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Abstract
This paper shows how scale vectors (which can represent either pitch or rhythmic patterns) can be written as a linear combination of columns of scale matrices, thus decomposing the scale into musically relevant intervals. When the scales or rhythms have different cardinalities, they can be compared using a canonical form closely related to Lyndon words. The eigenvalues of the scale matrix are equal to the Fourier coefficients, which leads to a number of relationships between the scale vectors and the decompositions. Overcomplete dictionaries of frame elements can be used for more convincing representations by finding sparse decompositions, a technique that can also be applied to tiling problems. Scale matrices are related to familiar theoretical properties such as the interval function, Z-relation or homometry, all of which can be efficiently studied within this framework. In many cases, the determinant of the scale matrix is key: singular scale matrices correspond to Lewin's special cases, regular matrices allow a simple method of recovering the argument of an interval function and elicit unique decompositions, large determinant values correspond to flat interval distributions.
Acknowledgements
The authors thank all the participants of the 2009 Bellair's workshop on Music and Mathematics for their comments and encouragement. Dmitri Tymoczko and Noam Elkies provided key insights into the varieties of possible representations of scale vectors. Mihalis Kolountzakis had independently come across the idea of a Linear Programming as a useful tool for tilings, and provided essential advice.
Notes
The domain of the s i could be extended by considering the characteristic functions of multisets.
In special cases, such a decomposition exists, but will not be unique: for instance the guidonian hexachord is a sum of three fifths, though the scale matrix of the fifth is singular.
The sum of all fifths beginning on one whole-tone scale is equal to the whole, as is the similar sum starting on the other whole-tone scale. So any perfect fifth is a linear combination of all the others.
Word theorists call these Lyndon words, computer scientists talk of circlists. Forte's basic form is almost identical but its definition is needlessly complicated.
For theoretical applications of downsampling in rhythmic canons, see Citation22.
It is the tensor product of S x and I m .
The transposition takes into account the minus sign before b. In full generality, it is necessary to use the transpose-conjugate of T, but the conjugacy is irrelevant in the present setting since T is real-valued.
The sum of those scales beginning on C, F, F♯, B is equal to the sum of the same scales beginning on D, D♯, G♯, A. Hence the scale matrix is singular.
For instance, the ‘tritone property’ states: ‘for any (0167)-set K, [it] has the same number of notes in common with T3(K), as it has in common with K’.
The original definition, both for musicology and for crystallography, precludes the simple case where the scales are identical under action of the T/I group.
The transpositions T j appear nicely as spectral units – they are powers of J – but any spectral unit that relates a major and a minor triad is of infinite order. Even more strangely, inversions cannot be mapped into the group of spectral units.
In his seminal paper, Rosenblatt Citation12 proves that this is true even for singular matrices.
The product of positive real numbers whose sum is fixed is maximal when they are equal.
Namely .
Specifically, that minimizing the linear form on the polyhedron defined by
and
by the simplex method yields a vector x with 0–1 coefficients satisfying Equation(20)
.
We denote here the gcd of d, n by (d, n) for concision.
Recall that circulating matrices are stable under multiplication and under transposition since the generating element J satisfies .