Extending a network-of-elaborations representation to polyphonic music: Schenker and species counterpoint.

A system of representing melodies as a network of elaborations has been developed, and used as the basis for software which generates melodies in response to the movements of a dancer. This paper examines the issues of extending this representation system to polyphonic music, and of deriving a structural representation of this kind from a musical score. The theories of Heinrich Schenker and of Species Counterpoint are proposed as potentially fruitful bases

Recognition of variations using automatic Schenkerian reduction.

Experiments on techniques to automatically recognise whether or not an extract of music is a variation of a given theme are reported, using a test corpus derived from ten of Mozart's sets of variations for piano. Methods which examine the notes of the 'surface' are compared with methods which make use of an automatically derived quasi-Schenkerian reduction of the theme and the extract in question. The maximum average F-measure achieved was 0.87. Unexpectedly, this was for a method of matching based on the surface alone, and in general the results for matches based on the surface were marginally better than those based on reduction, though the small number of possible test queries means that this result cannot be regarded as conclusive. Other inferences on which factors seem to be important in recognising variations are discussed. Possibilities for improved recognition of matching using reduction are outlined

Software for Schenkerian Analysis

Software developed to automate the process of Schen-kerian analysis is described. The current state of the art is that moderately good analyses of small extracts can be generated, but more information is required about the criteria by which analysts make decisions among alternative interpretations in the course of analysis. The software described here allows the procedure of reduction to be examined while in process, allowing decision points, and potentially criteria, to become clear

Music analysis by computer:ontology and epistemology

This chapter examines questions of what is to be analysed in computational music analysis, what is to be produced, and how one can have confidence in the results. These are not new issues for music analysis, but their consequences are here considered explicitly from the perspective of computational analysis. Music analysis without computers is able to operate with multiple or even indistinct conceptions of the material to be analysed because it can use multiple references whose meanings shift from context to context. Computational analysis, by contrast, must operate with definite inputs and produce definite outputs. Computational analysts must therefore face the issues of error and approximation explicitly. While computational analysis must retain contact with the music analysis as it is generally practised, I argue that the most promising approach for the development of computational analysis is not systems to mimic human analysis, but instead systems to answer specific music-analytical questions. The chapter concludes with several consequent recommendations for future directions in computational music analysis

Echoes in Plato's cave:ontology of sound objects in computer music and analysis

The sonic aspects of Plato's analogy of the cave is taken as a starting point for thought experiments to investigate the objective nature of sound, and the idea of quasi-Platonic forms in music. Sounds are found to be objects in a way that sights or appearances are not, and it is only in the presence of technology that they become artificial. When recognition, control and communication about sound come into play, abstract concepts emerge, but there is no reason to give these the priority status Plato affords to forms. Similar issues arise in discussion of the ontology of musical works, where the ideas of extension and intension prove useful for clarity about the nature of musical objects. They are also useful for strategies in the development of music software. Musical concepts are not fixed but arise from complex cultural interactions with sound. Music software should aim to use abstract concepts with are useful rather than correct

A Discrete Theory of Connections on Principal Bundles

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding to the choice of a splitting of the short exact sequence. Equivalent representations of a discrete connection are considered, and an extension of the pair groupoid composition, that takes into account the principal bundle structure, is introduced. Computational issues, such as the order of approximation, are also addressed. Discrete connections provide an intrinsic method for introducing coordinates on the reduced space for discrete mechanics, and provide the necessary discrete geometry to introduce more general discrete symmetry reduction. In addition, discrete analogues of the Levi-Civita connection, and its curvature, are introduced by using the machinery of discrete exterior calculus, and discrete connections.Comment: 38 pages, 11 figures. Fixed labels in figure

