Tonal Centrality in Music Theory

• Tonal centrality is a measure of how pitches in chords or scales cluster around a perceptual and mathematical center, defined using algebraic invariants and psychoacoustic mappings.
• It integrates methods from statistical physics, quantum models, and topology to reveal chord symmetry, harmonic tension, and melodic balance.
• Computational descriptors and machine learning approaches validate tonal centrality by correlating theoretical metrics with listener-based ratings and emotional responses.

Tonal centrality is a multifaceted concept in music theory and cognition, referring to how the constituent pitches of a chord, scale, or piece cluster around a perceptual, mathematical, or structural "center" in tonal space. Across mathematical harmony analysis, information-theoretic models, neurocomputational and quantum frameworks, and perceptual studies, it integrates invariant functions, psychoacoustic mappings, probabilistic syntax, network topology, and listener-dependent constructs to delineate the organizing principles of pitch and harmony.

1. Mathematical Invariants and the Structure of Chord Space

In algebraico-geometric approaches to harmony (Ryan, 2016), tonal centrality is captured by invariants that are unaffected by transposition. Each chord, when represented via integer frequency ratios, yields a set $CH$ from which the greatest common divisor (GCD) and lowest common multiple (LCM) are computed. The chord “Complexity” is given by:

$CY = \text{LCM}(CH) / \text{GCD}(CH)$

Low CY values correspond to chords that use small integers—modern mathematical reformulation of consonance.

The divisors of CY, called ComplexitySpace (CYS), form a lattice (analogous to the Tonnetz), where the arrangement of a chord's notes within this divisor lattice quantifies its centrality. Chords sit "centrally" when they are balanced geometrically within CYS. To accommodate the auditory logarithmic perception of pitch:

$$LCH(n) = \log_2[CH(n)], \quad LGCD = \log_2[\text{GCD}(CH)], \quad LLCM = \log_2[\text{LCM}(CH)], \ LCY = LLCM - LGCD$$

The LogMidpoint (LM), as the average position in LogComplexitySpace, mathematically represents the “central pitch” of the chord. Otonality and utonality, normalized coefficients based on LM and LCY, indicate how centrally each chord’s notes are distributed, with formulas such as:

$$OTC = (\text{AdjustmentFactor}) \cdot \frac{LCY - 2\,LM}{LCY}$$

Tonal centrality thus reflects a chord’s symmetrical placement within its mathematical invariant space, independently of absolute pitch.

2. Tonal Consonance, Entropy Principles, and Melodic Centrality

Statistical and physical approaches link microscopic properties of sound and interval energy to the macroscopic organization of melody (Useche et al., 2016). Tonal consonance for intervals is quantified by:

$f_j^2 - f_i^2 = (f_j + f_i)(f_j - f_i)$

and further linked to energy differences $$\epsilon_j - \epsilon_i = 2\pi^2 \rho T (f_j^2 - f_i^2)$$.

The statistical distribution of melodic intervals follows an exponential or asymmetric Laplace law:

$$P(\epsilon) = F^{H}_+ e^{-\epsilon / G^{H}_+}, \text{for}~\epsilon > 0,$$

and is modeled by minimizing relative entropy (Kullback-Leibler divergence) under constraints:

• normalization ($p_a + p_d + p_u = 1$),
• mean absolute interval size ($\langle|\epsilon|\rangle$),
• average asymmetry $\langle\epsilon\rangle$.

Lagrange multipliers ($\lambda_1$, $\lambda_2$) quantify centrality: $\lambda_1$ relates to register localization (and overall centrality), while $\lambda_2$ reflects asymmetry. Tonal centrality thus emerges from the density and optimization of interval sizes—musically, composers balance physical stimulus and psychoacoustic constraints to keep melodies “centered.”

3. Quantum, Geometric, and Topological Models of Tonal Attraction

Quantum cognition frameworks (Graben et al., 2017, Graben et al., 3 Apr 2024) encode tonal centrality using wave functions on symmetric spaces reflecting octave equivalence, fifth similarity, and transposition invariance (e.g., the circle of fifths, isomorphic to $\mathbb{Z}_{12}$). The core quantum model for static attraction is:

$$\psi(x) = \frac{1}{\sqrt{\pi}}\cos\left(\frac{x}{2}\right), \quad p(x) = |\psi(x)|^2$$

where the squared modulus gives tonal “attraction” (centrality) probabilities, maximized at the tonic (center of the key).

