Interior product

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The //interior product// is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or [[Wedgies and Multivals|n-map]], a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).

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 W = <<<1 2 -3 -2 1 -4 -5 12 9 -19|||, the wedgie for 11-limit marvel temperamment. To find W∨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 W∨441/440 will be W(2, 3, 441/440), the second element W(2, 5, 441/440) and so on down to the last element, W(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.

=Applications=
One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.

Another application is the use of the interior product to define the intervals of the [[abstract regular temperament]] given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q.

Original HTML content:

<html><head><title>Interior product</title></head><body>The <em>interior product</em> is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or <a class="wiki_link" href="/Wedgies%20and%20Multivals">n-map</a>, a multival of rank n induces on a list of n monzos. Let W be a multival of rank n, and m1, m2, ..., mn n monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of n vals, producing the multimonzo M. Treating both M and W as ordinary vectors, take the dot product. This is the value of W(m1, m2, ..., mn).<br />
<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 />
<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 W = &lt;&lt;&lt;1 2 -3 -2 1 -4 -5 12 9 -19|||, the wedgie for 11-limit marvel temperamment. To find W∨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 W∨441/440 will be W(2, 3, 441/440), the second element W(2, 5, 441/440) and so on down to the last element, W(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 />
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 />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:0 -->Applications</h1>
One very useful application is testing whether a wedgie defines a temperament which tempers out a particular comma. Any interval other than 1 is tempered out by the temperament defined by a rank r wedgie if the rank r-1 multival obtained by taking the interior product of the wedgie with the interval is the zero multival--that is, if all the coefficients are zero.<br />
<br />
Another application is the use of the interior product to define the intervals of the <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> given by a wedgie W. In this case, we use W∨q to define a multival which represents the tempered interval which q is tempered to. For this to make sense, we need a way to define the tuning for such multivals, which can be done in a variety of ways. One is as follows: let S be an element of tuning space defining a tuning for the abstract regular temperament denoted by W, and T a truncated version of S where S is shortened to only the first r commas, where r is the rank of W. Form the matrix [W∨2, W∨3, ... W∨R], where R is the r-th prime. Let U be the transpose of the pseudoinverse of this matrix, and let V = T∙U (the matrix product.) Then for any multival W∨q in the abstract regular temperament, the dot product (W∨q).V gives the tuning of W∨q.</body></html>