Quantum Models of Tonal Attraction

• Quantum models of tonal attraction are a mathematical formalism that unifies static (goodness-of-fit) and dynamic (predictive progression) aspects using Hilbert space and quantum amplitudes.
• They employ deformed wave functions and Gaussian mixture models to closely match empirical probe-tone experiments, achieving high correlations with tonal hierarchy data.
• Recent Hamiltonian dynamics models capture time-dependent key-deflection and oscillatory relaxation, offering a unified framework for both static and dynamic tonal phenomena.

Quantum models of tonal attraction employ mathematical structures inspired by quantum theory to formalize how listeners perceive the fit (static attraction) and predictive progression (dynamic attraction) of tones with respect to tonal contexts. This interdisciplinary area bridges music psychology, mathematical physics, and computational modeling, using Hilbert space, unitary symmetries, and wave function deformations to account for auditory phenomena such as the structure of the circle of fifths and the empirical profiles obtained in probe-tone experiments. Key models provide unified, transposition-invariant accounts of both static and dynamic tonal attraction, achieving or surpassing the predictive performance of traditional hierarchical and interval-cycle models. Recent developments extend these approaches by leveraging Gaussian mixture fits and time-dependent quantum oscillator theory, yielding frameworks that systematically encode both static key profiles and their dynamic modulation.

1. Distinguishing Static and Dynamic Tonal Attraction

Tonal attraction is probed via two experimental paradigms:

• Static tonal attraction quantifies "goodness-of-fit" in a key context, as operationalized by the Krumhansl & Kessler (1982) probe-tone experiment. After a tonal priming (e.g., scale, cadence, chord), subjects rate the context-fit of a probe tone. The resulting 12-dimensional profile over the chromatic pitch classes captures stable tonal hierarchies.
• Dynamic tonal attraction assesses "completion" or "resolution," following probe-tone methods with altered instructions. After a chord context, subjects rate how well a probe tone resolves or continues the context (e.g., the expectancy of F following C–E–G). This produces expectation profiles indicative of musical continuation (Graben et al., 2017).

Table 1 summarizes the paradigms:

Paradigm	Instruction	Empirical Profile
Static attraction	Rate "fit" of probe	Tonal hierarchy in key context
Dynamic attraction	Rate "completion"/"resolve"	Expectancy profile after chord

2. Tonal Configuration Space: Symmetries and the Circle of Fifths

Quantum models are founded on three key musical symmetries:

• Octave equivalence: Pitches related by frequency-doubling ($f$ and $2f$) belong to the same pitch class, forming a circular $\mathbb{Z}_{12}$ structure for the chromatic scale.
• Fifth similarity: The perfect fifth (frequency ratio $3:2$) underpins adjacency relationships; algebraically, repeated fifth-steps ($g^7$, $g$ a generator of $\mathbb{Z}_{12}$) generate the circle of fifths.
• Transposition symmetry: Transposing all pitch classes by $k$ (mod 12) preserves tonal structure; thus, musical space is best described as $S^1$ (the circle of fifths), enabling transposition-invariant modeling (Graben et al., 2017).

These auditory symmetries establish a natural configuration space, parameterized by $x \in [0,2\pi)$ where $2f$0 indexes the $2f$1-th fifth-step from a tonic.

3. Quantum Wave Function Models for Static and Dynamic Attraction

Hilbert Space Basis and Quantum Amplitudes

The foundational quantum model deploys the Hilbert space $2f$2 of square-integrable functions over the circle of fifths. The simplest ("free") model uses the subspace spanned by tonic and tritone basis states:

• $2f$3 (tonic)
• $2f$4 (tritone)

The static attraction probability between a context $2f$5 and probe $2f$6 is given by the quantum similarity:

$2f$7

For chord contexts $2f$8, the attraction profile is a discrete convolution:

$2f$9

This guarantees transposition invariance (Graben et al., 2017).

Deformed Wave Functions and Hierarchical Structure

Finer patterns, such as the major/minor asymmetry in empirical tonal hierarchies, require deforming the metric in tonal space. The cosine kernel is replaced:

$\mathbb{Z}_{12}$0

where $\mathbb{Z}_{12}$1 is a polynomial (e.g., symmetric fourth-order or general asymmetric fourth-order) with coefficients fixed by analytically imposed boundary conditions or by maximizing correlation with data. This framework recovers the Krumhansl–Shepard profile and adapts to empirical deformations in tonal structure.

Dynamic tonal attraction is analogously modeled by deforming $\mathbb{Z}_{12}$2 using interval-cycle proximity constraints from empirical data, typically via a symmetric sixth-order polynomial so that zero crossings of the wave kernel correspond to minima in dynamic attraction (Graben et al., 2017).

