This thesis contains a collection of results in computational geometry that are inspired from music theory literature. The solutions to the problems discussed are based on a representation of musical rhythms where pulses are viewed as points equally spaced around the circumference of a circle and onsets are a subset of the pulses. All our results for rhythms apply equally well to scales, and many of the problems we explore are interesting in their own right as distance geometry problems on the circle. / In this thesis, we characterize two families of rhythms called deep and Euclidean. We describe three algorithms that generate the unique Euclidean rhythm for a given number of onsets and pulses, and show that Euclidean rhythms are formed of repeating patterns of a Euclidean rhythm with fewer onsets, followed possibly by a different rhythmic pattern. We then study the conditions under which we can transform one Euclidean rhythm to another through five different operations. In the context of measuring rhythmic similarity, we discuss the necklace alignment problem where the goal is to find rotations of two rhythms and a perfect matching between the onsets that minimizes some norm of the circular distance between the matched points. We provide o (n2)-time algorithms to this problem using each of the ℓ1, ℓ2, and ℓinfinity norms as distance measures. Finally, we give a polynomial-time solution to the labeled beltway problem where we are given the ordering of a set of points around the circumference of a circle and a labeling of all distances defined by pairs of points, and we want to construct a rhythm such that two distances with a common onset as endpoint have the same length if and only if they have the same label.

This thesis studies measures of musical rhythm complexity. Informally, rhythm complexity may be thought of as the difficulty humans have performing a rhythm, listening to a rhythm, or recognizing its structure. The problem of understanding rhythm complexity has been studied in musicology and psychology, but there are approaches for its measurement from a variety of domains. This thesis aims to evaluate rhythm complexity measures based on how accurately they reflect human-based measures. Also, it aims to compare their performance using rhythms from Africa, India, and rhythms generated randomly. The results suggest that none of the measures accurately reflect the difficulty humans have performing or listening to rhythm; however, the measures do accurately reflect how humans recognize a rhythm's metrical structure. Additionally, the results suggest a need for normalization of the measures to account for variety among cultural rhythms.

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This thesis presents and analyzes tools and experiments that aim at achieving multiple yet related goals in the exploration and manipulation of musical cyclic rhythms. The work presented in this thesis may be viewed as a preliminary study for the ultimate future goal of developing a general computational theory of rhythm. Given a family of rhythms, how does one reconstruct its ancestral rhythms? How should one change a rhythm's cycle length while preserving its musicologically salient properties, and hence be able to confirm or disprove popular or historical beliefs regarding its origins and evolution? How should one compare musical rhythms? How should one automatically generate rhythmic patterns? All these questions are addressed and, to a certain extent, solved in our study, and serve as a basis for the development of novel general tools, implemented in Matlab, for the manipulation of rhythms.

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