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'' (n-k)-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual (n-k)-multival Mº. The dual may be defined in terms of the bracket 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'' (n-k)-multimonzo W°. Similarly, given a k-multimonzo M, there is a dual (n-k)-multival Mº. The dual may be defined in terms of the bracket 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 bracket product, ~~<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, W = 612∧441 = ~~<<~~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, M = |0 3 -2 0~~>~~∧|-2 1 -1 1~~>~~ = ||6 -4 0 -1 3 -~~2>>~~. Then ~~<W~~|~~M>~~ equals ~~<<~~18 27 18 1 -22 -34||6 -4 0 -1 3 -~~2>>~~ equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 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|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 or bracket product, ⟨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, W = 612∧441 = {{wedgie|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, M = {{monzo|0 3 -2 0}}∧{{monzo|-2 1 -1 1}} = [[6 -4 0 -1 3 -2⟩⟩. Then ⟨W|M⟩ equals {{wedgie|18 27 18 1 -22 -34}}[[6 -4 0 -1 3 -2⟩⟩ equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 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|interior product]], but then we must fuss about the sign.

=The dual=

Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that ~~<U~~|~~Vº>~~ = U∧V for every k-multival U.

Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual V° of V is simply the k-multimonzo which has the property that ⟨U|V°⟩ = U∧V for every k-multival U.

=Computing the dual=

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=Using the dual=

The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = |0 3 -2 0~~>~~∧|-2 1 -1 1~~>~~, which is ||6 -4 0 -1 3 -~~2>>~~, considered above, we may find the dual Mº as ||6 -4 0 -1 3 -~~2>>º~~ = ~~<<~~-2 -3 -1 0 4 6||. Normalizing this to a wedgie gives ~~<<~~2 3 1 0 -4 -6||, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, ~~W∧Mº~~ = ~~<W~~|~~M>~~ = 1. We can also take a multival, and use the dual to get a corresponding ~~mulitmonzo~~, and then use the same method described on the [[Abstract_regular_temperament|abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.

The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = {{monzo|0 3 -2 0}}∧{{monzo|-2 1 -1 1}}, which is [[6 -4 0 -1 3 -2⟩⟩, considered above, we may find the dual M° as [[6 -4 0 -1 3 -2⟩⟩° = {{wedgie|-2 -3 -1 0 4 6}}. Normalizing this to a wedgie gives {{wedgie|2 3 1 0 -4 -6}}, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, W∧M° = ⟨W|M⟩ = 1. We can also take a multival, and use the dual to get a corresponding multimonzo, and then use the same method described on the [[Abstract_regular_temperament|abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.

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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'' (n-k)-multimonzo Wº. Similarly, given a k-multimonzo M, there is a dual (n-k)-multival Mº. The dual may be defined in terms of the bracket 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'' (n-k)-multimonzo W°. Similarly, given a k-multimonzo M, there is a dual (n-k)-multival Mº. The dual may be defined in terms of the bracket 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 bracket product, &lt;W|M&gt;, 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, W = 612∧441 = &lt;&lt;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, M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt; = ||6 -4 0 -1 3 -2&gt;&gt;. Then &lt;W|M&gt; equals &lt;&lt;18 27 18 1 -22 -34||6 -4 0 -1 3 -2&gt;&gt; equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 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|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 or bracket product, ⟨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, W = 612∧441 = {{wedgie|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, M = {{monzo|0 3 -2 0}}∧{{monzo|-2 1 -1 1}} = [[6 -4 0 -1 3 -2⟩⟩. Then ⟨W|M⟩ equals {{wedgie|18 27 18 1 -22 -34}}[[6 -4 0 -1 3 -2⟩⟩ equals 18*6-27*4+18*0-1*1-22*3+34*2 equals 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|interior product]], but then we must fuss about the sign.
 =The dual=
-Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual Vº of V is simply the k-multimonzo which has the property that &lt;U|Vº&gt; = U∧V for every k-multival U.
+Given a k-multival U and an (n-k)-multival V, where n is the dimension (the number of coefficients, or length) of the vals, then U∧V is an n-multival. But the space of n-multivals is one-dimensional; if e2, e3, ..., ep is the standard basis of prime vals, then e2∧e3∧...∧ep is the sole basis vector for n-multivals. Hence by a slight abuse of notation it can be identified as a single scalar quantity. Given that identification, the dual V° of V is simply the k-multimonzo which has the property that ⟨U|V°⟩ = U∧V for every k-multival U.
 =Computing the dual=
@@ Line 23: / Line 23: @@
 =Using the dual=
-The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = |0 3 -2 0&gt;∧|-2 1 -1 1&gt;, which is ||6 -4 0 -1 3 -2&gt;&gt;, considered above, we may find the dual Mº as ||6 -4 0 -1 3 -2&gt;&gt;º = &lt;&lt;-2 -3 -1 0 4 6||. Normalizing this to a wedgie gives &lt;&lt;2 3 1 0 -4 -6||, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, W∧Mº = &lt;W|M&gt; = 1. We can also take a multival, and use the dual to get a corresponding mulitmonzo, and then use the same method described on the [[Abstract_regular_temperament|abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.
+The dual allows one to find the wedgie, which is a normalized multival, by wedging together monzos and then taking the dual. For instance from M = {{monzo|0 3 -2 0}}∧{{monzo|-2 1 -1 1}}, which is [[6 -4 0 -1 3 -2⟩⟩, considered above, we may find the dual M° as [[6 -4 0 -1 3 -2⟩⟩° = {{wedgie|-2 -3 -1 0 4 6}}. Normalizing this to a wedgie gives {{wedgie|2 3 1 0 -4 -6}}, the wedgie for bug temperament. Then if W is the wedgie for ennealimmal considered above, W∧M° = ⟨W|M⟩ = 1. We can also take a multival, and use the dual to get a corresponding multimonzo, and then use the same method described on the [[Abstract_regular_temperament|abstract regular temperament]] page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.