An algebra for periodic rhythms and scales

Abstract

This paper shows how scale vectors (which can represent either pitch or rhyth-mic 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 cardinality, 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 homom-etry, 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 matricesallow a simple method of recovering the ar-gument of an interval function and elicit unique decompositions; large determinant values correspond to flat interval distributions. 1