Probabilistic hierarchy is achieved through deformation functions:

$\psi(x) = A\cos(\gamma(x))$

allowing finer fitting to empirical tonal hierarchy profiles. Dynamic centrality is modeled through the evolution of displaced wavepackets governed by the time-dependent Schrödinger equation, such that central profiles relax toward equilibrium—conceptually corresponding to tonal resolution.

Geometric models (Himpel, 2022) deploy a stratified chord space $C$ endowed with a Riemannian metric, where the geodesic distance between chords quantifies voice-leading centrality:

$d(p,q) = \inf_{p} \int_a^b \|p'(t)\|dt$

Chords that minimize aggregate geodesic distance to the others serve as “centers”, while psychoacoustic height functions (e.g., periodicity, roughness) refine this by favoring chords perceived as more consonant.

Topological representations (Nardelli, 2020) treat compositions as score networks: chords are nodes, transitions are directed edges. Centrality maps to node degree distributions (hubs with elevated recurrence), modularity classes (tonal regions), and hierarchical levels, all naturally emergent from network topology.

4. Computational Descriptors and Predictive Models

Signal processing and machine learning perspectives extract descriptors from time-frequency representations, such as energy-based tonality $T(E)$ via gammachirp cochleagram pattern analysis (Elburg et al., 2017). Horizontal tract features isolate tonal centrality as regions of coherent harmonic energy:

$$T(E) = \log\left(\sum_{t,f}\sigma(T_-(t, f) - \Theta)E(S, t, f)\right) - \log\left(\sum_{t,f}E(S, t, f)\right)$$

Observed strong correlation with human tonality ratings indicates that enhanced horizontal energy patterns denote perceptually “central” tonal content.

In harmony prediction, finite-context (n-gram/PPM*) models outperform RNNs in capturing probabilistic syntactic centrality of chord progressions—especially where rare chord events are syntactically pivotal (Sears et al., 2018). Cross-entropy metrics:

$$H_m(p_m, e_1^j) = -\frac{1}{j}\sum_{i=1}^j \log_2 p_m(e_i|e_1^{i-1})$$

demonstrate improved predictive capacity for models that efficiently allocate probability mass to central harmonic structures.

5. Tonal Centrality, Tension, and Emotional Perception

Tension measures derived from geometric models (spiral arrays) (Cancino-Chacón et al., 2018, Guo et al., 2020)—cloud diameter and tensile strain—quantify centrality by evaluating spread and displacement from the key's center. Controlled music generation is then possible by manipulating latent tension vectors, offering actionable models for shifting tonal centrality while retaining rhythmic structure.

Experimental music cognition demonstrates that tonal centrality is integrally linked to emotional valence: stable, central scale degrees (tonic, 3rd, 5th) are associated with happier emotional responses both in children (Suberry et al., 2020) and adults (Maimon et al., 2021). These associations are evident in both explicit probe-tone matching and implicit reaction time tasks, revealing dual processing mechanisms—one reflective and conceptual, the other automatic and perceptual.

6. Perceptual and Philosophical Extensions

Recent phenomenological perspectives recast pitch—and thus tonal centrality—not as an objective invariant but as an emergent, context- and listener-dependent percept (Deruty, 17 Jun 2025). Concepts such as tonal fission (multiple pitches from a single quasi-harmonic tone) and multistability (perceptual shifting of pitch centers) reflect the dynamism and variability of tonal centers, paralleling ideas from statistical physics (coastline paradox analogy). This suggests tonal centrality in contemporary music may arise from the intricate interplay of acoustic structure and perceptual resolution, rather than fixed theoretical categories.

7. Implications for Theory, Analysis, and Generation

Unified across methodologies, tonal centrality acts as a nexus for understanding the organization, expectation, and emotional meaning of pitches:

• Invariant-based models promote algorithmic classification and optimization of consonant harmonies,
• Quantum/geometric/topological frameworks enable mathematically rigorous mapping of centrality,
• Machine-learned and signal-based descriptors facilitate analysis and synthesis of tonally focused music,
• Cognitive studies reveal centrality's foundational role in listener affect and expectation,
• Perceptual models challenge the assumption of discrete, objective tonal centers, recasting them as emergent features shaped by acoustic detail and listener dynamics.

Tonal centrality remains an analytic anchor point, simultaneously a quantifiable measure, a perceptual attractor, and an evolving construct in harmony, cognition, and music technology.