Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 199020142 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278334138 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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We may define an important function **class(i)** on the integers which gives the //generic intervals// of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have: | We may define an important function **class(i)** on the integers which gives the //generic intervals// of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have: | ||
**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 | **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) ∩ class(j) = ∅. In academic music theory, this is called the //partitioning property//. | ||
**[[http://en.wikipedia.org/wiki/Rothenberg_propriety|Propriety]]** : If s is monotone, and if i | **[[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. | **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. | ||
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We may define an important function <strong>class(i)</strong> on the integers which gives the <em>generic intervals</em> of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have:<br /> | We may define an important function <strong>class(i)</strong> on the integers which gives the <em>generic intervals</em> of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have:<br /> | ||
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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 | <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≠j implies class(i) ∩ class(j) = ∅. In academic music theory, this is called the <em>partitioning property</em>.<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 | <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 ≤ 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.<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 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Maximal_evenness" rel="nofollow">maximal evenness</a>.</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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Maximal_evenness" rel="nofollow">maximal evenness</a>.</body></html></pre></div> | ||