Quantum-Duality Models for Tonal Modulation

• Quantum/duality models for tonal modulation are mathematically rigorous frameworks that recast tonal spaces using quantum theory and symmetry groups.
• They integrate Hilbert spaces, unitary operators, and group representations to quantify key shifts, tonal attraction, and dynamic modulation.
• They offer dynamic simulation of modulation and empirical validation through quantum oscillators and probabilistically-timed models.

Quantum and Duality Models for Tonal Modulation offer a mathematically rigorous framework for describing, predicting, and analyzing the dynamic and structural aspects of tonal movement in Western music. These approaches recast tonal spaces, modulations, and attraction phenomena within the vocabulary of quantum theory, unitary group actions, and duality principles—enabling not only symbolic or algebraic but also probabilistically-timed models of key shifts and harmonic navigation. The theoretical underpinnings integrate Hilbert space formalism, groupoid representations (notably the affine TI and PLR groups), and quantum oscillators fit to empirical tonal profiles.

1. Mathematical Foundation: Hilbert Spaces, Operators, and Symmetry

Quantum/duality treatments of tonality rely on viewing the set of pitch classes, $\mathbb{Z}_{12}$, as a basis of a complex Hilbert space $$H_{\mathrm{pc}} = \mathrm{span}_{\mathbb{C}}\{\,|0\rangle,|1\rangle, \ldots, |11\rangle\,\}$$, with the standard orthonormal structure $\langle p|q\rangle=\delta_{p,q}$ (Agustín-Aquino et al., 21 Dec 2025, Graben et al., 2017). Chordal entities such as triads inhabit higher tensor powers $$H_{\mathrm{tri}}=H_{\mathrm{pc}}\otimes H_{\mathrm{pc}}\otimes H_{\mathrm{pc}}$$ or optimized subspaces for consonant triads (e.g., 24-dimensional for Major/Minor sets).

The primary symmetries are transpositions $T_n|p\rangle=|p+n\rangle$ and inversions $I_n|p\rangle=|-p+n\rangle$, structured as the affine group $\mathbb{Z}_{12}\rtimes\mathbb{Z}_2$ and acting diagonally on chord and interval states. The third functional group, $\langle P,L,R\rangle$, defines transformations on triads via the well-known PLR (parallel, leading-tone exchange, relative) operators, forming a group isomorphic to TI, with demonstrated commutation in chordal Hilbert space (Agustín-Aquino et al., 21 Dec 2025).

Dual-number extensions ($\mathbb{Z}_{12}[\varepsilon]$ with $\varepsilon^2=0$) model counterpoint intervals, while their symmetries act as unitaries on $H_{\mathrm{cp}}$—enabling a “quantum” formulation of contrapuntal move admissibility via overlap magnitudes $|\langle\eta|G|\xi\rangle|$ (Agustín-Aquino et al., 21 Dec 2025).

2. Quantum Oscillator Models: Static and Dynamic Tonal Attraction

A fully fledged quantum-theoretical model for tonal attraction emerges in the construction of a Schrödinger operator on the continuous cycle representing Regener's line of fifths ($q\in\mathbb{R}$, but periodic $\bmod\,2\pi$) (Graben et al., 3 Apr 2024, Graben et al., 2017). The central object is the Hamiltonian

$H = -\frac{d^2}{dq^2} + E_0^2 q^2 + W(q)$

where $E_0^2q^2$ encodes harmonic restoration and $W(q)$ is a perturbation selected so that the ground state probability $|\psi_0(q)|^2$ matches empirical Krumhansl–Kessler probe-tone distributions, fitted via a Gaussian mixture model (GMM).

Ground-state solutions ($H\psi_0=E_0\psi_0$) thus directly embody the static hierarchy of tonal attraction, while excited eigenstates $\psi_n(q)$, $n>0$, form a basis for expansion of arbitrary tonal distributions and their quantum time evolution.

3. Time-Dependent Dynamics and Modulation

To simulate modulation and dynamic phenomena, an initial “deflected key” state—formally, a transposed ground-state profile $\psi_0(q-a)$ for some $a$ (e.g., one perfect fifth, $a=\pm\pi/6$)—is expanded in the eigenbasis: $\Psi(q,0) = \sum_n R_n \psi_n(q)$. Under temporal evolution,

$$\Psi(q,t) = \sum_n R_n\,e^{-iE_n t/\hbar}\,\psi_n(q)$$

the profile $|\Psi(q,t)|^2$ oscillates and relaxes toward the original ground-state hierarchy. This quantifies the re-centering of key profiles after transposition and provides a unified explanation for the temporal aspects of tonal attraction and key modulation (Graben et al., 3 Apr 2024).

