Wedgie/Archived version: Difference between revisions
Wikispaces>genewardsmith **Imported revision 294106622 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 294106690 - 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>2012-01-21 12:36: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-21 12:36:34 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>294106690</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> | ||
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<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. | <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 [[ | The simplest kind of n-map is the 1-map, or [[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 product/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 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 septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, 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 if we take 5 as a period we find that four 3/2s give 5, so 5^(1/4) or equivalently 3/2 is the basic period. Using 3/2 as a period and 9/8 as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5^(1/4) gives us 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. | 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 septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, 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 if we take 5 as a period we find that four 3/2s give 5, so 5^(1/4) or equivalently 3/2 is the basic period. Using 3/2 as a period and 9/8 as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5^(1/4) gives us 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. | ||
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<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 <a class="wiki_link" href="/monzos">monzos</a> 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 /> | <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 <a class="wiki_link" href="/monzos">monzos</a> 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 | The simplest kind of n-map is the 1-map, or <a class="wiki_link" href="/val">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 product/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 /> | <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 septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, 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 if we take 5 as a period we find that four 3/2s give 5, so 5^(1/4) or equivalently 3/2 is the basic period. Using 3/2 as a period and 9/8 as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5^(1/4) gives us 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 /> | 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 septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, 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 if we take 5 as a period we find that four 3/2s give 5, so 5^(1/4) or equivalently 3/2 is the basic period. Using 3/2 as a period and 9/8 as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5^(1/4) gives us 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 /> | ||