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

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

Flipturning polygons

A flipturn is an operation that transforms a nonconvex simple polygon into another simple polygon, by rotating a concavity 180 degrees around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of flipturns eventually transforms any simple polygon into a convex polygon. This paper describes several new results about such flipturn sequences. We show that any orthogonal polygon is convexified after at most n-5 arbitrary flipturns, or at most 5(n-4)/6 well-chosen flipturns, improving the previously best upper bound of (n-1)!/2. We also show that any simple polygon can be convexified by at most n^2-4n+1 flipturns, generalizing earlier results of Ahn et al. These bounds depend critically on how degenerate cases are handled; we carefully explore several possibilities. We describe how to maintain both a simple polygon and its convex hull in O(log^4 n) time per flipturn, using a data structure of size O(n). We show that although flipturn sequences for the same polygon can have very different lengths, the shape and position of the final convex polygon is the same for all sequences and can be computed in O(n log n) time. Finally, we demonstrate that finding the longest convexifying flipturn sequence of a simple polygon is NP-hard.Comment: 26 pages, 32 figures, see also http://www.uiuc.edu/~jeffe/pubs/flipturn.htm

Bounded-Degree Polyhedronization of Point Sets

Abstract In 1994 Grünbaum showed that, given a point set S in R 3 , it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n log 6 n) expected time that constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al

Machine recognition of independent and contextually constrained contour-traced handprinted characters

A contour-tracing technique originally divised by Clemens and Mason was modified and used with several different classifiers of varying complexity to recognize upper case handprinted alphabetic characters. An analysis and comparison of the various classifiers, with the modifications introduced to handle variable length feature vectors, is presented. On independent characters, one easily realized suboptimum parametric classifier yielded recognition accuracies which compare favourably with other published results. Additional simple tests on commonly confused characters improved results significantly as did use of contextual constraints. In addition, the above classifier uses much less storage capacity than a non-parametric optimum Bayes classifier and performs significantly better than the optimum classifier when training and testing data are limited. The optimum decision on a string of m contextually constrained characters, each having a variable-length feature vector, is derived. A computationally efficient algorithm, based on this equation, was developed and tested with monogram, bigram and trigram contextual constraints of English text. A marked improvement in recognition accuracy was noted over the case when contextual constraints were not used, and a trade-off was observed not only between the order of contextual information used and the number of measurements taken, but also between the order of context and the value of a parameter ds which indicates the complexity of the classification algorithm.Applied Science, Faculty ofElectrical and Computer Engineering, Department ofGraduat

A COMPARISON OF RHYTHMIC SIMILARITY MEASURES

Traditionally, rhythmic similarity measures are compared according to how well rhythms may be recognized with them, how efficiently they can be retrieved from a data base, or how well they model human perception and cognition. In contrast, here similarity measures are compared on the basis of how much insight they provide about the structural inter-relationships that exist within families of rhythms, when phylogenetic trees and graphs are computed from the distance matrices determined by these similarity measures. Phylogenetic analyses yield insight into the evolution of rhythms and may uncover interesting ancestral rhythms

On Separating Two Simple Polygons by a Single Translation

Let P and Q be two disjoint simple polygons having n sides each. We present an algorithm which determines whether Q can be moved by a single translation to a position sufficiently far from P, and which produces all such motions if they exist. The algorithm runs in time O(t(n)) where t(n) is the time needed to triangulate an n-sided polygon. Since Tarjan and Van Wyk have recently shown that t(n) = O(n log log n) this improves the previous best result for this problem which was O(n log n) even after triangulation. 1. Introduction Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout considerable attention has been devoted recently to the problem of moving polygons in the plane without collisions [1]-[11]. A typical problem in robotics is the FIND-PATH problem [12], where a robot must determine if an object, modeled as a polygon in the plane, can be moved from a starting position to a goal state without collisions occurring between the object being m..

A Simple Linear Algorithm for Intersecting Convex Polygons

Let P and Q be two convex polygons with m and n vertices, respectively, which are specified by their cartesian coordinates in order. A simple O(m+n) algorithm is presented for computing the intersection of P and Q. Unlike previous algorithms, the new algorithm consists of a two-step combination of two simple algorithms for finding convex hulls and triangulations of polygons. Key words: Algorithms - Complexity - Computational geometry - Convex polygons - Intersection 1. Introduction Let P = {p 1 , p 2 ,..., p m } and Q = {q 1 , q 2 ,..., q n } be two convex polygons whose vertices are specified by their cartesian coordinates in clockwise order. It is assumed that the polygons are in standard form, i.e., the vertices of each polygon are distinct and no three consecutive vertices are collinear [7]. A common problem in computer graphics, image processing and many sub-problems in computational geometry is that of computing the intersection of P and Q which is itself another c..