In operator-theoretic (Hilbert-space) representations, modulation via transpositions and inversions is realized strictly as application of unitary operators $U_m$ to tonic or cadence-state vectors. Thus, modulation paths, including classical functional routes (I→V, I→IV, I→vi), are mapped to continuous-time quantum transitions, with probabilities for “arrival” in a new tonal region given by Born densities and explicit transition amplitudes (Graben et al., 2017, Agustín-Aquino et al., 21 Dec 2025).

4. Group-Theoretic Duality and the PLR/TI Formalism

Complementing quantum models, the duality perspective treats modulation and counterpoint through the interplay of symmetry groups. The action of the PLR group on triads, and its commutativity with TI on $H_{\mathrm{tri}}$, provides a dual action—one by functional harmonic transformations and one by transpositional-inversional geometry. This dual framework, grounded in the work of Mazzola and elaborated by Agustín-Aquino and Curiel-López, enables characterization of both classical modulations and “modulation quanta” (sets $M\subset\mathbb{Z}_{12}$) (Agustín-Aquino et al., 21 Dec 2025).

Group action allows construction of rigidity metrics (e.g., $r(S\to T) = |M\cap T|/|M|$) which quantify the smoothness or strength of a modulation, corresponding to the presence and role of pivot chords along traditional voice-leading paths.

In counterpoint, quantizing the space of intervals leads to the formulation of an admissibility indicator $A_g(\xi,\eta) = \langle\eta|G|\xi\rangle$ measuring the nonzero overlap under a given counterpoint symmetry, and the definition of parsimony indices. Steps with “ultra-parsimonious” single symmetry (P=1) are formally distinguished from higher cardinality intersections required in homophonic or mixed-texture contexts (Agustín-Aquino et al., 21 Dec 2025).

5. Quantum-Cognition Interpretations and Empirical Fits

Quantum-cognition models embed tonal processing within the architecture of a 12-dimensional Hilbert space, exploiting the symmetry structure of the chromatic system (notably the dihedral group $D_{12}$ generated by $T$, $I$), and encoding attraction and preference hierarchies as outputs of wave function evolution (Graben et al., 2017). Static attraction profiles are expressed as Born densities $|\psi_a(\theta)|^2$ for angle $\theta$ along the circle of fifths, with model fits (e.g., $P_0(\theta)=\cos^2(\theta/2)$ for C major) reproducing empirical data.

Dynamic modulation is modeled by time-evolution under a Hamiltonian encoding diatonic couplings: $$H = -J_d \sum_k (|k\rangle\langle k+7|+\text{h.c.}) - J_m \sum_k (|k\rangle\langle k+4|+\text{h.c.}) - J_s \sum_k (|k\rangle\langle k+5|+\text{h.c.}),$$ where $J_d>J_m>J_s>0$. The relative rates and probabilities for I→V, I→IV, and I→vi modulations follow immediately from the resulting quantum mechanical transition amplitudes, providing quantitative confirmation of classical function theory (Graben et al., 2017).

A direct analogy is drawn between position–momentum complementarity and the Fourier-dual bases: maximal tonal coherence implies pitch-class delocalization and vice versa. This complementarity grounds the duality between discrete pitch focus and global key perception.

6. Application to Historical Repertoires and Model Validation

The formal methodology enables explicit analysis of historical works such as Monteverdi’s L’Orfeo. Cadence structures and modulation chains (e.g., $C \to D \to A \to C$) are represented as sequences of unitary transformations acting on Hilbert-state vectors of tonalities or chords, with pivot structures and parsimony readily quantified via overlap and rigity indices (Agustín-Aquino et al., 21 Dec 2025).

Example computations for counterpoint utilize the dual-number Hilbert space and symmetry actions to determine admissibility and maximal intersection conditions at critical contrapuntal junctures. Empirical parsimony measures for specific interval progressions (e.g., $5$–$3$ with $P=1$) align precisely with classical descriptions of consonance and harmonic simplicity in the literature.

7. Synthesis and Prospects

Quantum and duality models for tonal modulation synthesize data-driven, psychological, and algebraic approaches within a parsimonious, predictive, and mathematically explicit formalism. They endow traditional harmonic and contrapuntal structures with operator-theoretic and dynamical meaning, unify empirical data fitting (via GMM and quantum oscillators), and connect musical phenomena to principles of symmetry, group action, and time-evolution in Hilbert space (Graben et al., 3 Apr 2024, Agustín-Aquino et al., 21 Dec 2025, Graben et al., 2017).

• A plausible implication is that generalized operator algebras and their spectral properties could support more complex analyses of voice-leading, modulations in chromatic music, and entanglement phenomena in polyphony. Future research may extend these frameworks to Fock-space models of multi-voice interactions and further integration with cognitive and computational musicology.