Periodic scale: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 400874228 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 400877058 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2013-01-23 | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-01-23 16:03:44 UTC</tt>.<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 scals were apparently first considered by [[Yves Hellegouarch]] and later again by Gene Smith. | **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. | ||
**Epimorphic (Mike's Definition):** <span style="line-height: 1.5;">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."</span> | **Epimorphic (Mike's Definition):** <span style="line-height: 1.5;">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]].</span> | ||
**[[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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<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. Epimorphic scals were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a> and later again by Gene Smith.<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. Epimorphic scals were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a> and later again by Gene Smith.<br /> | ||
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<strong>Epimorphic (Mike's Definition):</strong> <span style="line-height: 1.5;">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 &quot;weakly epimorphic&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 &quot;epimorphic.&quot;</span><br /> | <strong>Epimorphic (Mike's Definition):</strong> <span style="line-height: 1.5;">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 &quot;weakly epimorphic&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 &quot;epimorphic.&quot; A related and more general notion is that of <a class="wiki_link" href="/abstract%20epimorphicity">abstract epimorphicity</a>.</span><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.</body></html></pre></div> | ||