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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-22 12:55:22 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-22 17:36:18 UTC</tt>.

: The original revision id was <tt>~~535807084~~</tt>.

: The original revision id was <tt>535825630</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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 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.

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 via the [[dual list]] function, this defines both join and meet as operations on normal val lists.

There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it <JI>; and in the temperament defined by G^, nothing is tempered out, and we may also call it <1>.

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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.

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.

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 via the <a class="wiki_link" href="/dual%20list">dual list</a> function, this defines both join and meet as operations on normal val lists.

There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it &lt;JI&gt;; and in the temperament defined by G^, nothing is tempered out, and we may also call it &lt;1&gt;.

@@ 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>2014-12-22 12:55:22 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-22 17:36:18 UTC</tt>.<br>
-: The original revision id was <tt>535807084</tt>.<br>
+: The original revision id was <tt>535825630</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 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.
+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 via the [[dual list]] function, this defines both join and meet as operations on normal val lists.
 There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it &lt;JI&gt;; and in the temperament defined by G^, nothing is tempered out, and we may also call it &lt;1&gt;.
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 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 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;
+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 via the &lt;a class="wiki_link" href="/dual%20list"&gt;dual list&lt;/a&gt; function, this defines both join and meet as operations on normal val lists.&lt;br /&gt;
 &lt;br /&gt;
 There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it &amp;lt;JI&amp;gt;; and in the temperament defined by G^, nothing is tempered out, and we may also call it &amp;lt;1&amp;gt;.&lt;br /&gt;