Wedgie/Archived version: Difference between revisions
Wikispaces>genewardsmith **Imported revision 294106366 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 294106622 - 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: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-21 12:36:11 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>294106622</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 taking a certain number n of [[ | <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 [[|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). | 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). | ||
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These reduced n-vals, and particularly reduced bivals, are called **wedgies**, and the fact that they are reduced both makes the name unique and tells us that wedgies are [[http://en.wikipedia.org/wiki/Projective_space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = <24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called //contorted//. Wedgies do not name or signify contorted temperaments.</pre></div> | These reduced n-vals, and particularly reduced bivals, are called **wedgies**, and the fact that they are reduced both makes the name unique and tells us that wedgies are [[http://en.wikipedia.org/wiki/Projective_space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, E24 = <24 38 56| is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called //contorted//. Wedgies do not name or signify contorted temperaments.</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 taking a certain number n of <a class="wiki_link" href="/ | <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 [[|val]]. 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 /> | 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 <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 /> | ||