7-limit symmetrical lattices: Difference between revisions
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-24 17:17:36 UTC</tt>.<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">[[toc|flat]] | <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">[[toc|flat]] | ||
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=Norms and lattices= | =Norms and lattices= | ||
Of the various [[http://mathworld.wolfram.com/VectorNorm.html|norms]] which can be put on [[Monzos and Interval Space|interval space]] which then make the monzos into a lattice, the most useful seem to be the L1 and L2 norms on the coordinates weighted by log2 of the primes. However, in the 5 and 7 limit cases, it is sometimes convenient, when emphasizing symmetry properties, to put a Euclidean norm on //unweighted// monzos, so that | Of the various [[http://mathworld.wolfram.com/VectorNorm.html|norms]] which can be put on [[Monzos and Interval Space|interval space]] which then make the monzos into a lattice, the most useful seem to be the L1 and L2 norms on the coordinates weighted by log2 of the primes. However, in the 5 and 7 limit cases, it is sometimes convenient, when emphasizing symmetry properties, to put a Euclidean norm on //unweighted// monzos, so that | ||
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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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>The Seven Limit Symmetrical Lattices</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Norms and lattices">Norms and lattices</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#The cubic lattice of tetrads">The cubic lattice of tetrads</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Dual lattices">Dual lattices</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Transformations">Transformations</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Articles">Articles</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <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>The Seven Limit Symmetrical Lattices</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Norms and lattices">Norms and lattices</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#The cubic lattice of tetrads">The cubic lattice of tetrads</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Dual lattices">Dual lattices</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Transformations">Transformations</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Articles">Articles</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Norms and lattices"></a><!-- ws:end:WikiTextHeadingRule:0 -->Norms and lattices</h1> | |||
Of the various <a class="wiki_link_ext" href="http://mathworld.wolfram.com/VectorNorm.html" rel="nofollow">norms</a> which can be put on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> which then make the monzos into a lattice, the most useful seem to be the L1 and L2 norms on the coordinates weighted by log2 of the primes. However, in the 5 and 7 limit cases, it is sometimes convenient, when emphasizing symmetry properties, to put a Euclidean norm on <em>unweighted</em> monzos, so that<br /> | Of the various <a class="wiki_link_ext" href="http://mathworld.wolfram.com/VectorNorm.html" rel="nofollow">norms</a> which can be put on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a> which then make the monzos into a lattice, the most useful seem to be the L1 and L2 norms on the coordinates weighted by log2 of the primes. However, in the 5 and 7 limit cases, it is sometimes convenient, when emphasizing symmetry properties, to put a Euclidean norm on <em>unweighted</em> monzos, so that<br /> | ||
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