Meet and join: Difference between revisions
Wikispaces>genewardsmith **Imported revision 535775446 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 535775600 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2014-12-21 22: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-12-21 22:50:22 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>535775600</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 14: | Line 14: | ||
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, 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. | 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. | ||
=Examples= | =Examples= | ||
| Line 24: | Line 24: | ||
Meantone⋎Marvel = 31, Meantone⋏Marvel = <225/224> | Meantone⋎Marvel = 31, Meantone⋏Marvel = <225/224> | ||
Meantone⋎Magic = | Meantone⋎Magic = G, the maximal temperament. | ||
Meantone⋏Magic = <225/224>. | 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 = | 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 = G, then A⋏B is represented by A∧B.</pre></div> | ||
<h4>Original HTML content:</h4> | <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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1> | <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><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1> | ||
| Line 36: | Line 36: | ||
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 /> | 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 /> | <br /> | ||
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. <br /> | 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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:4 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:4 -->Examples</h1> | ||
| Line 46: | Line 46: | ||
Meantone⋎Marvel = 31, Meantone⋏Marvel = &lt;225/224&gt;<br /> | Meantone⋎Marvel = 31, Meantone⋏Marvel = &lt;225/224&gt;<br /> | ||
<br /> | <br /> | ||
Meantone⋎Magic = | Meantone⋎Magic = G, the maximal temperament.<br /> | ||
Meantone⋏Magic = &lt;225/224&gt;.<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 = | 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 = G, then A⋏B is represented by A∧B.</body></html></pre></div> | ||