Scale diversity
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author genewardsmith and made on 2014-01-10 13:39:41 UTC.
- The original revision id was 481921730.
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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.
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 Avela]].Original HTML content:
<html><head><title>Scale Diversity</title></head><body><strong>Diversity</strong> 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.<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:<h1> --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition:</h1>
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)<br/>[[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]]<br/>
X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))<br/>[[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%20Avela">Ryan Avela</a>.</body></html>