Interior product: Difference between revisions
Wikispaces>genewardsmith **Imported revision 278559412 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278575980 - 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>2011-11-23 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-23 15:31:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278575980</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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For example, suppose W = <<6 -7 -2 -25 -20 15||, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely |1 0 0 0> and |-1 1 1 1>, then wedging them together gives the bimonzo ||1 1 -1 0 0 0>>. The dot product with W is <<6 -7 -2 -25 -20 15||1 1 -1 0 0 0>>, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0>, |-1 1 1 1>) = 1. The fact that the result is ∓1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament. | For example, suppose W = <<6 -7 -2 -25 -20 15||, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely |1 0 0 0> and |-1 1 1 1>, then wedging them together gives the bimonzo ||1 1 -1 0 0 0>>. The dot product with W is <<6 -7 -2 -25 -20 15||1 1 -1 0 0 0>>, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0>, |-1 1 1 1>) = 1. The fact that the result is ∓1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament. | ||
If W is a multival of rank n and m is a monzo of the same prime limit p, then form a list of (n-1) tuples of primes less than or equal to p in alphabetical order. Taking these in order, the ith element of W∨m, which we may also write W∨q where q is the rational number with monzo m, will be W(s1, s2, s3 ... s_(n-1), q), where [s1, s2, ..., s_(n-1)] is the ith tuple on the list of (n-1)-tuples of primes. This will result in W∨m, a multival of rank n-1. For instance, let | If W is a multival of rank n and m is a monzo of the same prime limit p, then form a list of (n-1) tuples of primes less than or equal to p in alphabetical order. Taking these in order, the ith element of W∨m, which we may also write W∨q where q is the rational number with monzo m, will be W(s1, s2, s3 ... s_(n-1), q), where [s1, s2, ..., s_(n-1)] is the ith tuple on the list of (n-1)-tuples of primes. This will result in W∨m, a multival of rank n-1. For instance, let Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19|||, the wedgie for 11-limit marvel temperamment. To find Marv∨441/440, we form the list [[2, 3], [2, 5], [2, 7], [2, 11], [3, 5], [3, 7], [3, 11], [5, 7], [5, 11], [7, 11]]. The first element of Marv∨441/440 will be Marv(2, 3, 441/440), the second element Marv(2, 5, 441/440) and so on down to the last element, Marv(7, 11, 441/440). This gives us <<6 -7 -2 15 -25 -20 3 15 59 49||, which is the wedgie for 11-limit miracle. The interior product has added a comma to marvel to produce miracle. | ||
If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign. | If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign. | ||
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For example, suppose W = &lt;&lt;6 -7 -2 -25 -20 15||, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely |1 0 0 0&gt; and |-1 1 1 1&gt;, then wedging them together gives the bimonzo ||1 1 -1 0 0 0&gt;&gt;. The dot product with W is &lt;&lt;6 -7 -2 -25 -20 15||1 1 -1 0 0 0&gt;&gt;, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0&gt;, |-1 1 1 1&gt;) = 1. The fact that the result is ∓1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.<br /> | For example, suppose W = &lt;&lt;6 -7 -2 -25 -20 15||, the wedgie for 7-limit miracle. If our two monzos are the monzos for 2 and 15/14, namely |1 0 0 0&gt; and |-1 1 1 1&gt;, then wedging them together gives the bimonzo ||1 1 -1 0 0 0&gt;&gt;. The dot product with W is &lt;&lt;6 -7 -2 -25 -20 15||1 1 -1 0 0 0&gt;&gt;, which is 6 - 7 - (-2) = 1, so W(2, 15/14) = W(|1 0 0 0&gt;, |-1 1 1 1&gt;) = 1. The fact that the result is ∓1 tells us that 2 and 15/14 can serve as a pair of generators for miracle; if the absolute value of the n-map is N, then the monzos it was applied to, when tempered, generate a subgroup of index N of the whole group of intervals of the temperament.<br /> | ||
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If W is a multival of rank n and m is a monzo of the same prime limit p, then form a list of (n-1) tuples of primes less than or equal to p in alphabetical order. Taking these in order, the ith element of W∨m, which we may also write W∨q where q is the rational number with monzo m, will be W(s1, s2, s3 ... s_(n-1), q), where [s1, s2, ..., s_(n-1)] is the ith tuple on the list of (n-1)-tuples of primes. This will result in W∨m, a multival of rank n-1. For instance, let | If W is a multival of rank n and m is a monzo of the same prime limit p, then form a list of (n-1) tuples of primes less than or equal to p in alphabetical order. Taking these in order, the ith element of W∨m, which we may also write W∨q where q is the rational number with monzo m, will be W(s1, s2, s3 ... s_(n-1), q), where [s1, s2, ..., s_(n-1)] is the ith tuple on the list of (n-1)-tuples of primes. This will result in W∨m, a multival of rank n-1. For instance, let Marv = &lt;&lt;&lt;1 2 -3 -2 1 -4 -5 12 9 -19|||, the wedgie for 11-limit marvel temperamment. To find Marv∨441/440, we form the list [[2, 3], [2, 5], [2, 7], [2, 11], [3, 5], [3, 7], [3, 11], [5, 7], [5, 11], [7, 11]]. The first element of Marv∨441/440 will be Marv(2, 3, 441/440), the second element Marv(2, 5, 441/440) and so on down to the last element, Marv(7, 11, 441/440). This gives us &lt;&lt;6 -7 -2 15 -25 -20 3 15 59 49||, which is the wedgie for 11-limit miracle. The interior product has added a comma to marvel to produce miracle.<br /> | ||
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If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign.<br /> | If we like, we can take the wedge product m∨W from the front by using W(q, s1, s2, s3 ... s_(n-1)) instead of W(s1, s2, s3 ... s_(n-1), q), but this can only lead to a difference in sign.<br /> | ||