Interior product: Difference between revisions
Wikispaces>genewardsmith **Imported revision 302946056 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 310761664 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-13 19:50:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>310761664</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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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 temperament. 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 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 temperament. 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. We can also define the interior product of W with a multimonzo M of rank r < n, by forming a list of (n-r)-tuples of primes in alphabetical order, wedging these together with M, and taking the dot product with W to get a coefficient of W∨M. | ||
=Applications= | =Applications= | ||
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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 Marv = &lt;&lt;&lt;1 2 -3 -2 1 -4 -5 12 9 -19|||, the wedgie for 11-limit marvel temperament. 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 /> | 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 temperament. 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 /> | ||
<br /> | <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 /> | 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. We can also define the interior product of W with a multimonzo M of rank r &lt; n, by forming a list of (n-r)-tuples of primes in alphabetical order, wedging these together with M, and taking the dot product with W to get a coefficient of W∨M.<br /> | ||
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