7-limit symmetrical lattices: Difference between revisions
Wikispaces>clumma **Imported revision 245784087 - Original comment: spelling and linebreaks** |
Wikispaces>genewardsmith **Imported revision 245792679 - Original comment: ** |
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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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-13 17:02:22 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>245792679</tt>.<br> | ||
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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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">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 // | <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">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 | ||
|| |e2 e2 e5 e7> || = sqrt(e2^2 + e3^3 + e5^2 + e7^2) | || |e2 e2 e5 e7> || = sqrt(e2^2 + e3^3 + e5^2 + e7^2) | ||
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* [[http://x31eq.com/lattice.htm#7limit|Octave Equivalent Music Lattices]] by Graham Breed [[http://www.webcitation.org/5xeJ48wUh|Permalink]]</pre></div> | * [[http://x31eq.com/lattice.htm#7limit|Octave Equivalent Music Lattices]] by Graham Breed [[http://www.webcitation.org/5xeJ48wUh|Permalink]]</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>The Seven Limit Symmetrical Lattices</title></head><body>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> | <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>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 /> | ||
<br /> | <br /> | ||
|| |e2 e2 e5 e7&gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)<br /> | || |e2 e2 e5 e7&gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)<br /> | ||