Scale diversity - Revision history

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

2022-02-26T16:32:52Z

Moving from Category:Scale theory to Category:Scale using Cat-a-lot

← Older revision		Revision as of 16:32, 26 February 2022
Line 27:		Line 27:

	[[Category:Math]]		[[Category:Math]]
	[[Category:Scale ~~theory~~]]		[[Category:Scale]]

	{{todo\|clarify}}		{{todo\|clarify}}

Xenwolf: recat, todo

2021-01-02T00:20:58Z

recat, todo

← Older revision		Revision as of 00:20, 2 January 2021
Line 3:		Line 3:
	Properties:		Properties:

	~~<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>~~		* 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:=		=Definition:=
Line 16:		Line 18:
	N is the cardinality of S.		N is the cardinality of S.

	dia(S) is the [[~~Diamonds\|~~diamond]] ~~function~~.		dia(S) is the [[diamond function]].

	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.
Line 22:		Line 24:
	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

	The scale diversity measure was proposed by [[~~Ryan_Avella\|~~Ryan Avella]].		The scale diversity measure was proposed by [[Ryan Avella]].
	~~[[Category:math]]~~
	[[Category:~~scale~~]]		[[Category:Math]]
	[[Category:theory]]		[[Category:Scale theory]]

			{{todo\|clarify}}

Xenwolf: Xenwolf moved page Scale Diversity to Scale diversity: rule (for normal terms): only first letter should be capitalized for easy linking

2020-07-24T16:26:58Z

Xenwolf moved page Scale Diversity to Scale diversity: rule (for normal terms): only first letter should be capitalized for easy linking

← Older revision	Revision as of 16:26, 24 July 2020
(No difference)

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

@@ Line 1: / Line 1: @@
-<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.
 Properties:
-=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]]
+=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]]
 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]].</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">&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;
+[[Category:scale]]
-&lt;br /&gt;
+[[Category:theory]]

Wikispaces>genewardsmith: Imported revision 481921922 - Original comment:

2014-01-10T13:40:43Z

**Imported revision 481921922 - Original comment: **

Wikispaces>genewardsmith: Imported revision 481921730 - Original comment:

2014-01-10T13:39:41Z

**Imported revision 481921730 - Original comment: **

Wikispaces>genewardsmith: Imported revision 481915814 - Original comment:

2014-01-10T13:17:45Z

**Imported revision 481915814 - Original comment: **

← Older revision		Revision as of 13:17, 10 January 2014
Line 1:		Line 1:
	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 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>		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:		Properties:
	* Degenerate scales (scales with no intervals smaller than an octave and larger than a unison) have a diversity of 0.		* 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) ~~<= 2.~~
	* Similarly, it can be shown that 0 ~~<= Div(S) by noting that~~ there are no intervals larger than an octave.

	=Definition:=		=Definition:=
Line 34:		Line 32:
	<br />		<br />
	Properties:<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) ~~&lt;= 2.</li><li>Similarly, it can be shown that~~ 0 ~~&lt;= Div(S) by noting that~~ there are no intervals larger than an octave.</li></ul><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:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition:</h1>
	<!-- ws:start:WikiTextMathRule:0:		<!-- ws:start:WikiTextMathRule:0:

Wikispaces>Sarzadoce: Imported revision 477653516 - Original comment:

2013-12-15T21:01:15Z

**Imported revision 477653516 - Original comment: **

New page

<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>
: The original revision id was <tt>477653516</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>
<h4>Original Wikitext 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;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.

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 T</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"><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>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) &lt;= 2.</li><li>Similarly, it can be shown that 0 &lt;= Div(S) by noting that there are no intervals larger than an octave.</li></ul><br />
<h1 id="toc0"><a name="Definition:"></a>Definition:</h1>
<script type="math/tex">\mathrm{Div} (S) = - \log_N \left( \sum \limits_{x \in X} x^2 \right)</script><br />
<br />
<script type="math/tex">X = \mathrm{steps} ( \mathrm{sort} ( \mathrm{dia} (S)))</script><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></pre></div>

← Older revision		Revision as of 13:40, 10 January 2014
Line 1:		Line 1:
	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 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>		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		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>		<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"><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 />		<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 />
Line 53:		Line 53:
	steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T<br />		steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T<br />
	<br />		<br />
	The scale diversity measure was proposed by <a class="wiki_link" href="/Ryan%~~20Avela~~">Ryan ~~Avela~~</a>.</body></html></pre></div>		The scale diversity measure was proposed by <a class="wiki_link" href="/Ryan%20Avella">Ryan Avella</a>.</body></html></pre></div>

← Older revision		Revision as of 13:39, 10 January 2014
Line 1:		Line 1:
	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 13:17:45 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2014-01-10 13:39:41 UTC</tt>.<br>
	: The original revision id was <tt>~~481915814~~</tt>.<br>		: The original revision id was <tt>481921730</tt>.<br>
	: The revision comment was: <tt></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>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 27:		Line 27:
	dia(S) is the [[Diamonds\|diamond]] function.		dia(S) is the [[Diamonds\|diamond]] function.
	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.
	steps(T) returns a multiset whose elements are the consecutive differences of elements in a tuple T</pre></div>		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>
	<h4>Original HTML content:</h4>		<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"><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 />		<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 />
Line 49:		Line 51:
	dia(S) is the <a class="wiki_link" href="/Diamonds">diamond</a> function.<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 />		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></pre></div>		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></pre></div>

Scale diversity - Revision history

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

Xenwolf: recat, todo

Xenwolf: Xenwolf moved page Scale Diversity to Scale diversity: rule (for normal terms): only first letter should be capitalized for easy linking

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>genewardsmith: **Imported revision 481921922 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 481921730 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 481915814 - Original comment: **

Wikispaces>Sarzadoce: **Imported revision 477653516 - Original comment: **

Wikispaces>genewardsmith: Imported revision 481921922 - Original comment:

Wikispaces>genewardsmith: Imported revision 481921730 - Original comment:

Wikispaces>genewardsmith: Imported revision 481915814 - Original comment:

Wikispaces>Sarzadoce: Imported revision 477653516 - Original comment: