On Linear Algebraic Representation of Time-span and Prolongational Trees

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

Multiple levels of structure in language and music

Item does not contain fulltextA forum devoted to the relationship between music and language begins with an implicit assumption: There is at least one common principle that is central to all human musical systems and all languages, but that is not characteristic of (most) other domains. Why else should these two categories be paired together for analysis? We propose that one candidate for a common principle is their structure. In this chapter, we explore the nature of that structure—and its consequences for psychological and neurological processing mechanisms—within and across these two domain