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Wikispaces>genewardsmith **Imported revision 141655607 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 141656005 - 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:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-13 01:37:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>141656005</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">An alternating [[http://en.wikipedia.org/wiki/Multilinear_map|multilinear map]] which is a multilinear function | <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. | 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> | ||
<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>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 | <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 /> | ||
<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. | 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).</body></html></pre></div> | ||
Revision as of 01:37, 13 May 2010
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2010-05-13 01:37:54 UTC.
- The original revision id was 141656005.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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).
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).</body></html>