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Periodic scale: Difference between revisions

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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>
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-24 04:27:35 UTC</tt>.<br>
: The original revision id was <tt>400877172</tt>.<br>
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**[[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 &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 //coherence//. 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 &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 //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 scals were apparently first considered by [[Yves Hellegouarch]] and later again by Gene Smith.
The set {s[i]|i∈ℤ} generates a group G, the **group of the scale**; this is a free, finitely generated subgroup of the reals ℝ.


**Epimorphic (Mike's Definition):** &lt;span style="line-height: 1.5;"&gt;For any periodic scale p, there exists a group G which is completely generated by taking Z-linear sums of the elements of Im(p). We can say that p is "weakly epimorphic" if there exists an element h in Hom(G,Z) such that Im(p) is a transversal of h. If p is monotone and weakly epimorphic, then it is also "epimorphic." A related and more general notion is that of [[Abstract Epimorphicity|abstract epimorphicity]].&lt;/span&gt;
**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]] and later again by Gene Smith.


**[[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.</pre></div>
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&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow"&gt;Propriety&lt;/a&gt;&lt;/strong&gt; : 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 &amp;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 &lt;em&gt;coherence&lt;/em&gt;. Note that strict propriety implies constant structure.&lt;br /&gt;
&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow"&gt;Propriety&lt;/a&gt;&lt;/strong&gt; : 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 &amp;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 &lt;em&gt;coherence&lt;/em&gt;. Note that strict propriety implies constant structure.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Epimorphic&lt;/strong&gt;: If the image of s consists of rational numbers, that is, if s[i] is rational for every i, and if there exists a &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; 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 scals were apparently first considered by &lt;a class="wiki_link" href="/Yves%20Hellegouarch"&gt;Yves Hellegouarch&lt;/a&gt; and later again by Gene Smith.&lt;br /&gt;
The set {s[i]|i∈ℤ} generates a group G, the &lt;strong&gt;group of the scale&lt;/strong&gt;; this is a free, finitely generated subgroup of the reals ℝ.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;Epimorphic (Mike's Definition):&lt;/strong&gt; &lt;span style="line-height: 1.5;"&gt;For any periodic scale p, there exists a group G which is completely generated by taking Z-linear sums of the elements of Im(p). We can say that p is &amp;quot;weakly epimorphic&amp;quot; if there exists an element h in Hom(G,Z) such that Im(p) is a transversal of h. If p is monotone and weakly epimorphic, then it is also &amp;quot;epimorphic.&amp;quot; A related and more general notion is that of &lt;a class="wiki_link" href="/Abstract%20Epimorphicity"&gt;abstract epimorphicity&lt;/a&gt;.&lt;/span&gt;&lt;br /&gt;
&lt;strong&gt;Epimorphic&lt;/strong&gt;: 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 &lt;a class="wiki_link" href="/Yves%20Hellegouarch"&gt;Yves Hellegouarch&lt;/a&gt; and later again by Gene Smith.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow"&gt;Myhill's property&lt;/a&gt;&lt;/strong&gt; : 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 &lt;strong&gt;trivalence property&lt;/strong&gt;. If every class has less than three elements, it has the property of distributional evenness.&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;strong&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow"&gt;Myhill's property&lt;/a&gt;&lt;/strong&gt; : 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 &lt;strong&gt;trivalence property&lt;/strong&gt;. If every class has less than three elements, it has the property of distributional evenness.&lt;/body&gt;&lt;/html&gt;</pre></div>
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