Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 508629270 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 509653074 - Original comment: ** |
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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>2014-05-13 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:42:20 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509653074</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> | 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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<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">[[toc]] | <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">[[toc]] | ||
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=Definition= | =Definition= | ||
A **periodic scale** may be defined in mathematical language as a type of [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the [[http://en.wikipedia.org/wiki/Integers|integers]] to musical intervals; the integers in this case formalize the notion of "scale degrees." Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale **s** has a nonzero quasiperiod **P** and repetition interval **O** satisfying the following conditions | A **periodic scale** may be defined in mathematical language as a type of [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the [[http://en.wikipedia.org/wiki/Integers|integers]] to musical intervals; the integers in this case formalize the notion of "scale degrees." Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale **s** has a nonzero quasiperiod **P** and repetition interval **O** satisfying the following conditions | ||
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==Numerical properties== | ==Numerical properties== | ||
[[Scale diversity]] | [[Scale diversity]] | ||
[[Lumma stability]] | [[Lumma stability]]</pre></div> | ||
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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Periodic scale</title></head><body><!-- ws:start:WikiTextTocRule:27:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | <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>Periodic scale</title></head><body><!-- ws:start:WikiTextTocRule:27:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
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<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:3 -->Definition</h1> | <!-- ws:start:WikiTextLocalImageRule:41:&lt;div style=&quot;text-align: center&quot;&gt;&lt;img src=&quot;/file/view/mathhazard.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt;&lt;/div&gt; --><div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div><!-- ws:end:WikiTextLocalImageRule:41 --><!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:3 -->Definition</h1> | ||
A <strong>periodic scale</strong> may be defined in mathematical language as a type of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> from the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integers" rel="nofollow">integers</a> to musical intervals; the integers in this case formalize the notion of &quot;scale degrees.&quot; Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale <strong>s</strong> has a nonzero quasiperiod <strong>P</strong> and repetition interval <strong>O</strong> satisfying the following conditions<br /> | A <strong>periodic scale</strong> may be defined in mathematical language as a type of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> from the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integers" rel="nofollow">integers</a> to musical intervals; the integers in this case formalize the notion of &quot;scale degrees.&quot; Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real numbers with values in cents. In this case, a periodic scale <strong>s</strong> has a nonzero quasiperiod <strong>P</strong> and repetition interval <strong>O</strong> satisfying the following conditions<br /> | ||
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