Quantum Tonality: A Mathemusical Playground

Abstract: We adopt some basic ideas on quantum-theoretical modeling of tonal attraction and develop them further in an alternative direction. Fitting Gaussian Mixture Models (GMM) to the Krumhansl-Kessler (KK) probe tone profiles for static attraction opens the possibility to investigate the underlying wave function as the stationary ground state of an anharmonic quantum oscillator with a schematic Hamiltonian involving a perturbation potential. We numerically verify the fulfilment of the associated stationary Schr\"odinger equation and also inspect its excited states as a solution basis for the corresponding time-dependent Schr\"odinger equation. With their help we calculate the temporal evolution of any initial state. As an example, we study the dynamics of transpositions of the stationary KK wave function across Regener's line of fifths. This offers potential models for dynamic tonal attraction and also for the behavior of deflected key profiles within the Hamiltonian dynamics of a given tonality.