Computational Geometric Aspects of Musical Rhythm

For our purpose a rhythm is represented as a cyclic binary string. Consider the following three 12/8 time ternary rhythms expressed in box-like notation: [x . x . x . x . x . x .], [x . x . x x . x . x . x] and [x . . . x x . . x x x .]. Here “x” denotes the striking of a percussion instrument, and “.” denotes a silence. It is intuitively clear that the first rhythm is the most even (well spaced) of the three. Traditional rhythms have a tendency to exhibit such properties of evenness. Therefore mathematical measures of evenness find application in the new field of mathematical ethnomusicology [2], [17], where they may help to identify, if not explain, cultural preferences of rhythms in traditional music. Clough and Duthett [3] introduced the notion of maximally even sets with respect to pitch scales represented on a circle. Block and Douthett [1] went further by constructing several mathematical measures of the amount of evenness contained in a scale. One of their measures simply adds all the interval arc-lengths (geodesics along the circle) determined by all pairs of pitches in the scale. However, this measure is too coarse to be useful for comparing rhythm timelines such as those studied in [13] and [15]. Using interval chord-lengths (as opposed to geodesic distances), proposed by Block and Douthet [1], yields a more discriminating measure, and is therefore a function that receives more attention. In fact, this problem had been investigated by Fejes Tóth [12] some forty years earlier without the restriction of placing the points on the circular lattice. He showed that the sum of the pairwise distances determined by n points on a circle is maximized when the points are the vertices of a regular n-gon. One may also examine the spectrum of the frequencies with which all the durations are present in a rhythm. In music theory this spectrum is called the interval vector (or full-interval vector) [7]. For example, the interval vector for the clave Son pattern [x . . x . . x . . . x . x . . .] is given by [0,1,2,2,0,3,2,0].