4. Quantum-Oscillator Models and Hamiltonian Formalism

Recent work formalizes tonal attraction as the ground state of an anharmonic quantum oscillator. The configuration variable $\mathbb{Z}_{12}$3 parameterizes the real line of fifths. The Hamiltonian is:

$\mathbb{Z}_{12}$4

where $\mathbb{Z}_{12}$5 (following Lerdahl), and $\mathbb{Z}_{12}$6 is a perturbation potential determined by requiring that the Gaussian mixture model (GMM) density fitted to the Krumhansl–Kessler (KK) data be the oscillator's ground-state probability. For $\mathbb{Z}_{12}$7 major:

• Means: $\mathbb{Z}_{12}$8, $\mathbb{Z}_{12}$9, $3:2$0
• Weights: $3:2$1, $3:2$2, $3:2$3
• The ground state is $3:2$4, with $3:2$5, $3:2$6

The Schrödinger equation

$3:2$7

is verified numerically by diagonalizing the Hamiltonian in the Hermite basis. The ground-state wave function achieves $3:2$8 (constrained means) to $3:2$9 (free means) correlation with major key KK data (Graben et al., 2024).

5. Modeling Dynamic Phenomena: Time-Dependence and Key Deflection

Excited state solutions $g^7$0 of the oscillator Hamiltonian $g^7$1 form an orthonormal basis:

$g^7$2

An arbitrary initial state, such as a key-deflected profile $g^7$3, is expanded as

$g^7$4

Its time evolution follows

$g^7$5

For upward or downward deflection along Regener’s line of fifths, the probability density $g^7$6 (computed with $g^7$7 states) exhibits oscillatory relaxation toward the tonic, analogous to coherent-state oscillations in an anharmonic oscillator. This dynamical treatment provides a principled model of key-deflection and time-dependent phenomena in tonal music, extending the framework to both static and dynamic attraction (Graben et al., 2024).

6. Parameter Estimation and Comparative Predictive Performance

• Static, symmetric deformation: The two-parameter model is fully specified via imposed boundary conditions ($g^7$8, $g^7$9) without direct least-squares fitting.
• Static, asymmetric deformation: Asymmetric coefficients are selected by maximizing correlation with empirical data, notably achieving $g$0 with C-major KK data.
• Dynamic deformation: Parameters are set algebraically to interpolate zeros of interval-cycle proximity (ICP) patterns.
• Hamiltonian model: GMM parameters are fit to KK profiles; the ground-state density provides the static attraction model.

Comparative correlations (with empirical data) are summarized below:

Model	Static $g$1 (C Maj/C Min)	Dynamic $g$2 (Diverse Contexts)
Hierarchical (Lerdahl–Jackendoff)	0.98 / 0.95	—
Free pure-state cosine	0.70 / 0.68	—
Free mixed-state cosine	0.78 / 0.72	—
Symmetric deformation pure	0.89 / 0.80	—
Symmetric deformation mixed	0.97 / 0.93	—
Asymmetric deformation (pure)	0.95 / —	—
Asymmetric deformation (hierarchical)	0.93 / —	—
Quantum GMM ground state (Graben et al., 2024)	0.989–0.995 (major)	—
ICP (dynamic, Woolhouse)	— / —	0.69–0.89 (contexts)
Quantum dynamic (deformation)	— / —	0.44–0.93 (contexts)

The quantum model outperforms or matches classical accounts across static (hierarchical, symbolic) and dynamic (ICP) prediction tasks, and does so within a unified general formalism (Graben et al., 2017, Graben et al., 2024).

7. Theoretical Advantages and Interpretative Implications

Quantum models of tonal attraction offer several structural and practical benefits:

• Unified structural and probabilistic account: Both symmetry-based musical structure and probabilistic distributional predictions are subsumed into a Hilbert-space framework, leveraging quantum amplitudes and unitarity.
• Hierarchy generation: The full range of musical hierarchies (fifths, triads, diatonic/chromatic distinctions) emerges from minimal wave function deformation, rather than requiring ad hoc level specification.
• Transposition invariance: All models naturally implement global symmetry via unitary rotations—context-independent encoding of musical relationships.
• Gestalt-field interpretation: Musical similarity and consonance are encapsulated by quantum-probability amplitudes, realized as "wave-fields" on $g$3.
• Hamiltonian dynamics: Time-dependent key-deflection and nonstationary phenomena are modeled systematically as quantum dynamics of the ground state and excited states.
• Parsimony and fit: Small numbers of analytically-constrained parameters yield high correlations with empirical measurements, demonstrating parsimony and empirical adequacy.

A plausible implication is that quantum models capture both the stable structural core of tonal cognition and its dynamical flexibility, providing a mechanistically interpretable and empirically robust formalism for diverse tonal phenomena (Graben et al., 2017, Graben et al., 2024).