Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 508603452 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 508611358 - 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 11: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-13 11:43:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>508611358</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> | ||
<h4>Original Wikitext content:</h4> | <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">[[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]] | ||
=Definition= | =Definition= | ||
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[[math]] | [[math]] | ||
Scales written in the widely used [[http://www.huygens-fokker.org/scala/scl_format.html|Scala format]] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Neither Scala nor the above definition assumes that the scales are [[http://en.wikipedia.org/wiki/Monotonic_function|monotonically strictly increasing]], but this condition, giving a **monotone periodic scale**, is often important to add: | Scales written in the widely used [[http://www.huygens-fokker.org/scala/scl_format.html|Scala format]] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala scl file is intended to denote. Of course, since arbitrarily high and low pitches go beyond the range of human hearing, this definition is a mathematical idealization, but it is much simpler to adopt the idealization than to worry about that. Neither Scala nor the above definition assumes that the scales are [[http://en.wikipedia.org/wiki/Monotonic_function|monotonically strictly increasing]], but this condition, giving a **monotone periodic scale**, is often important to add: | ||
[[math]] | [[math]] | ||
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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@@ | <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> | ||
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><div style="margin-left: 2em;"><a href="#Definition-Rotations">Rotations</a></div> | |||
<!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 2em;"><a href="#Definition-Classes">Classes</a></div> | |||
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#Scale properties">Scale properties</a></div> | |||
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Scale properties-Constant Structure">Constant Structure</a></div> | |||
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 2em;"><a href="#Scale properties-Propriety">Propriety</a></div> | |||
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Scale properties-Epimorphic">Epimorphic</a></div> | |||
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Scale properties-Myhill's property">Myhill's property</a></div> | |||
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Scale properties-Distributional evenness">Distributional evenness</a></div> | |||
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Scale properties-Convexity">Convexity</a></div> | |||
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 2em;"><a href="#Scale properties-Maximal evenness">Maximal evenness</a></div> | |||
<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 2em;"><a href="#Scale properties-Numerical properties">Numerical properties</a></div> | |||
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --></div> | |||
<!-- ws:end:WikiTextTocRule:40 --><br /> | <!-- ws:end:WikiTextTocRule:40 --><br /> | ||
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Scales written in the widely used <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/scl_format.html" rel="nofollow">Scala format</a> are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Neither Scala nor the above definition assumes that the scales are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotonically strictly increasing</a>, but this condition, giving a <strong>monotone periodic scale</strong>, is often important to add:<br /> | Scales written in the widely used <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/scl_format.html" rel="nofollow">Scala format</a> are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala scl file is intended to denote. Of course, since arbitrarily high and low pitches go beyond the range of human hearing, this definition is a mathematical idealization, but it is much simpler to adopt the idealization than to worry about that. Neither Scala nor the above definition assumes that the scales are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotonically strictly increasing</a>, but this condition, giving a <strong>monotone periodic scale</strong>, is often important to add:<br /> | ||
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