On the Size-Dependence of the Inclination Distribution of the Main Kuiper Belt

We present a new analysis of the currently available orbital elements for the known Kuiper belt objects. In the non-resonant, main Kuiper belt we find a statistically significant relationship between an object's absolute magnitude (H) and its inclination (i). Objects with H~170 km for a 4% albedo) have higher inclinations than those with H>6.5 (radii >~ 170 km). We have shown that this relationship is not caused by any obvious observational bias. We argue that the main Kuiper belt consists of the superposition of two distinct distributions. One is dynamically hot with inclinations as large as \~35 deg and absolute magnitudes as bright as 4.5; the other is dynamically cold with i6.5. The dynamically cold population is most likely dynamically primordial. We speculate on the potential causes of this relationship.Comment: 14 pages, 6 postscript figure

Calculus Unlimited

Purpose: This book is intended to supplement our text, Calculus (Benjamin/Cummings, 1980), or virtually any other calculus text (see page vii, How To Use This Book With Your Calculus Text). As the title Calculus Unlimited implies, this text presents an alternative treatment of calculus using the method of exhaustion for the derivative and integral in place of limits. With the aid of this method, a definition of the derivative may be introduced in the first lecture of a calculus course for students who are familiar with functions and graphs. Approach: Assuming an intuitive understanding of real numbers, we begin in Chapter 1 with the definition of the derivative. The axioms for real numbers are presented only when needed, in the discussion of continuity. Apart from this, the development is rigorous and contains complete proofs. As you will note, this text has a more geometric flavor than the usual analytic treatment of calculus. For example, our definition of completeness is in terms of convexity rather than least upper bounds, and Dedekind cuts are replaced by the notion of a transition point. Who Should Use This Book: This book is for calculus instructors and students interested in trying an alternative to limits. The prerequisites are a knowledge of functions, graphs, high school algebra and trigonometry. How To Use This Book: Because the "learning-by-doing" technique introduced in Calculus has proved to be successful, we have adapted the same format for this book. The solutions to "Solved Exercises" are provided at the back of the book; however readers are encouraged to try solving each example before looking up the solution. The Origin Of The Definition of The Derivative: Several years ago while reading Geometry and the Imagination, by Hilbert and Cohn-Vossen (Chelsea, 1952, p. 176), we noticed a definition of the circle of curvature for a plane curve C. No calculus, as such, was used in this definition. This suggested that the same concept could be used to define the tangent line and thus serve as a limit-free foundation for the differential calculus. We introduced this new definition of the derivative into our class notes and developed it in our calculus classes for several years. As far as we know, the definition has not appeared elsewhere. If our presumption of originality is ill-founded, we welcome your comments. Jerrold Marsden Alan Weinstein Berkeley, C

A standard format proposal for hierarchical analyses and representations

In the realm of digital musicology, standardizations efforts to date have mostly concentrated on the representation of music. Analyses of music are increasingly being generated or communicated by digital means. We demonstrate that the same arguments for the desirability of standardization in the representation of music apply also to the representation of analyses of music: proper preservation, sharing of data, and facilitation of digital processing. We concentrate here on analyses which can be described as hierarchical and show that this covers a broad range of existing analytical formats. We propose an extension of MEI (Music Encoding Initiative) to allow the encoding of analyses unambiguously associated with and aligned to a representation of the music analysed, making use of existing mechanisms within MEI's parent TEI (Text Encoding Initiative) for the representation of trees and graphs

Hamiltonian systems with symmetry, coadjoint orbits and plasma physics

The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the Lie-Poisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the Maxwell-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. First, the Poisson-Vlasov equations are shown to be in Hamiltonian form relative to the Lie-Poisson bracket on the dual of the (nite dimensional) Lie algebra of innitesimal canonical transformations. Then we write Maxwell's equations in Hamiltonian form using the canonical symplectic structure on the phase space of the electromagnetic elds, regarded as a gauge theory. In the last step we couple these two systems via the reduction procedure for interacting systems. We also show that two other standard models in plasma physics, ideal MHD and two- uid electrodynamics, can be written in Hamiltonian form using similar group theoretic techniques