Wedgie/Archived version: Difference between revisions

Revision as of 02:11, 13 May 2010

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This revision was by author genewardsmith and made on 2010-05-13 02:11:00 UTC.

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An alternating [[http://en.wikipedia.org/wiki/Multilinear_map|multilinear map]] which is a multilinear function taking a certain number n of monzos as arguments and returning an integer as a value we may call an **n-map**. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory. 

The simplest kind of n-map is the 1-map, or [[Vals and Tuning Space|val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [[http://en.wikipedia.org/wiki/Group_homomorphism|group homomorphism]] and a [[http://mathworld.wolfram.com/ModuleHomomorphism.html|module homomorphism]]. Vals are [[http://en.wikipedia.org/wiki/Linear_map|linear]]: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).

One use for such things is as "machines" for measuring complexity. If we consdier the 1-map which is the val for 11-limit 31et, we find we have <31 49 72 87 107|. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 5 in this system. Now consider a 2-map meantone(u,v) which tells us, roughly speaking, how many generator steps it takes to get to v assuming u is being used as a period in spetimal meantone. Using 2 as a generator we can take (the approximate) 3/2 as a period, in which case we have
meantone(2,3)=1, meantone(2,5)=4, meatone(2,7)=10. With 3 as a period and 2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=

Original HTML content:

<html><head><title>Wedgies and Multivals</title></head><body>An alternating <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multilinear_map" rel="nofollow">multilinear map</a> which is a multilinear function taking a certain number n of monzos as arguments and returning an integer as a value we may call an <strong>n-map</strong>. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory. <br />
<br />
The simplest kind of n-map is the 1-map, or <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a>. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow">group homomorphism</a> and a <a class="wiki_link_ext" href="http://mathworld.wolfram.com/ModuleHomomorphism.html" rel="nofollow">module homomorphism</a>. Vals are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Linear_map" rel="nofollow">linear</a>: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).<br />
<br />
One use for such things is as &quot;machines&quot; for measuring complexity. If we consdier the 1-map which is the val for 11-limit 31et, we find we have &lt;31 49 72 87 107|. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 5 in this system. Now consider a 2-map meantone(u,v) which tells us, roughly speaking, how many generator steps it takes to get to v assuming u is being used as a period in spetimal meantone. Using 2 as a generator we can take (the approximate) 3/2 as a period, in which case we have<br />
meantone(2,3)=1, meantone(2,5)=4, meatone(2,7)=10. With 3 as a period and 2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=</body></html>

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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:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 01:37:54 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 02:11:00 UTC</tt>.<br>
-: The original revision id was <tt>141656005</tt>.<br>
+: The original revision id was <tt>141658795</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>
@@ Line 8: / Line 8: @@
 <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">An alternating [[http://en.wikipedia.org/wiki/Multilinear_map|multilinear map]] which is a multilinear function taking a certain number n of monzos as arguments and returning an integer as a value we may call an **n-map**. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.
-The simplest kind of n-map is the 1-map, or [[Vals and Tuning Space|val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [[http://en.wikipedia.org/wiki/Group_homomorphism|group homomorphism]] and a [[http://mathworld.wolfram.com/ModuleHomomorphism.html|module homomorphism]]. Vals are [[http://en.wikipedia.org/wiki/Linear_map|linear]]: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).</pre></div>
+The simplest kind of n-map is the 1-map, or [[Vals and Tuning Space|val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [[http://en.wikipedia.org/wiki/Group_homomorphism|group homomorphism]] and a [[http://mathworld.wolfram.com/ModuleHomomorphism.html|module homomorphism]]. Vals are [[http://en.wikipedia.org/wiki/Linear_map|linear]]: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).
+One use for such things is as "machines" for measuring complexity. If we consdier the 1-map which is the val for 11-limit 31et, we find we have &lt;31 49 72 87 107|. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 5 in this system. Now consider a 2-map meantone(u,v) which tells us, roughly speaking, how many generator steps it takes to get to v assuming u is being used as a period in spetimal meantone. Using 2 as a generator we can take (the approximate) 3/2 as a period, in which case we have
+meantone(2,3)=1, meantone(2,5)=4, meatone(2,7)=10. With 3 as a period and 2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=</pre></div>
 <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;Wedgies and Multivals&lt;/title&gt;&lt;/head&gt;&lt;body&gt;An alternating &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multilinear_map" rel="nofollow"&gt;multilinear map&lt;/a&gt; which is a multilinear function taking a certain number n of monzos as arguments and returning an integer as a value we may call an &lt;strong&gt;n-map&lt;/strong&gt;. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory. &lt;br /&gt;
 &lt;br /&gt;
-The simplest kind of n-map is the 1-map, or &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt;. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow"&gt;group homomorphism&lt;/a&gt; and a &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/ModuleHomomorphism.html" rel="nofollow"&gt;module homomorphism&lt;/a&gt;. Vals are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Linear_map" rel="nofollow"&gt;linear&lt;/a&gt;: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).&lt;/body&gt;&lt;/html&gt;</pre></div>
+The simplest kind of n-map is the 1-map, or &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt;. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow"&gt;group homomorphism&lt;/a&gt; and a &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/ModuleHomomorphism.html" rel="nofollow"&gt;module homomorphism&lt;/a&gt;. Vals are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Linear_map" rel="nofollow"&gt;linear&lt;/a&gt;: if you take the product of two p-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions f(u,v), linear for u fixing v, and linear for v fixing u, and alternating. meaning that f(u,u)=0 and f(u,v)=-f(v,u).&lt;br /&gt;
+&lt;br /&gt;
+One use for such things is as &amp;quot;machines&amp;quot; for measuring complexity. If we consdier the 1-map which is the val for 11-limit 31et, we find we have &amp;lt;31 49 72 87 107|. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 5 in this system. Now consider a 2-map meantone(u,v) which tells us, roughly speaking, how many generator steps it takes to get to v assuming u is being used as a period in spetimal meantone. Using 2 as a generator we can take (the approximate) 3/2 as a period, in which case we have&lt;br /&gt;
+meantone(2,3)=1, meantone(2,5)=4, meatone(2,7)=10. With 3 as a period and 2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=&lt;/body&gt;&lt;/html&gt;</pre></div>

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