Wedgie/Archived version: Difference between revisions
Wikispaces>guest **Imported revision 199011616 - Original comment: ** |
Wikispaces>guest **Imported revision 199011802 - 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:guest|guest]] and made on <tt>2011-02-05 21: | : This revision was by author [[User:guest|guest]] and made on <tt>2011-02-05 21:36:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>199011802</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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[[math]] | [[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)) | 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]] | [[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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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). Let's consider a specific example. Suppose E19 = <19 30 44 53| is the equal temperament val for septimal 19et, and E31 = <31 49 72 87| is the val for septimal 31et. Then writing intervals multiplicatively, we have | 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). Let's consider a specific example. Suppose E19 = <19 30 44 53| is the equal temperament val for septimal 19et, and E31 = <31 49 72 87| is the val for septimal 31et. Then writing intervals multiplicatively, we have | ||
E19 | [[math]] | ||
(E19\wedge E31)(2,3) = E19(2)E31(3) - E19(3)E31(2) = 19*49 - 31*30 = 1 | |||
[[math]] | |||
We may continue in this way to consider (2,5), (2,7), (3,5), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields <<1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a **bival**. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19^E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone. | We may continue in this way to consider (2,5), (2,7), (3,5), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields <<1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a **bival**. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19^E31 is the same object we were calling "meantone(u,v)" which gives us complexity measurements for meantone. | ||
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<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[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]] | 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 /> | --><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 /> | ||
| Line 47: | Line 49: | ||
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). Let's consider a specific example. Suppose E19 = &lt;19 30 44 53| is the equal temperament val for septimal 19et, and E31 = &lt;31 49 72 87| is the val for septimal 31et. Then writing intervals multiplicatively, we have <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). Let's consider a specific example. Suppose E19 = &lt;19 30 44 53| is the equal temperament val for septimal 19et, and E31 = &lt;31 49 72 87| is the val for septimal 31et. Then writing intervals multiplicatively, we have <br /> | ||
<br /> | <br /> | ||
E19 | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | |||
(E19\wedge E31)(2,3) = E19(2)E31(3) - E19(3)E31(2) = 19*49 - 31*30 = 1&lt;br/&gt;[[math]] | |||
--><script type="math/tex">(E19\wedge E31)(2,3) = E19(2)E31(3) - E19(3)E31(2) = 19*49 - 31*30 = 1</script><!-- ws:end:WikiTextMathRule:1 --><br /> | |||
<br /> | <br /> | ||
We may continue in this way to consider (2,5), (2,7), (3,5), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields &lt;&lt;1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a <strong>bival</strong>. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to <a class="wiki_link" href="/rank%20two%20temperament">rank two temperament</a>s such as <a class="wiki_link" href="/meantone">meantone</a>, trivals to <a class="wiki_link" href="/rank%20three%20temperament">rank three temperament</a>s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19^E31 is the same object we were calling &quot;meantone(u,v)&quot; which gives us complexity measurements for meantone.<br /> | We may continue in this way to consider (2,5), (2,7), (3,5), (3,5), (3,7) and (5,7), and writing them in this alphabetical order yields &lt;&lt;1 4 10 4 13 12||. Here the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a <strong>bival</strong>. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to <a class="wiki_link" href="/rank%20two%20temperament">rank two temperament</a>s such as <a class="wiki_link" href="/meantone">meantone</a>, trivals to <a class="wiki_link" href="/rank%20three%20temperament">rank three temperament</a>s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, E19^E31 is the same object we were calling &quot;meantone(u,v)&quot; which gives us complexity measurements for meantone.<br /> | ||