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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | '''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. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-01-10 13:40:43 UTC</tt>.<br>
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| : The original revision id was <tt>481921922</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| 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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| <h4>Original Wikitext content:</h4>
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| <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;white-space: pre-wrap ! important" class="old-revision-html">**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.
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| Properties: | | Properties: |
| * Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.
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| * EDOs have a diversity of 1.
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| * Div(S) ≥ 0 since there are no intervals larger than an octave.
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| =Definition:=
| | <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> |
| [[math]]
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| \mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)
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| [[math]]
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| [[math]]
| | =Definition:= |
| X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S))) | | <math>\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)</math> |
| [[math]]
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| | <math>X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))</math> |
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| | <u>Where:</u> |
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| __Where:__
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| S is a multiset. | | S is a multiset. |
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| N is the cardinality of S. | | N is the cardinality of S. |
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| dia(S) is the [[Diamonds|diamond]] function. | | dia(S) is the [[Diamonds|diamond]] function. |
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| sort(S) returns a tuple with all of the elements of S in non-decreasing order. | | sort(S) returns a tuple with all of the elements of S in non-decreasing order. |
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| steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T | | steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T |
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| The scale diversity measure was proposed by [[Ryan Avella]].</pre></div> | | The scale diversity measure was proposed by [[Ryan_Avella|Ryan Avella]]. |
| <h4>Original HTML content:</h4>
| | [[Category:math]] |
| <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"><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 />
| | [[Category:scale]] |
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| | [[Category:theory]] |
| Properties:<br />
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| <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 />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition:</h1>
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| <!-- ws:start:WikiTextMathRule:0:
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| [[math]]&lt;br/&gt; | |
| \mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)&lt;br/&gt;[[math]]
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| --><script type="math/tex">\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)</script><!-- ws:end:WikiTextMathRule:0 --><br />
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| <br />
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| <!-- ws:start:WikiTextMathRule:1:
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| [[math]]&lt;br/&gt;
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| X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))&lt;br/&gt;[[math]]
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| --><script type="math/tex">X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))</script><!-- ws:end:WikiTextMathRule:1 --><br />
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| <br />
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| <u>Where:</u><br />
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| S is a multiset.<br />
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| N is the cardinality of S.<br />
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| dia(S) is the <a class="wiki_link" href="/Diamonds">diamond</a> function.<br />
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| sort(S) returns a tuple with all of the elements of S in non-decreasing order.<br />
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| steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T<br />
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| <br />
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| The scale diversity measure was proposed by <a class="wiki_link" href="/Ryan%20Avella">Ryan Avella</a>.</body></html></pre></div>
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