Normal forms: 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:<br>

: This revision was by author [[User:~~mbattaglia1~~|~~mbattaglia1~~]] and made on <tt>~~2012~~-07-~~31 03~~:19:59 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:56:30 UTC</tt>.<br>

: The original revision id was <tt>~~355646652~~</tt>.<br>

: The original revision id was <tt>509655008</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>

<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">[[toc]]

[[image:mathhazard.jpg align="center"]]

An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is [[http://en.wikipedia.org/wiki/Hermite_normal_form|Hermite normal form]], and by using Hermite normal form we may define normalized forms of lists of [[Harmonic Limit|p-limit]] musical intervals (or [[Monzos and Interval Space|monzos]]) or lists of [[Vals and Tuning Space|vals]], the normal interval (or monzo) list and the normal val list.

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<div style="margin-left: 2em;"><a href="#x-Maple code">Maple code</a></div>

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An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hermite_normal_form" rel="nofollow">Hermite normal form</a>, and by using Hermite normal form we may define normalized forms of lists of <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> musical intervals (or <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a>) or lists of <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a>, the normal interval (or monzo) list and the normal val list.<br />

<br />

<div style="text-align: center"><img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /></div>An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hermite_normal_form" rel="nofollow">Hermite normal form</a>, and by using Hermite normal form we may define normalized forms of lists of <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> musical intervals (or <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a>) or lists of <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">vals</a>, the normal interval (or monzo) list and the normal val list.<br />

<br />

There are slightly different definitions of Hermite normal form in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.<br />

@@ 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:mbattaglia1|mbattaglia1]] and made on <tt>2012-07-31 03:19:59 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-18 13:56:30 UTC</tt>.<br>
-: The original revision id was <tt>355646652</tt>.<br>
+: The original revision id was <tt>509655008</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>
 <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">[[toc]]
+[[image:mathhazard.jpg align="center"]]
 An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is [[http://en.wikipedia.org/wiki/Hermite_normal_form|Hermite normal form]], and by using Hermite normal form we may define normalized forms of lists of [[Harmonic Limit|p-limit]] musical intervals (or [[Monzos and Interval Space|monzos]]) or lists of [[Vals and Tuning Space|vals]], the normal interval (or monzo) list and the normal val list.
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 &lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x-Maple code"&gt;Maple code&lt;/a&gt;&lt;/div&gt;
 &lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:10 --&gt;An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hermite_normal_form" rel="nofollow"&gt;Hermite normal form&lt;/a&gt;, and by using Hermite normal form we may define normalized forms of lists of &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; musical intervals (or &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt;) or lists of &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;vals&lt;/a&gt;, the normal interval (or monzo) list and the normal val list.&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:55:&amp;lt;div style=&amp;quot;text-align: center&amp;quot;&amp;gt;&amp;lt;img src=&amp;quot;/file/view/mathhazard.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt;&amp;lt;/div&amp;gt; --&gt;&lt;div style="text-align: center"&gt;&lt;img src="/file/view/mathhazard.jpg" alt="mathhazard.jpg" title="mathhazard.jpg" /&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextLocalImageRule:55 --&gt;An integral matrix is a matrix whose entries are all integers. An important normalized form for integral matrices is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Hermite_normal_form" rel="nofollow"&gt;Hermite normal form&lt;/a&gt;, and by using Hermite normal form we may define normalized forms of lists of &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; musical intervals (or &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt;) or lists of &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;vals&lt;/a&gt;, the normal interval (or monzo) list and the normal val list.&lt;br /&gt;
 &lt;br /&gt;
 There are slightly different definitions of Hermite normal form in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.&lt;br /&gt;