Tenney–Euclidean metrics: Difference between revisions

Suppose now A is a matrix whose rows are vals defining a ''p''-limit regular temperament. Then the corresponding weighted matrix is V = AW. The [[Tenney-Euclidean Tuning|TE tuning]] projection matrix is then V⁺V, where V⁺ is the [[Tenney-Euclidean Tuning|pseudoinverse]]. If the rows of V (or equivalently, A) are linearly independent, then we have V⁺ = V^T(VV^T)^-1, where V^T denotes the transpose. In terms of vals, the tuning projection matrix is P = V⁺V = V^T(VV^T)^-1V = WA^T(AW²A^T)^-1AW. P is a [[Wikipedia: Positive-definite matrix|positive semidefinite matrix]], so it defines a [[Wikipedia: Definite bilinear form|positive semidefinite bilinear form]]. In terms of weighted monzos m₁ and m₂, m₁^TPm₂ defines the semidefinite form on weighted monzos, and hence b₁^TW^-1PW^-1b₂ defines a semidefinite form on unweighted monzos, in terms of the matrix W^-1WA^T(AW²A^T)^-1AWW^-1 = A^T(AW²A^T)^-1A = '''P'''. From the semidefinite form we obtain an associated [[Wikipedia: Definite quadratic form|semidefinite quadratic form]] b^T'''P'''b and from this the [[Wikipedia: Norm (mathematics)|seminorm]] sqrt (b^T'''P'''b).

It may be noted that (VV^T)^-1 = (AW²A^T)^-1 is the inverse of the [[Wikipedia: Gramian matrix|Gram matrix]] used to compute [[TE complexity]], and hence is the corresponding Gram matrix for the dual space. Hence '''P''' represents a change of basis defined by the mapping given by the vals combined with an [[Wikipedia: Inner product space|inner product]] on the result. Given a monzo b, Ab represents the tempered interval corresponding to b in a basis defined by the mapping A, and ''P'' = (AW²A^T)^-1 defines a positive-definite quadratic form, and hence a norm, on the tempered interval space with basis defined by A.

Denoting the temperament-defined, or temperamental, seminorm by T(x), the subspace of interval space such that T(x) = 0 contains a lattice consisting of the commas of the temperament, which is a sublattice of the lattice of monzos. The [[Wikipedia: Quotient space (linear algebra)|quotient space]] of the full vector space by the commatic subspace such that T(x) = 0 is now a [[Wikipedia: Normed vector space|normed vector space]] with norm given by T, in which the intervals of the regular temperament define a lattice. The norm T on these lattice points is the '''temperamental norm''' or '''temperamental complexity''' of the intervals of the regular temperament; in terms of the basis defined by A, it is sqrt (t^T''P''t) where t is the image of a monzo b by t = Ab.