Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 401005304 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 401008094 - 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>2013-01-24 04: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-24 04:51:11 UTC</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 there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by [[Yves Hellegouarch]]. | **Epimorphic**: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by [[Yves Hellegouarch]]. | ||
**[[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]]** : A monotone scale in which every class but classes nP have 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 distributional evenness.</pre></div> | **[[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]]** : A monotone scale in which every class but classes nP have 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 distributional evenness. | ||
**Convexity**: The scale is [[Convex scale|convex]] if it is a ℤ-polytope in the group G; that is, if every convex combination, meaning every ℕ-linear combination of scale notes, is a scale note.</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>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 /> | <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; 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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<strong>Epimorphic</strong>: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a>.<br /> | <strong>Epimorphic</strong>: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a>.<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 classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the <strong>trivalence property</strong>. If every class has less than three elements, it has the property of distributional evenness.</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 classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the <strong>trivalence property</strong>. If every class has less than three elements, it has the property of distributional evenness.<br /> | ||
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<strong>Convexity</strong>: The scale is <a class="wiki_link" href="/Convex%20scale">convex</a> if it is a ℤ-polytope in the group G; that is, if every convex combination, meaning every ℕ-linear combination of scale notes, is a scale note.</body></html></pre></div> | |||