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
Full text not available from this repository.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.