Generating Sonic Phantoms with Quadratic Difference Tone Spectrum Synthesis

Quadratic difference tones belong to a family of perceptual phenomena that arise from the neuromechanics of the auditory system in response to particular physical properties of sound. Long deployed as “ghost” or “phantom” tones by sound artists, improvisers, and computer musicians, in this article we address an entirely new topic: How to create a quadratic difference tone spectrum (QDTS) in which a target fundamental and harmonic overtone series are specified and in which the complex tone necessary to evoke it is synthesized. We propose a numerical algorithm that solves the problem of how to synthesize a QDTS for a target distribution of amplitudes. The algorithm aims to find a solution that matches the desired spectrum as closely as possible for an arbitrary number of target harmonics. Results from experiments using different parameter settings and target distributions show that the algorithm is effective in the majority of cases, with at least 99% of the cases being solvable in real time. An external object for the visual programming language Max is described. We discuss musical and perceptual considerations for using the external, and we describe a range of audio examples that demonstrate the synthesis of QDTSs across different cases. As we show, the method makes possible the matching of QDTSs to particular instrumental timbres with surprising efficiency. Also included is a discussion of a musical work by composer Marcin Pietruszewski that makes use of QDTS synthesis.