Modelling Motivic Processes in Music: A Mathematical Approach

This thesis proposes a new model for motivic analysis which, being based on the metaphor of a web or network and expanded using the mathematical field of graph theory, balances the polar concerns prevalent in analytical writing to date: those of static, out-of-time category membership and dynamic, in-time process. The concepts that constitute the model are presented in the third chapter, both as responses to a series of analytical observations (using the worked example of Beethoven’s Piano Sonata in F minor, Op. 2, No. 1), and as rigorously defined mathematical formalisms. The other chapters explore in further detail the disciplines and methodologies on which this model impinges, and serve both to motivate, and to reflect upon, its development. Chapter 1 asks what it means to make mathematical statements about music, and seeks to disentangle mathematics (as a tool or language) from science (as a method), arguing that music theory’s aims can be met by the former without presupposing its commonly assumed inextricability from the latter. Chapter 2 provides a thematic overview of the field of motivic theory and analysis, proposing four archetypal models that combine to underwrite much thought on the subject before outlining the problems inherent in a static account and the creative strategies that can be used to construct a dynamic account. Finally, Chapter 4 applies these strategies, together with Chapter 3’s model and the piece’s extensive existing scholarly literature, to the analysis of the first and last movements of Mahler’s Sixth Symphony. The central theme throughout – as it relates to mathematical modelling, music theory, and music analysis – is that of potential, invitation, openness, and dialogic engagement.