On Linear Algebraic Representation of Time-span and Prolongational Trees

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.

No longer 'somewhat arbitrary'

“…Our interest in the topic originates in our use of matrices to represent tree structures [19]. The connections in a tree can be represented in a matrix in which each row and column corresponds to a pitch event in the music.…”

“…We speculate that a more useful matrix representation is possible using not just the values 0 and 1, but values related to the MTS for each event. In [19] we demonstrated how placing the MTS values on the diagonal of the matrix allows matrix multiplication to be used (as in matrices using only 0 and 1) to determine reachability without destroying the MTS information in the matrix. What value should be used off the diagonal to indicate connection between events?…”

“…This value should give some information about the relationship between the events, just as the height of branching in a tree diagram gives information about the relative structural importance of events, by indicating the 'level' of an event. In [19] we proposed to use the difference between the MTS of the superordinate event and the MTS of the subordinate event. This allowed a simple relationship between MTS and branch height (Figure 3).…”