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An Abbreviated Survey

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Mathematical and Computational Modeling of Tonality

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 204))

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Abstract

This chapter weaves together a backdrop of related work in music theory, cognitive science, and operations research that has inspired and influenced the design of the Spiral Array and its associated algorithms. The chapter begins with an overview of the Spiral Array model and its core ideas. This is followed by a review of some spatial models for musical pitch that have informed the model’s design, and an overview of the Harmonic Network (a.k.a. the tonnetz) and some of its applications. The idea of the center of effect (CE) is central to the Spiral Array and its associated algorithms. The idea of the CE draws inspiration from interior point methods in linear optimization. The second part of the chapter describes the von Neumann Center of Gravity algorithm and Dantzig’s bracketing technique to speed convergence, and then draws analogies between the algorithm and the CEG method.

This chapter is based, in part, on “Dantzig’s Indirect Contribution to Music Research: How the von Neumann Center of Gravity Algorithm Influenced the Center of Effect Generator Key Finding Algorithm” by Elaine Chew, an article in the INFORMS Computing Society Newsletter (Spring 2008), and on the Background (Chapter 2) of “Towards a Mathematical Modeling of Tonality” by Elaine Chew, an MIT PhD dissertation, Cambridge, Massachusetts (2000) https://dspace.mit.edu/handle/1721.1/9139

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Authors and Affiliations

  1. Centre for Digital Music, Queen Mary University of London, London, UK

    Elaine Chew

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  1. Elaine Chew
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Correspondence to Elaine Chew .

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Chew, E. (2014). An Abbreviated Survey. In: Mathematical and Computational Modeling of Tonality. International Series in Operations Research & Management Science, vol 204. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-9475-1_2

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