Symmetry and symmetry reduction in music

Research Doctorate - Doctor of Philosophy (PhD)There is a significant body of literature on the subject of symmetry in music, revealing many important insights into its role, particularly in music of the twentieth century. However, the scope of symmetry research is limited by the nature of symmetry itself. The self-referential nature of symmetry, associated with stasis and conservation, is at odds with the dynamic nature of music and its need to develop and evolve, prompting Stravinsky’s assertion that “to be perfectly symmetrical is to be perfectly dead”. By contrast, the study of symmetry reduction is the study of transitions, bifurcations and structural transformations, and despite its importance in a wide range of disciplines, including physics, biology and information technology, its role in music has not been addressed in a systematic way. Exemplified by the process of crystallisation, symmetry reduction occurs when a structure or process makes a transition from a state of high symmetry to one of relatively low symmetry, and is associated with self-organisation, the emergence of information, differentiation and hierarchy, and the appearance of patterns and periodicity. Order increases when symmetry is reduced. This exegesis offers the first systematic study of symmetry reduction in music. An analytical framework is established, which includes the use of clock-face diagrams and a novel modification of Allen Forte’s classification of pitch-class sets, allowing the symmetry of any set to be described. A number of works are analysed, including detailed studies of the opening of Wagner’s <i>Tristan</i> Prelude, the first movement of Ravel’s String Quartet, Schubert’s song “Der Neugierige” (from <i>Die schöne Müllerin</i>) and the second movement of Webern’s Symphony. The analyses demonstrate the place of symmetry reduction in each work, and show that the principles and consequences of symmetry reduction described in the scientific literature play a significant role in music, demonstrating that the dynamic nature of music is derived, at least in part, from the tension between symmetry and symmetry reduction, stasis and evolution, ambiguity and information. In the final part of the thesis, the role of symmetry and symmetry reduction is considered in a range of original compositions. The portfolio includes music for string orchestra, wind quintet, and symphony orchestra, ranging from pedagogical to professional works. In some of the compositions, symmetry, rather than symmetry reduction, is the most important feature, while in the others symmetry and symmetry reduction interact, demonstrating a role for these ideas in the creation of new music. The analysis of the works will include reflections on the compositional process, in which analysis of symmetry and symmetry reduction lead to important creative decisions in determining various elements of the music. This research presents opportunities for several avenues of further research including the relationship between symmetry reduction and thermodynamics, raising the possibility of a better understanding of the elusive concept of musical energy