Scale diversity
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author Sarzadoce and made on 2013-12-15 21:01:15 UTC.
- The original revision id was 477653516.
- The revision comment was:
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.
* EDO's have a diversity of 1.
* Perfect Cyclic Difference Sets have a diversity of 2.
* 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:=
[[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 TOriginal 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>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) <= 2.</li><li>Similarly, it can be shown that 0 <= Div(S) by noting that 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</body></html>