The Edit Distance as a Measure of Perceived Rhythmic Similarity

The ‘edit distance’ (or ‘Levenshtein distance’) measure of distance between two data sets is defined as the minimum number of editing operations – insertions, deletions, and substitutions – that are required to transform one data set to the other (Orpen and Huron, 1992). This measure of distance has been applied frequently and successfully in music information retrieval, but rarely in predicting human perception of distance. In this study, we investigate the effectiveness of the edit distance as a predictor of perceived rhythmic dissimilarity under simple rhythmic alterations. Approaching rhythms as a set of pulses that are either onsets or silences, we study two types of alterations. The first experiment is designed to test the model’s accuracy for rhythms that are relatively similar; whether rhythmic variations with the same edit distance to a source rhythm are also perceived as relatively similar by human subjects. In addition, we observe whether the salience of an edit operation is affected by its metric placement in the rhythm. Instead of using a rhythm that regularly subdivides a 4/4 meter, our source rhythm is a syncopated 16-pulse rhythm, the son. Results show a high correlation between the predictions by the edit distance model and human similarity judgments (r = 0.87); a higher correlation than for the well-known generative theory of tonal music (r = 0.64). In the second experiment, we seek to assess the accuracy of the edit distance model in predicting relatively dissimilar rhythms. The stimuli used are random permutations of the son’s inter-onset intervals: 3-3-4-2-4. The results again indicate that the edit distance correlates well with the perceived rhythmic dissimilarity judgments of the subjects (r = 0.76). To gain insight in the relationships between the individual rhythms, the results are also presented by means of graphic phylogenetic trees

ISAMA The International Society of the Arts, Mathematics, and Architecture BRIDGES Mathematical Connections in Art, Music, and Science Classification and Phylogenetic Analysis of African Ternary Rhythm Timelines

Abstract A combinatorial classification 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 classification are proposed based on measures of rhythmic oddity and off-beatness. These classifications reveal several new uniqueness properties of the Bembe bell pattern that may explain its widespread popularity. A new distance measure called the swap-distance is introduced to measure the non-similarity of two rhythms that have the same number of strokes (onsets). A swap in a sequence of notes and rests of equal duration is the location interchange of a note and a rest that are adjacent in the sequence. The swap distance between two rhythms is defined as the minimum number of swaps required to transform one rhythm to the other. A phylogenetic analysis using Splits Graphs with the swap distance shows that each of the ten bell patterns can be derived from one of two "canonical" patterns with at most four swap operations, or from one with at most five swap operations. Furthermore, the phylogenetic analysis suggests that for these ten bell patterns there are no "ancestral" rhythms not contained in this set

An O(n log n)-Time Algorithm for the Restricted Scaffold Assignment

The assignment problem takes as input two finite point sets S and T and establishes a correspondence between points in S and points in T, such that each point in S maps to exactly one point in T, and each point in T maps to at least one point in S. In this paper we show that this problem has an O(n log n)-time solution, provided that the points in S and T are restricted to lie on a line (linear time, if S and T are presorted).Comment: 13 pages, 8 figure

Perception of Rhythmic Similarity in Flamenco Music: Comparing Musicians and Non-Musicians.

Background in Music Psychology. Previous research on rhythmic perception has highlighted differences between novice listeners and musicians in their ability to label perceived differences as well as strategies for representing musical structures. Novice listeners tend to focus on “surface” features while musicians tend to focus on the underlying rhythmic structure and develop a specific vocabulary. Furthermore, there is evidence that changes in tempo affect novices’ perception of rhythm

On Reconfiguring Tree Linkages: Trees can Lock

It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an $N$-link tree can have $2^{\Omega(N)}$ equivalence classes of configurations.Comment: 16 pages, 6 figures Introduction reworked and references added, as the main open problem was recently close

The Distance Geometry of Music

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG '05), University of Windsor, Canada, 200

A Novel Approach for Ellipsoidal Outer-Approximation of the Intersection Region of Ellipses in the Plane

In this paper, a novel technique for tight outer-approximation of the intersection region of a finite number of ellipses in 2-dimensional (2D) space is proposed. First, the vertices of a tight polygon that contains the convex intersection of the ellipses are found in an efficient manner. To do so, the intersection points of the ellipses that fall on the boundary of the intersection region are determined, and a set of points is generated on the elliptic arcs connecting every two neighbouring intersection points. By finding the tangent lines to the ellipses at the extended set of points, a set of half-planes is obtained, whose intersection forms a polygon. To find the polygon more efficiently, the points are given an order and the intersection of the half-planes corresponding to every two neighbouring points is calculated. If the polygon is convex and bounded, these calculated points together with the initially obtained intersection points will form its vertices. If the polygon is non-convex or unbounded, we can detect this situation and then generate additional discrete points only on the elliptical arc segment causing the issue, and restart the algorithm to obtain a bounded and convex polygon. Finally, the smallest area ellipse that contains the vertices of the polygon is obtained by solving a convex optimization problem. Through numerical experiments, it is illustrated that the proposed technique returns a tighter outer-approximation of the intersection of multiple ellipses, compared to conventional techniques, with only slightly higher computational cost

Proximity-graph-based tools for DNA clustering

There are more than one billion documents on the Web, with the count continually rising at a pace of over one million new documents per day. As information increases, the motivation and interest in data warehousing and mining research and practice remains high in organizational interest. The Encyclopedia of Data Warehousing and Mining, Second Edition, offers thorough exposure to the issues of importance in the rapidly changing field of data warehousing and mining. This essential reference source informs decision makers, problem solvers, and data mining specialists in business, academia, government, and other settings with over 300 entries on theories, methodologies, functionalities, and applications

Cauchy's Arm Lemma on a Growing Sphere

We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma

Similaridad y evolución en la rítmica del flamenco: una incursión de la matemática computacional

Presentamos un artículo que es singular por muchas razones. Por un lado, por la procedencia y características variadas de sus autores (profesores de Matemática Aplicada de las Universidades de Sevilla y Politécnica de Madrid, de Computación de Queen’s y McGill University, una concertista de piano de la Real Escuela Profesional de Danza de Madrid), incluyendo, entre ellos, la figura –señera en Geometría Discreta y Algorítmica– de Godfried Toussaint, que ha desarrollado, desde hace varios a˜nos, una estrecha relación con la pujante escuela española de Geometría Computacional. Por otra lado, por la temática elegida, el ritmo flamenco, a cuyo análisis se quiere contribuir (y permítaseme subrayar este término: contribución) aquí aportando determinadas herramientas matemáticas. Quisiera, como editor de esta Sección de La Gaceta, dar las gracias a los autores por su original contribución y, también, a los recensores (cuyo nombre, desgraciadamente, no puedo revelar), que han desarrollado una labor crítica y constructiva extraordinaria, en un tema tan alejado, aparentemente, de los conocimientos de la mayoría de los matemáticos de mi personal base de datos. Unos y otros me permiten constatar con orgullo que en España, hoy, es posible encontrar matemáticos de primer nivel que son capaces de aportar comentarios autorizados sobre temas tan singulares e interesantes como este