A combinatorial classi cation and a phylogenetic analysis of the ten 12/8 time, seven-stroke bell rhythm timelines in African and Afro-American music are presented. New methods for rhythm classi cation are proposed based on measures of rhythmic oddity and o-beatness. These combinatorial classi cations reveal several new uniqueness properties of the Bembe bell pattern that may explain its widespread popularity and preference among the other patterns in this class.

Musical rhythm is considered from the point of view of geometry.

e 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]. This research was partially supported by NSERC and FCAR. e-mail: godfried@cs.mcgill.ca Examination of such rhythm histograms leads to questions of interest in a variety of fields of enquiry: musicology, geometry, combinatorics, and number theory. For example, David Locke [9] has given musicological explanations for the characterization of the Gahu bell pattern, given by [x . . x . . x . . . x . . . x .], as "rhythmically potent ", exhibiting a "tricky" quality, creating a "spiralling e#ect", causing "ambiguity of phrasing" leading to "aural illusions." Comparing the full-interval histogram of the Gahu pattern with the his

Music is composed of tension and resolution, and one of the most interesting resources to create rhythmic tension is syncopation. Several attempts have been made to mathematically define a measure of syncopation that captures its essence. A first approach could be to consider the rhythmic oddity property used by Simha Arom to analyze rhythms from the Aka pygmies. Although useful for other purposes, this property has its limitations as a measure of syncopation. More elaborate ideas come from the works by Michael Keith (based on combinatorial methods) and Godfried Toussaint (based on group theory). In this paper we propose a new measure, called the weighted note-to-beat distance measure, which overcomes certain drawbacks of the previous measures. We also carry out a comparison among the three measures. In order to properly compare these measures of syncopation, we have tested them on a number of rhythms taken from several musical traditions. 1

The heart of an African rhythm is the timeline, a beat that cyclically repeats thoughout a piece, and is often performed with an iron bell that all performers can hear. Such rhythms can be represented as sequences of points on a circular lattice, where the position of the points indicates the time in the cycle at which the instrument is struck. Whereas in theory there are thousands of possible choices for such timeline patterns, in practice only a few of these are ever used. This brings up the question of how these few patterns were selected over all the others, and of those selected, why some are preferred (have more widespread use) than others. Simha Arom discovered that the rhythms used in the traditional music of the Aka Pygmies of Central Africa possess what he calls the rhythmic oddity property. A rhythm has the rhythmic oddity property if it does not contain two onsets that partition the cycle into two half-cycles. Here a broader spectrum of rhythms from West, Central and South Africa are analysed. A mathematical property of rhythms is proposed, dubbed "O#-Beatness", that is based on group theory, and it is argued that it is superior to the rhythmic oddity property as a measure of preference among Sub-Saharan African rhythm timelines. The "O#-Beatness" measure may also serve as a mathematical definition of syncopation, a feature for music recognition in general, and it is argued that it is superior to the mathematical syncopation measure proposed by Michael Keith.

Computational music theory offers a wide variety of interesting geometric, combinatoric, and algorithmic problems. Some of these problems are illustrated for the special cases of rhythm and melody. In particular, several techniques useful for the teaching, analysis, generation and automated recognition of the rhythmic components of music are reviewed. A new measure of rhythm-evenness is described and shown to be better than previous measures for discriminating between rhythm timelines. It may also be more efficiently computed. Several open problems are discussed.

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.

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

in partial fullfilment of the requirements for the degree of

The wide range of vocal styles, musical textures and recording techniques found in ethnomusicological field recordings leads us to consider the problem of automatically labeling the content to know whether a recording is a song or instrumental work. Furthermore, if it is a song, we are interested in labeling aspects of the vocal texture: e.g. solo, choral, acapella or singing with instruments. We present evidence to suggest that automatic annotation is feasible for recorded collections exhibiting a wide range of recording techniques and representing musical cultures from around the world. Our experiments used the Alan Lomax Cantometrics training tapes data set, to encourage future comparative evaluations. Experiments were conducted with a labeled subset consisting of several hundred tracks, annotated at the track and frame levels, as acapella singing, singing plus instruments or instruments only. We trained frame-by-frame SVM classifiers using MFCC features on positive and negative exemplars for two tasks: per-frame labeling of singing and acapella singing. In a further experiment, the frame-by-frame classifier outputs were integrated to estimate the predominant content of whole tracks. Our results show that frame-byframe classifiers achieved 71 % frame accuracy and whole track classifier integration achieved 88 % accuracy. We conclude with an analysis of classifier errors suggesting avenues for developing more robust features and classifier strategies for large ethnographically diverse collections. 1.