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 12:22:05 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-20 13:46:32 UTC</tt>.<br>

: The original revision id was <tt>~~535730076~~</tt>.<br>

: The original revision id was <tt>535732446</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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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, "Verband" 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 defined by intersection of the commas of A and B. On the ~~other hand, if~~ A and B are defined by 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 join 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 join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet 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.

=Examples=

Suppose we take G to be the 11-limit group. Then we have the following:

Meantone⋎Meanpop = [<31 49 72 87 107|] = 31, where "31" is the shorthand notation for the 31edo patent val.

Meantone⋏Meanpop = [<1 0 -4 -13 0|, <0 1 4 10 0|, <0 0 0 0 1|] = <81/80, 126/125>, where <S> for a set of commas S denotes the temperament of the group G tempering out the given commas.

Meantone⋎Marvel = 31, Meantone⋏Marvel = <225/224>

Meantone⋎Magic = [<0 0 0 0 0|] which we will denote "0".

Meantone⋏Magic = <225/224>.

Note that in terms of wedgies, Meantone∧Magic = <<<<0 1 2 -2 -5||||, which represents Meantone⋏Magic. This is an instance of the general proposition that if A⋎B = 0, then A⋏B is represented by A∧B.</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><h1 id="toc0"><a name="Introduction"></a>Introduction</h1>

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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;Verband&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 defined by intersection of the commas of A and B. On the ~~other hand, if~~ A and B are defined by 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 join 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 join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet 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.<br />

<br />

<h1 id="toc2"><a name="Examples"></a>Examples</h1>

Suppose we take G to be the 11-limit group. Then we have the following:<br />

<br />

Meantone⋎Meanpop = [&lt;31 49 72 87 107|] = 31, where &quot;31&quot; is the shorthand notation for the 31edo patent val.<br />

Meantone⋏Meanpop = [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0|, &lt;0 0 0 0 1|] = &lt;81/80, 126/125&gt;, where &lt;S&gt; for a set of commas S denotes the temperament of the group G tempering out the given commas.<br />

<br />

Meantone⋎Marvel = 31, Meantone⋏Marvel = &lt;225/224&gt;<br />

<br />

Meantone⋎Magic = [&lt;0 0 0 0 0|] which we will denote &quot;0&quot;.<br />

Meantone⋏Magic = &lt;225/224&gt;.<br />

Note that in terms of wedgies, Meantone∧Magic = &lt;&lt;&lt;&lt;0 1 2 -2 -5||||, which represents Meantone⋏Magic. This is an instance of the general proposition that if A⋎B = 0, then A⋏B is represented by A∧B.</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 12:22:05 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-20 13:46:32 UTC</tt>.<br>
-: The original revision id was <tt>535730076</tt>.<br>
+: The original revision id was <tt>535732446</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 12: / Line 12: @@
 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, "Verband" 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 defined by intersection of the commas of A and B. On the other hand, if A and B are defined by 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 join 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 join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet 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.
+=Examples=
+Suppose we take G to be the 11-limit group. Then we have the following:
+Meantone⋎Meanpop = [&lt;31 49 72 87 107|] = 31, where "31" is the shorthand notation for the 31edo patent val.
+Meantone⋏Meanpop = [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0|, &lt;0 0 0 0 1|] = &lt;81/80, 126/125&gt;, where &lt;S&gt; for a set of commas S denotes the temperament of the group G tempering out the given commas.
+Meantone⋎Marvel = 31, Meantone⋏Marvel = &lt;225/224&gt;
+Meantone⋎Magic = [&lt;0 0 0 0 0|] which we will denote "0".
+Meantone⋏Magic = &lt;225/224&gt;.
+Note that in terms of wedgies, Meantone∧Magic = &lt;&lt;&lt;&lt;0 1 2 -2 -5||||, which represents Meantone⋏Magic. This is an instance of the general proposition that if A⋎B = 0, then A⋏B is represented by A∧B.</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;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Introduction&lt;/h1&gt;
@@ Line 20: / Line 32: @@
 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;Verband&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 defined by intersection of the commas of A and B. On the other hand, if A and B are defined by 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 join 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 join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet 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;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Examples&lt;/h1&gt;
+Suppose we take G to be the 11-limit group. Then we have the following:&lt;br /&gt;
+&lt;br /&gt;
+Meantone⋎Meanpop = [&amp;lt;31 49 72 87 107|] = 31, where &amp;quot;31&amp;quot; is the shorthand notation for the 31edo patent val.&lt;br /&gt;
+Meantone⋏Meanpop = [&amp;lt;1 0 -4 -13 0|, &amp;lt;0 1 4 10 0|, &amp;lt;0 0 0 0 1|] = &amp;lt;81/80, 126/125&amp;gt;, where &amp;lt;S&amp;gt; for a set of commas S denotes the temperament of the group G tempering out the given commas.&lt;br /&gt;
+&lt;br /&gt;
+Meantone⋎Marvel = 31, Meantone⋏Marvel = &amp;lt;225/224&amp;gt;&lt;br /&gt;
+&lt;br /&gt;
+Meantone⋎Magic = [&amp;lt;0 0 0 0 0|] which we will denote &amp;quot;0&amp;quot;.&lt;br /&gt;
+Meantone⋏Magic = &amp;lt;225/224&amp;gt;.&lt;br /&gt;
+Note that in terms of wedgies, Meantone∧Magic = &amp;lt;&amp;lt;&amp;lt;&amp;lt;0 1 2 -2 -5||||, which represents Meantone⋏Magic. This is an instance of the general proposition that if A⋎B = 0, then A⋏B is represented by A∧B.&lt;/body&gt;&lt;/html&gt;</pre></div>