Linking sonic aesthetics with mathematical theories

Mathematics describes relationships between objects: collections of rules that determine how one or more object may be transformed into one or more other object, and the rich patterns that follow from these relationships. By a judicious mapping from mathematical structures to musical features, it becomes possible to imbue the latter with similarly rich structure and patterning" to link sonic aesthetics with mathematical theories. In the following section, I explore the application of mathematical techniques to mould the raw materials of music into interesting latent (as yet unrealized) structures. The focus will be on musical scales and metres, both periodic and nonperiodic. In the section after that, I explore some mathematically informed procedures that can produce musical realizations of these latent structures. These include musical canons, methods for generating self-similar and fractal-like forms, and the use of the Fourier transform to dynamically change pitch, timbre, and rhythm. At the risk of perpetuating the hegemonic 'three-dimensional lattice' of discrete pitches, times, and timbres (Wishart 1983), most of the examples use discrete events; despite that, many of the techniques described here are also applicable to smooth and dynamic changes of musical variable. In the third section, prior to the conclusion, I ground some of the abstractions by discussing a real-world musical realization.