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The European Physical Journal Special Topics

Exploring quantum tonality

Thomas Noll, Felix Pastor Olives, Pedro González-Fernández, Peter beim Graben

Published Oct 28, 2026• 1 citation
Publisher

Abstract

Abstract This article reports on a series of mathe-musical experiments based on the dynamic behavior of a musical quantum model of C-major tonality. The model reinterprets empirical pitch-class profiles of tonal attraction in C-major as the probability wave function of a one-dimensional stationary ground state within a perturbed quantum harmonic oscillator. By solving the associated time-dependent Schrödinger equation, we assign specific temporal evolutions to any given initial state. We explore two distinct sonification strategies: First, the time-evolutions of various initial states—in particular deflections of the ground state and higher excited states—are used to drive a stroboscopic stratification of musical hierarchies and to set them into Minimal Music-inspired travelling note patterns. Secondly, we implement rich continuous sound processes that rely on the uninterrupted flow of the wave function. Crucially, these approaches bypass the simulation of quantum measurements; instead, they leverage the full theoretical information of the wave function’s evolution.

DOI
10.1140/epjs/s11734-026-02614-2

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Abstract This article reports on a series of mathe-musical experiments based on the dynamic behavior of a musical quantum model of C-major tonality. The model reinterprets empirical pitch-class profiles of tonal attraction in C-major as the probability wave function of a one-dimensional stationary ground state within a perturbed quantum harmonic oscillator. By solving the associated time-dependent Schrödinger equation, we assign specific temporal evolutions to any given initial state. We explore two distinct sonification strategies: First, the time-evolutions of various initial states—in particular deflections of the ground state and higher excited states—are used to drive a stroboscopic stratification of musical hierarchies and to set them into Minimal Music-inspired travelling note patterns. Secondly, we implement rich continuous sound processes that rely on the uninterrupted flow of the wave function. Crucially, these approaches bypass the simulation of quantum measurements; instead, they leverage the full theoretical information of the wave function’s evolution.
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This work investigates a case study of using physical-based sonification of Quadratic Unconstrained Binary Optimization (QUBO) problems, optimized by the Variational Quantum Eigensolver (VQE) algorithm. The VQE approximates the solution of the problem by using an iterative loop between the quantum computer and a classical optimization routine. This work explores the intermediary statevectors found in each VQE iteration as the means of sonifying the optimization process itself. The implementation was realised in the form of a musical interface prototype named Variational Quantum Harmonizer (VQH), providing potential design strategies for musical applications, focusing on chords, chord progressions, and arpeggios. The VQH can be used both to enhance data visualization or to create artistic pieces. The methodology is also relevant in terms of how an artist would gain intuition towards achieving a desired musical sound by carefully designing QUBO cost functions. Flexible mapping strategies could supply a broad portfolio of sounds for QUBO and quantum-inspired musical compositions, as demonstrated in a case study composition, 'Dependent Origination' by Peter Thomas and Paulo Itaborai.
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