Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 144119323 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 144121435 - Original comment: ** |
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<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>2010-05-23 22: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-23 22:50:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>144121435</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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**Epimorphic**: If the image of s consists of rational numbers, that is, if s[i] is rational for every i, and if there exists a [[Vals and Tuning Space|val]] V such that V(s[i]) = i for every integer i, then s is weakly epimorphic with val V. If s is monotone, then s is simply called epimorphic. Weakly epimorphic also implies constant structure, but not propriety. | **Epimorphic**: If the image of s consists of rational numbers, that is, if s[i] is rational for every i, and if there exists a [[Vals and Tuning Space|val]] V such that V(s[i]) = i for every integer i, then s is weakly epimorphic with val V. If s is monotone, then s is simply called epimorphic. Weakly epimorphic also implies constant structure, but not propriety. | ||
**[[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]] **: A monotone scale in which every class but class(0) has exactly two elements has Myhill's property. If every such class has exactly three elements, it has the trivalence property.</pre></div> | **[[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]] **: A monotone scale in which every class but class(0) has exactly two elements has Myhill's property. If every such class has exactly three elements, it has the trivalence property. If every class has less than three elements, it has the property of [[http://en.wikipedia.org/wiki/Maximal_evenness|maximal evenness]].</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>Periodic scale</title></head><body>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. 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 nunmbers with values in cents. In this case, a periodic scale s has a nonzero quasiperiod P and repetition interval O satisfying the following conditions<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>Periodic scale</title></head><body>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. 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 nunmbers with values in cents. In this case, a periodic scale s has a nonzero quasiperiod P and repetition interval O satisfying the following conditions<br /> | ||
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<strong>Epimorphic</strong>: If the image of s consists of rational numbers, that is, if s[i] is rational for every i, and if there exists a <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> V such that V(s[i]) = i for every integer i, then s is weakly epimorphic with val V. If s is monotone, then s is simply called epimorphic. Weakly epimorphic also implies constant structure, but not propriety.<br /> | <strong>Epimorphic</strong>: If the image of s consists of rational numbers, that is, if s[i] is rational for every i, and if there exists a <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> V such that V(s[i]) = i for every integer i, then s is weakly epimorphic with val V. If s is monotone, then s is simply called epimorphic. Weakly epimorphic also implies constant structure, but not propriety.<br /> | ||
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<strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a> </strong>: A monotone scale in which every class but class(0) has exactly two elements has Myhill's property. If every such class has exactly three elements, it has the trivalence property.</body></html></pre></div> | <strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a> </strong>: A monotone scale in which every class but class(0) has exactly two elements has Myhill's property. If every such class has exactly three elements, it has the trivalence property. If every class has less than three elements, it has the property of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Maximal_evenness" rel="nofollow">maximal evenness</a>.</body></html></pre></div> | ||