Learned Continuous Synthesis of Quadratic Difference Tone Spectra

Abstract: Quadratic difference tones (QDTs) are a species of auditory distortion product in which a "phantom" pure tone, absent from the acoustic signal, is clearly audible to listeners. Exploiting this phenomenon, one can synthesize harmonically rich tones for musical purposes, a technique called Quadratic Difference Tone Spectrum (QDTS) synthesis. Previous works have introduced numerical methods to synthesize QDTS based on the distortion function, which links a target QDTS and an overtone-structured carrier signal. While accurate, these methods were stochastic and discontinuous, making them difficult to control for musical purposes and effectively limiting them to stationary signals. This paper proposes a neural network-based approach that learns an approximate inverse of the distortion mapping in an autoencoder-like configuration, producing a continuous approximation that addresses prior limitations. Experimental results show that, although slightly less numerically precise, the method is sufficient for perceptual and musical applications. We also implement a real-time version in Max and evaluate its performance. Various sound examples demonstrate its expressive and musical potential. The source code, audio examples, tutorials, and software accompanying this work are available at https://cordutie.github.io/projects/qdts.html