Hodge dual

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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 W°. Similarly, given a k-multimonzo M, there is a dual (n − k)-multival Mº. The dual may be defined in terms of the 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, ⟨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, 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⟩ = ⟨⟨18 27 18 1 -22 -34]] ⋅ [[6 -4 0 -1 3 -2⟩⟩ = 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

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, …, e_p is the standard basis of prime vals, then e₂ ∧ e₃ ∧ … ∧ e_p 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

Again with a basis of dimension n, suppose we have a k-multival V and wish to find its dual multimonzo M. The elements of V are associated with k-combinations, and of M with (n − k)-combinations, of the basis elements. Because of the symmetry of binomial coefficients, V and M will have the same length. To find M we adjust the signs of V with the following procedure

1. Let C be the k-combinations of the numbers 1 through n in lexicographic order

2. C will have the same length as V and M

3. Sum the numbers in each combination C_i with ⌈k⁄2⌉ to find S_i

4. Multiply the ith element of V by −1^S_i

and then reverse the elements of V.

To find an unknown V from a known M, first reverse M and then adjust the signs.

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 M° as [[6 -4 0 -1 3 -2⟩⟩° = ⟨⟨-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 ∧ 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 page for extracting a normal val list from a multival to get a normal comma list from the multimonzo.