /&gt;
 &lt;br /&gt;
-The scale diversity measure was proposed by &lt;a class="wiki_link" href="/Ryan%20Avela"&gt;Ryan Avela&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
+The scale diversity measure was proposed by &lt;a class="wiki_link" href="/Ryan%20Avella"&gt;Ryan Avella&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>

Scale diversity: Difference between revisions

Revision as of 13:40, 10 January 2014

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