Gödel, Plato, Xenakis : eternal forms, numbers and rules : a portfolio of compositions

Abstract

This written commentary describes how pieces were composed from individual strands of music, independent parts combined in a polyphonic style. A study of the rules of species counterpoint in Fux’s Gradus Ad Parnassum inspired the first compositions; a modified set of rules, developed from the writings of Fux controlled initial pitch-organisation. This set of rules underwent modifications informed by innovations introduced in subsequent compositions. The theories discussed in Xenakis’ Symbolic Music were utilised to generate pre-compositional pitch-sets and duration-sets. The pitch-sets were organised vertically to produce chords. These developments led to considerations of form, proportion and structure in the research. Changes in tonality with an associated contrast of chords articulated formal sections, bearing similarity to the use of the contrast of key found in traditional tonality. The pre-composed duration-sets gave more impetus and rhythmic momentum to the music and expanded the overall rhythmic content. Speculation on the ideas of musical Platonism raised two questions: Which musical elements might be considered universal? Is the act of composition invention or discovery? Furthermore, does a composer discover a copy of an eternal Platonic Form? Contemplation upon the metaphysics of Plato and Platonic Forms inspired an extension of Xenakis’ Symbolic Music using numbers, specifically integers, similar in construction to Godel Numbers. Once music has been encoded symbolically as an integer the fundamental elements of the music may be manipulated, to produce an alternative copy of an eternal Platonic Form. Compositions by John Dowland and Palestrina were encoded symbolically and the encoding was used to control the rhythm, pitch and harmonic structure of new compositions.