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7-limit symmetrical lattices: Difference between revisions

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Wikispaces>xenwolf
**Imported revision 141116167 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 141329065 - Original comment: **
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<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:xenwolf|xenwolf]] and made on <tt>2010-05-11 10:50:03 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-12 00:46:15 UTC</tt>.<br>
: The original revision id was <tt>141116167</tt>.<br>
: The original revision id was <tt>141329065</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>
<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">//the following extracted from http://lumma.org/tuning/gws/sevlat.htm//
<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 //unwieghted// monzos, so that
 
|| |e2 e2 e5 e7&gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)
 
When we do this, we can induce a norm on the
 
 


The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as triples of integers (a b c). We can make this into a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]]   
The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as triples of integers (a b c). We can make this into a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]]   
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temperament.</pre></div>
temperament.</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">&lt;html&gt;&lt;head&gt;&lt;title&gt;The Seven Limit Symmetrical Lattices&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;the following extracted from &lt;!-- ws:start:WikiTextUrlRule:87:http://lumma.org/tuning/gws/sevlat.htm --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/gws/sevlat.htm" rel="nofollow"&gt;http://lumma.org/tuning/gws/sevlat.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:87 --&gt;&lt;/em&gt;&lt;br /&gt;
<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;The Seven Limit Symmetrical Lattices&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Of the various &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/VectorNorm.html" rel="nofollow"&gt;norms&lt;/a&gt; which can be put on &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt; 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 &lt;em&gt;unwieghted&lt;/em&gt; monzos, so that&lt;br /&gt;
&lt;br /&gt;
|| |e2 e2 e5 e7&amp;gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)&lt;br /&gt;
&lt;br /&gt;
When we do this, we can induce a norm on the &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as triples of integers (a b c). We can make this into a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattice&lt;/a&gt;  &lt;br /&gt;
The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as triples of integers (a b c). We can make this into a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattice&lt;/a&gt;  &lt;br /&gt;
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