Hodge dual: Difference between revisions

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Given ''n'' basis elements (i.e. the number of primes in a prime limit) and a ''k''-multival '''W''' in this basis, there is a ''dual'' {{nowrap|(''n'' − ''k'')}}-multimonzo '''W'''°. Similarly, given a k-multimonzo '''M''', there is a dual {{nowrap|(''n'' − ''k'')}}-multival '''M'''º. The dual may be defined in terms of the {{w|dot product}} relating multivals and multimonzos, which we discuss first.

Given ''n'' basis elements (i.e. the number of primes in a prime limit) and a ''k''-multival '''W''' in this basis, there is a ''dual'' {{nowrap|(''n'' − ''k'')}}-multimonzo '''W'''°. Similarly, given a k-multimonzo '''M''', there is a dual {{nowrap|(''n'' − ''k'')}}-multival '''M'''º. The dual may be defined in terms of bracket product (which is similar to the {{w|dot product}}) relating multivals and multimonzos, which we discuss first.

== The bracket ==

Given a ''k''-multival '''W''' and a ''k''-multimonzo '''M''' (in which we may include sums of ''k''-fold wedge products of vals or monzos), the bracket ~~or dot~~ product, {{vmp|'''W'''|'''M'''}}, acts just the same as the ~~dot~~ product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, {{nowrap| '''W''' {{=}} 612 ∧ 441}} {{nowrap|{{=}} {{multival| 18 27 18 1 -22 -34 }} }}, which is the wedgie for ennealimmal temperament, and is a 2-val. Then suppose we take the wedge product of the monzos for 27/25 and 21/20, {{nowrap| '''M''' {{=}} {{monzo| 0 3 -2 0 }} ∧ {{monzo| -2 1 -1 1}} }} {{nowrap| {{=}} {{multimonzo| 6 -4 0 -1 3 -2 }} }}. Then {{nowrap|{{vmp|'''W'''|'''M'''}} {{=}} {{multival| 18 27 18 1 -22 -34 }}{{dot}}{{multimonzo| 6 -4 0 -1 3 -2 }} }} {{nowrap|{{=}} 18 × 6 − 27 × 4 + 18 × 0 − 1 × 1 − 22 × 3 + 34 × 2}} = 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.

Given a ''k''-multival '''W''' and a ''k''-multimonzo '''M''' (in which we may include sums of ''k''-fold wedge products of vals or monzos), the bracket product, {{vmp|'''W'''|'''M'''}}, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, {{nowrap| '''W''' {{=}} 612 ∧ 441}} {{nowrap|{{=}} {{multival| 18 27 18 1 -22 -34 }} }}, which is the wedgie for ennealimmal temperament, and is a 2-val. Then suppose we take the wedge product of the monzos for 27/25 and 21/20, {{nowrap| '''M''' {{=}} {{monzo| 0 3 -2 0 }} ∧ {{monzo| -2 1 -1 1}} }} {{nowrap| {{=}} {{multimonzo| 6 -4 0 -1 3 -2 }} }}. Then {{nowrap|{{vmp|'''W'''|'''M'''}} {{=}} {{multival| 18 27 18 1 -22 -34 }}{{dot}}{{multimonzo| 6 -4 0 -1 3 -2 }} }} {{nowrap|{{=}} 18 × 6 − 27 × 4 + 18 × 0 − 1 × 1 − 22 × 3 + 34 × 2}} = 1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.

== The dual ==

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 {{inacc}}
-Given ''n'' basis elements (i.e. the number of primes in a prime limit) and a ''k''-multival '''W''' in this basis, there is a ''dual'' {{nowrap|(''n'' − ''k'')}}-multimonzo '''W'''°. Similarly, given a k-multimonzo '''M''', there is a dual {{nowrap|(''n'' − ''k'')}}-multival '''M'''º. The dual may be defined in terms of the {{w|dot product}} relating multivals and multimonzos, which we discuss first.
+Given ''n'' basis elements (i.e. the number of primes in a prime limit) and a ''k''-multival '''W''' in this basis, there is a ''dual'' {{nowrap|(''n'' − ''k'')}}-multimonzo '''W'''°. Similarly, given a k-multimonzo '''M''', there is a dual {{nowrap|(''n'' − ''k'')}}-multival '''M'''º. The dual may be defined in terms of bracket product (which is similar to the {{w|dot product}}) relating multivals and multimonzos, which we discuss first.
 == The bracket ==
-Given a ''k''-multival '''W''' and a ''k''-multimonzo '''M''' (in which we may include sums of ''k''-fold wedge products of vals or monzos), the bracket or dot product, {{vmp|'''W'''|'''M'''}}, acts just the same as the dot product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, {{nowrap| '''W''' {{=}} 612 ∧ 441}} {{nowrap|{{=}} {{multival| 18 27 18 1 -22 -34 }} }}, which is the wedgie for ennealimmal temperament, and is a 2-val. Then suppose we take the wedge product of the monzos for 27/25 and 21/20, {{nowrap| '''M''' {{=}} {{monzo| 0 3 -2 0 }} ∧ {{monzo| -2 1 -1 1}} }} {{nowrap| {{=}} {{multimonzo| 6 -4 0 -1 3 -2 }} }}. Then {{nowrap|{{vmp|'''W'''|'''M'''}} {{=}} {{multival| 18 27 18 1 -22 -34 }}{{dot}}{{multimonzo| 6 -4 0 -1 3 -2 }} }} {{nowrap|{{=}} 18 × 6 − 27 × 4 + 18 × 0 − 1 × 1 − 22 × 3 + 34 × 2}} =&nbsp;1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.
+Given a ''k''-multival '''W''' and a ''k''-multimonzo '''M''' (in which we may include sums of ''k''-fold wedge products of vals or monzos), the bracket product, {{vmp|'''W'''|'''M'''}}, acts just the same as the bracket product of a val with a monzo. Suppose, for example, we take the wedge product of the 7-limit patent vals 612 and 441, {{nowrap| '''W''' {{=}} 612 ∧ 441}} {{nowrap|{{=}} {{multival| 18 27 18 1 -22 -34 }} }}, which is the wedgie for ennealimmal temperament, and is a 2-val. Then suppose we take the wedge product of the monzos for 27/25 and 21/20, {{nowrap| '''M''' {{=}} {{monzo| 0 3 -2 0 }} ∧ {{monzo| -2 1 -1 1}} }} {{nowrap| {{=}} {{multimonzo| 6 -4 0 -1 3 -2 }} }}. Then {{nowrap|{{vmp|'''W'''|'''M'''}} {{=}} {{multival| 18 27 18 1 -22 -34 }}{{dot}}{{multimonzo| 6 -4 0 -1 3 -2 }} }} {{nowrap|{{=}} 18 × 6 − 27 × 4 + 18 × 0 − 1 × 1 − 22 × 3 + 34 × 2}} =&nbsp;1. In fact, we can compute the same result just using the vals and monzos we wedge together to get the bivals and bimonzos, by taking the determinant of the matrix which is the product of the matrix with rows the vals with the matrix with monzos the columns. We can also define it via the [[interior product]], but then we must fuss about the sign.
 == The dual ==