On Linear Algebraic Representation of Time-span and Prolongational Trees

Marsden, Alan Alexander and Tojo, Satoshi and Hirata, Keiji (2017) On Linear Algebraic Representation of Time-span and Prolongational Trees. In: Proceedings of the 13th International Symposium on Computer Music Multidisciplinary Research. Les éditions de PRISM, Marseille, pp. 126-136. ISBN 9791097498009

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In constructive music theory, such as Schenkerian analysis and the Generative Theory of Tonal Music (GTTM), the hierarchical importance of pitch events is conveniently represented by a tree structure. Although a tree is intuitive and visible, such a graphic representation cannot be treated in mathematical formalization. Especially in the GTTM, the conjunction height of two branches is often arbitrary, contrary to the notion of hierarchy. As even a tree is a kind of graph, and a graph is often represented by a matrix, we show the linear algebraic representation of a tree, specifying the conjunction heights. Thereafter, we explain the ‘reachability’ between pitch events (corresponding to information about reduction) by the multiplication of matrices. In addition we discuss multiplication with vectors representing a sequence of harmonic functions, and suggest the notion of stability. Finally, we discuss operations between matrices with the objective of modelling compositional processes with simple algebraic operations.

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04 Jul 2018 10:10
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27 Nov 2020 09:34