Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 142097281 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 144116891 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-23 22:27:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>144116891</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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**Constant Structure**: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by [[http://en.wikipedia.org/wiki/Erv_Wilson|Erv Wilson]]) means that i<>j implies class(i) intersect class(j) = {}. | **Constant Structure**: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by [[http://en.wikipedia.org/wiki/Erv_Wilson|Erv Wilson]]) means that i<>j implies class(i) intersect class(j) = {}. | ||
**[[http://en.wikipedia.org/wiki/Rothenberg_propriety|Propriety]] **: If s is monotone, and if i <= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. Note that strict propriety implies constant structure. | **[[http://en.wikipedia.org/wiki/Rothenberg_propriety|Propriety]] **: If s is monotone, and if i <= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called //coherence//. Note that strict propriety implies constant structure. | ||
**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.</pre></div> | **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.</pre></div> | ||
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<strong>Constant Structure</strong>: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Erv Wilson</a>) means that i&lt;&gt;j implies class(i) intersect class(j) = {}.<br /> | <strong>Constant Structure</strong>: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Erv Wilson</a>) means that i&lt;&gt;j implies class(i) intersect class(j) = {}.<br /> | ||
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<strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Propriety</a> </strong>: If s is monotone, and if i &lt;= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i &lt; j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. Note that strict propriety implies constant structure. <br /> | <strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Propriety</a> </strong>: If s is monotone, and if i &lt;= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i &lt; j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called <em>coherence</em>. Note that strict propriety implies constant structure. <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.</body></html></pre></div> | <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.</body></html></pre></div> | ||