Meet and join: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2014-12-20 01:52:22 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-20 01:56:11 UTC</tt>.<br>

: The original revision id was <tt>~~535718844~~</tt>.<br>

: The original revision id was <tt>535718884</tt>.<br>

: The revision comment was: <tt></tt><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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<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;white-space: pre-wrap ! important" class="old-revision-html">Meet and join are a pair of binary operations which combine two [[Abstract regular temperament|abstract regular temperaments]] on a JI group G into another temperament on G. The operations are commutative and associative. More concretely, for any of the standard ways of representing an abstract regular temperament (normal val lists, normal comma lists, wedgies, Frobenius projection maps, and reduced row echelon form) we can regard them as taking any pair of such defined on G and producing another also defined on G.

Meet and join are defined in terms of the [[https://en.wikipedia.org/wiki/Lattice_of_subgroups|lattice of subgroups]] of G, consisting of groups of [[Smonzos and svals|smonzos]], or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here "lattice" means [[https://en.wikipedia.org/wiki/Lattice_(order)|lattice in the order theory sense]]; "trellis" in French, "Gitter" in German. Either of these subgroup lattices serves to define the temperaments of G, but since the canonical map defined by the normal val list or the multisval defined by taking the wedge product of this are the two most common ways of denoting a temperament, we adopt the val point of view.

Meet and join are defined in terms of the [[https://en.wikipedia.org/wiki/Lattice_of_subgroups|lattice of subgroups]] of G, consisting of groups of [[Smonzos and svals|smonzos]] defining the commas of the temperaments of G, or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here "lattice" means [[https://en.wikipedia.org/wiki/Lattice_(order)|lattice in the order theory sense]]; "trellis" in French, "Gitter" in German. Either of these subgroup lattices serves to define the temperaments of G.

Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B, On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.</pre></div>

Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B. On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.</pre></div>

<h4>Original HTML content:</h4>

<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>Meet and Join</title></head><body>Meet and join are a pair of binary operations which combine two <a class="wiki_link" href="/Abstract%20regular%20temperament">abstract regular temperaments</a> on a JI group G into another temperament on G. The operations are commutative and associative. More concretely, for any of the standard ways of representing an abstract regular temperament (normal val lists, normal comma lists, wedgies, Frobenius projection maps, and reduced row echelon form) we can regard them as taking any pair of such defined on G and producing another also defined on G.<br />

<br />

Meet and join are defined in terms of the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_of_subgroups" rel="nofollow">lattice of subgroups</a> of G, consisting of groups of <a class="wiki_link" href="/Smonzos%20and%20svals">smonzos</a>, or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here &quot;lattice&quot; means <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_(order)" rel="nofollow">lattice in the order theory sense</a>; &quot;trellis&quot; in French, &quot;Gitter&quot; in German. Either of these subgroup lattices serves to define the temperaments of G, but since the canonical map defined by the normal val list or the multisval defined by taking the wedge product of this are the two most common ways of denoting a temperament, we adopt the val point of view.<br />

Meet and join are defined in terms of the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_of_subgroups" rel="nofollow">lattice of subgroups</a> of G, consisting of groups of <a class="wiki_link" href="/Smonzos%20and%20svals">smonzos</a> defining the commas of the temperaments of G, or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here &quot;lattice&quot; means <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_(order)" rel="nofollow">lattice in the order theory sense</a>; &quot;trellis&quot; in French, &quot;Gitter&quot; in German. Either of these subgroup lattices serves to define the temperaments of G.<br />

<br />

Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B, On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.</body></html></pre></div>

Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B. On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2014-12-20 01:52:22 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-20 01:56:11 UTC</tt>.<br>
-: The original revision id was <tt>535718844</tt>.<br>
+: The original revision id was <tt>535718884</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 8: / Line 8: @@
 <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;white-space: pre-wrap ! important" class="old-revision-html">Meet and join are a pair of binary operations which combine two [[Abstract regular temperament|abstract regular temperaments]] on a JI group G into another temperament on G. The operations are commutative and associative. More concretely, for any of the standard ways of representing an abstract regular temperament (normal val lists, normal comma lists, wedgies, Frobenius projection maps, and reduced row echelon form) we can regard them as taking any pair of such defined on G and producing another also defined on G.
-Meet and join are defined in terms of the [[https://en.wikipedia.org/wiki/Lattice_of_subgroups|lattice of subgroups]] of G, consisting of groups of [[Smonzos and svals|smonzos]], or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here "lattice" means [[https://en.wikipedia.org/wiki/Lattice_(order)|lattice in the order theory sense]]; "trellis" in French, "Gitter" in German. Either of these subgroup lattices serves to define the temperaments of G, but since the canonical map defined by the normal val list or the multisval defined by taking the wedge product of this are the two most common ways of denoting a temperament, we adopt the val point of view.
+Meet and join are defined in terms of the [[https://en.wikipedia.org/wiki/Lattice_of_subgroups|lattice of subgroups]] of G, consisting of groups of [[Smonzos and svals|smonzos]] defining the commas of the temperaments of G, or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here "lattice" means [[https://en.wikipedia.org/wiki/Lattice_(order)|lattice in the order theory sense]]; "trellis" in French, "Gitter" in German. Either of these subgroup lattices serves to define the temperaments of G.
-Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B, On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.</pre></div>
+Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B. On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.</pre></div>
 <h4>Original HTML content:</h4>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Meet and Join&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Meet and join are a pair of binary operations which combine two &lt;a class="wiki_link" href="/Abstract%20regular%20temperament"&gt;abstract regular temperaments&lt;/a&gt; on a JI group G into another temperament on G. The operations are commutative and associative. More concretely, for any of the standard ways of representing an abstract regular temperament (normal val lists, normal comma lists, wedgies, Frobenius projection maps, and reduced row echelon form) we can regard them as taking any pair of such defined on G and producing another also defined on G.&lt;br /&gt;
 &lt;br /&gt;
-Meet and join are defined in terms of the &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_of_subgroups" rel="nofollow"&gt;lattice of subgroups&lt;/a&gt; of G, consisting of groups of &lt;a class="wiki_link" href="/Smonzos%20and%20svals"&gt;smonzos&lt;/a&gt;, or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here &amp;quot;lattice&amp;quot; means &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_(order)" rel="nofollow"&gt;lattice in the order theory sense&lt;/a&gt;; &amp;quot;trellis&amp;quot; in French, &amp;quot;Gitter&amp;quot; in German. Either of these subgroup lattices serves to define the temperaments of G, but since the canonical map defined by the normal val list or the multisval defined by taking the wedge product of this are the two most common ways of denoting a temperament, we adopt the val point of view.&lt;br /&gt;
+Meet and join are defined in terms of the &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_of_subgroups" rel="nofollow"&gt;lattice of subgroups&lt;/a&gt; of G, consisting of groups of &lt;a class="wiki_link" href="/Smonzos%20and%20svals"&gt;smonzos&lt;/a&gt; defining the commas of the temperaments of G, or equivalently and dually, the lattice of subgroups of the dual group G^ of svals, where here &amp;quot;lattice&amp;quot; means &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Lattice_(order)" rel="nofollow"&gt;lattice in the order theory sense&lt;/a&gt;; &amp;quot;trellis&amp;quot; in French, &amp;quot;Gitter&amp;quot; in German. Either of these subgroup lattices serves to define the temperaments of G.&lt;br /&gt;
 &lt;br /&gt;
-Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B, On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Given two temperaments A and B defined in terms of normal val lists, the meet A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The meet in terms of vals is the subgroup of G generated by A and B, and in terms of intervals is the intersection of the commas of A and B. On the other hand, if A and B are defined as normal interval lists, then the join A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again, this defines both join and meet as operations on normal val lists.&lt;/body&gt;&lt;/html&gt;</pre></div>