Wedgie/Archived version: 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:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 02:11:00 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 03:40:22 UTC</tt>.<br>

: The original revision id was <tt>~~141658795~~</tt>.<br>

: The original revision id was <tt>141665907</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>

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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

One use for such things is as "machines" for measuring complexity. If we consider 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 72 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, meantone(2,7)=10. With 3 as a period and 3/2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=13. Finally, with 5 as a period and 7/5 as a generator we get meantone(5,7)=12. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.

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>

Given an n-map f and an m-map g we may define a new (n+m)-map, the [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge product]] of f and g, written f^g, as follows:

f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))

where the sum is taken over S(n,m), the set of all [[http://en.wikipedia.org/wiki/Permutation|permutations]] of the first n+m integers which are an [[http://en.wikipedia.org/wiki/%28p,q%29_shuffle|(n,m) shuffles]], and sgn(t) is the [[http://en.wikipedia.org/wiki/Parity_of_a_permutation|parity of the permutation]] t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.

If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).

</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"><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 />

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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 />~~

One use for such things is as &quot;machines&quot; for measuring complexity. If we consider 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 72 in this system. Now consider a 2-map &quot;meantone(u,v)&quot; 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, meantone(2,7)=10. With 3 as a period and 3/2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=13. Finally, with 5 as a period and 7/5 as a generator we get meantone(5,7)=12. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.<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></pre></div>

<br />

Given an n-map f and an m-map g we may define a new (n+m)-map, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">wedge product</a> of f and g, written f^g, as follows:<br />

<br />

f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))<br />

<br />

where the sum is taken over S(n,m), the set of all <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation" rel="nofollow">permutations</a> of the first n+m integers which are an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/%28p,q%29_shuffle" rel="nofollow">(n,m) shuffles</a>, and sgn(t) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Parity_of_a_permutation" rel="nofollow">parity of the permutation</a> t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.<br />

<br />

If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).</body></html></pre></div>

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 02:11:00 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 03:40:22 UTC</tt>.<br>
-: The original revision id was <tt>141658795</tt>.<br>
+: The original revision id was <tt>141665907</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 10: / Line 10: @@
 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
+One use for such things is as "machines" for measuring complexity. If we consider 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 72 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, meantone(2,7)=10. With 3 as a period and 3/2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=13. Finally, with 5 as a period and 7/5 as a generator we get meantone(5,7)=12. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.
-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>
+Given an n-map f and an m-map g we may define a new (n+m)-map, the [[http://en.wikipedia.org/wiki/Exterior_algebra|wedge product]] of f and g, written f^g, as follows:
+f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))
+where the sum is taken over S(n,m), the set of all [[http://en.wikipedia.org/wiki/Permutation|permutations]] of the first n+m integers which are an [[http://en.wikipedia.org/wiki/%28p,q%29_shuffle|(n,m) shuffles]], and sgn(t) is the [[http://en.wikipedia.org/wiki/Parity_of_a_permutation|parity of the permutation]] t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.
+If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).
+</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;
@@ Line 17: / Line 26: @@
 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;
+One use for such things is as &amp;quot;machines&amp;quot; for measuring complexity. If we consider 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 72 in this system. Now consider a 2-map &amp;quot;meantone(u,v)&amp;quot; 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, meantone(2,7)=10. With 3 as a period and 3/2 as a generator, we get meantone(3,5)=4 and meantone(3,7)=13. Finally, with 5 as a period and 7/5 as a generator we get meantone(5,7)=12. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.&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>
+&lt;br /&gt;
+Given an n-map f and an m-map g we may define a new (n+m)-map, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow"&gt;wedge product&lt;/a&gt; of f and g, written f^g, as follows:&lt;br /&gt;
+&lt;br /&gt;
+f^g = sum_s sgn(s) f(x_s(1), x_s(2)...x_s_n)g(x_s(n+1)...x_s(n+m))&lt;br /&gt;
+&lt;br /&gt;
+where the sum is taken over S(n,m), the set of all &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Permutation" rel="nofollow"&gt;permutations&lt;/a&gt; of the first n+m integers which are an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/%28p,q%29_shuffle" rel="nofollow"&gt;(n,m) shuffles&lt;/a&gt;, and sgn(t) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Parity_of_a_permutation" rel="nofollow"&gt;parity of the permutation&lt;/a&gt; t, which is +1 is even meaning an even number of transpositions of two numbers will get to t, and -1 if it is odd.&lt;br /&gt;
+&lt;br /&gt;
+If f and g are both vals (1-maps) then this becomes especially easy: f^g(u,v) = f(u)g(v) - f(v)g(u).&lt;/body&gt;&lt;/html&gt;</pre></div>