Abstract. Let S and T be two sets of points with total cardinality n. The minimum-cost many-to-many matching problem matches each point in S to at least one point in T and each point in T to at least one point in S, such that sum of the matching costs is minimized. Here we examine the special case where both S and T lie on the line and the cost of matching s ∈ S to t ∈ T is equal to the distance between s and t. In this context, we provide an algorithm that determines a minimum-cost many-to-many matching in O(n log n) time, improving the previous best time complexity of O(n 2) for the same problem. 1.

Thirty two measures of rhythm complexity are compared using three widely different rhythm data sets. Twenty-two of these measures have been investigated in a limited context in the past, and ten new measures are explored here. Some of these measures are mathematically inspired, some were designed to measure syncopation, some were intended to predict various measures of human performance, some are based on constructs from music theory, such as Pressing’s cognitive complexity, and others are direct measures of different aspects of human performance, such as perceptual complexity, meter complexity, and performance complexity. In each data set the rhythms are ranked either according to increasing complexity using the judgements of human subjects, or using calculations with the computational models. Spearman rank correlation coefficients are computed between all pairs of rhythm rankings. Then phylogenetic trees are used to visualize and cluster the correlation coefficients. Among the many conclusions evident from the results, there are several observations common to all three data sets that are worthy of note. The syncopation measures form a tight cluster far from other clusters. The human performance measures fall in the same cluster as the syncopation measures. The complexity measures based on statistical properties of the inter-onset-interval histograms are poor predictors of syncopation or human performance complexity. Finally, this research suggests several open problems.

Abstract. We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p = 2, and p = ∞. For p = 2, we reduce the problem to standard convolution, while for p = ∞ and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time. 1

Abstract. The Hexachordal Theorem may be interpreted in terms of scales, or rhythms, or as abstract mathematics. In terms of scales it claims that the complement of a chord that uses half the pitches of a scale is homometric to—i.e., has the same interval structure as—the original chord. In terms of onsets it claims that the complement of a rhythm with the same number of beats as rests is homometric to the original rhythm. We generalize the theorem in two directions: from points on a discrete circle (the mathematical model encompassing both scales and rhythms) to a continuous domain, and simultaneously from the discrete presence or absence of a pitch/onset to a continuous strength or weight of that pitch/onset. Athough this is a significant generalization of the Hexachordal Theorem, having all discrete versions as corollaries, our proof is arguably simpler than some that have appeared in the literature. We also establish the natural analog of what is sometimes known as Patterson’s second theorem: if two equal-weight rhythms are homometric, so are their complements. 1

Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed. 1

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Abstract—Several methods for the mathematical notation, representation, and visualization of musical rhythm at the symbolic level are illustrated and compared in terms of their advantages and drawbacks, as well as their suitability for particular applications.

While it is difficult to define precisely what makes a “good ” rhythm good, it is not hard to specify properties that contribute to a rhythm’s goodness. One such property is that the mirrorsymmetric transformation of the rhythm about some axis of the rhythm’s cycle, represented as a circle, be the same as its complementary rhythm. Rhythms that have this property are called interlocking reflection rhythms. Another family of rhythms termed toggle rhythms are those cyclic rhythms that when played using the alternating-hands method, have their onsets in one cycle divided into two consecutive sets such that first set is played consecutively with one hand, and the second set is played consecutively with the other hand. Several simple rhythmgeneration methods that yield good rhythm timelines with these properties are presented and illustrated with examples. 1.

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