Tenney–Euclidean metrics: Difference between revisions

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Let us define the val weighting matrix W to be the [[wikipedia:diagonal matrix|diagonal matrix]] with values 1, 1/log₂3, 1/log₂5 … 1/log₂p along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa^T where the ^T denotes the transpose. Then the dot product of weighted vals is aW²a^T, which makes the Euclidean metric on vals, a measure of complexity, to be || <a₂ a₃ … a_''p''| ||₂ = sqrt (a₂² + a₃²/(log₂3)² + … + a_p²/(log₂''p'')²); dividing this by sqrt (''n''), where ''n'' = π(''p'') is the number of primes to ''p'' gives the Tenney-Euclidean, or TE, norm. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^-1, and the dot product is bW^-2b^T, leading to sqrt (b₂² + (log₂3)²b₃² + … + (log₂''p'')²b_''p''²); multiplying this by sqrt (''n'') gives the dual RMS norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.

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 [http://en.wikipedia.org/wiki/Positive-definite_matrix positive semidefinite matrix], so it defines a [http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Definite_quadratic_form semidefinite quadratic form] b^T'''P'''b and from this the [http://en.wikipedia.org/wiki/Norm_%28mathematics%29 seminorm] sqrt (b^T'''P'''b).

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 [http://en.wikipedia.org/wiki/Quotient_space_%28linear_algebra%29 quotient space] of the full vector space by the commatic subspace such that T(x) = 0 is now a [http://en.wikipedia.org/wiki/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.  

Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^-1 is the corresponding weighted matrix. Q = M⁺M is a projection matrix dual to P = I - Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M^T(MM^T)^-1M = I - W^-1B^T(BW^-2B^T)^-1BW^-1, and mPm^T = bW^-1PW^-1b^T, or b(W^-2 - W^-2B^T(BW^-2B^T)^-1BW^-2)b^T, so that the terms inside the parenthesis define a formula for '''P''' in terms of the matrix of monzos B.

To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a row |1 0 0 … 0> representing 2 to the matrix B. An alternative proceedure is to find the [[Normal_lists|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by A.

Consider the temperament defined by the 5-limit [[Patent_val|patent vals]] for 15 and 22 equal. From the vals, we may construct a 2×3 matrix A = [<15 24 35|, <22 35 51|]. From this we may obtain the matrix '''P''' as A^T(AW²A^T)^-1A, approximately

If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals R, then the inner product on note classes in this basis is defined by the symmetric matrix S = (RW²R^T)^-1. In the case of marvel, we obtain S = [[(p3)²(4(p5)²+(p7)²) -4(p3)²(p5)²], [-4(p3)²(p5)² (p5)²(4(p3)²+(p7)²)]]/H. If k = [a b] is a note class of marvel in the coordinates defined by the truncated val list R, which in this case has a basis corresponding to tempered 3 and 5, then sqrt (kSk^T) gives the OE complexity of the note class.

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[[Category:metric]]

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