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  • TojoMarsdenHirataLNCS2018AAM

    Rights statement: The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-030-01692-0_14

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    Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

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On Linear Algebraic Representation of Time-span and Prolongational Trees

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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 easy to recognize and has high visibility, such an intuitive representation can hardly be treated in mathematical formalization. Especially in GTTM, the conjunction height of two branches is often arbitrary, contrary to the notion of hierarchy. Since a tree is a kind of graph, and a graph is often represented by a matrix, we show the linear algebraic representation of trees, specifying 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 to model compositional processes with simple algebraic operations.

Bibliographic note

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-030-01692-0_14