Wedgie/Archived version: Difference between revisions
Wikispaces>xenwolf **Imported revision 148245059 - Original comment: Wikipedia link replaced by an internal reference (where I moved the Wikipedia link)** |
Wikispaces>guest **Imported revision 199011616 - 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: | : This revision was by author [[User:guest|guest]] and made on <tt>2011-02-05 21:34:53 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>199011616</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
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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: | 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 | [[math]] | ||
f\wedge g = \sum_s sgn(s)f(x_s(1),x_s(2),...,x_s(n))g(x_s(n+1),...,x_s(n+m)) | |||
[[math]] | |||
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 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd. | 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 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd. | ||
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This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[http://en.wikipedia.org/wiki/Greatest_common_divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call //reduced//, and reduced n-vals can be used to give unique names to [[Regular Temperaments|regular temperaments]]. | This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[http://en.wikipedia.org/wiki/Greatest_common_divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call //reduced//, and reduced n-vals can be used to give unique names to [[Regular Temperaments|regular temperaments]]. | ||
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. | 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> | ||
</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="/monzo">monzo</a>s 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="/monzo">monzo</a>s 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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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 /> | 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 /> | <br /> | ||
f | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | |||
f\wedge 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;[[math]] | |||
--><script type="math/tex">f\wedge g = \sum_s sgn(s)f(x_s(1),x_s(2),...,x_s(n))g(x_s(n+1),...,x_s(n+m))</script><!-- ws:end:WikiTextMathRule:0 --><br /> | |||
<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 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.<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 if t is even meaning an even number of transpositions of two numbers will get to t, and -1 if t is odd.<br /> | ||