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On Linear Algebraic Representation of Time-span and Prolongational Trees

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Publication date25/09/2017
Host publicationProceedings of the 13th International Symposium on Computer Music Multidisciplinary Research
EditorsR. Kronkland-Martinet, S. Ystad, M. Aramaki
Place of PublicationMarseille
PublisherLes éditions de PRISM
Pages126-136
Number of pages11
ISBN (Electronic)9791097498009
<mark>Original language</mark>English

Abstract

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.