Fourier Oracles for Computer-Aided Improvisation

The mathematical study of the diatonic and chromatic universes in the tradition of David Lewin (9) and John Clough (6) is a point of departure for several recent investigations. Surprisingly, Lewin’s original idea to apply finite Fourier transform to musical structures has not been further investigated for four decades. It turns out that several music-theoretically interesting properties of certain types of musical structures, as the partial symmetry of Fourier balances, maximal evenness (5) and well-formedness (4), allow alternative characterizations in terms of their Fourier transforms. The paper explores two particularly interesting cases: vanishing Fourier coefficients as an expression of chord symmetry and maximal Fourier coefficients as a reinterpretation of maximally even scales. In order to experimentally explore the Fourier approach we design an interactive playground for rhythmic loops. We propose a Fourier-based approach to be integrated as an ”Scratching”-interface in the OMAX environment (built on OpenMusic and MaxMSP) which allows to interactively change a rhythm through a gestural control of its Fourier image. A collection of theoretical tools in OpenMusic visual programming language helps the improviser to explore some new musical situations by inspecting mathematical and visual characteristics of the Fourier image. 1 David Lewin’s call for Fourier In a few lines at the end of his very first paper (9), David Lewin tantalizingly alludes to Fourier transforms and convolution products to explain how he was led to the special symmetries he considers in his paper about the relative intervallic content of two chords. In the present paper we elaborate some of Lewin’s ideas by applying the Finite Fourier Transform (FT) to more several of musical structures. To begin with, we model the set of notes modulo octave by Zc (c = 12 in the equal division of the octave in twelve parts) and define, in this section, the Fourier transform of a subset A as a FT of its characteristic function: FA : t !→ ∑