Diachronic Modeling of Tonal Coherence on the Tonnetz Across Classical and Popular Repertoires

Weilun Xu; Edward Hall; Martin Rohrmeier

cs.SDMar 27, 2026

Diachronic Modeling of Tonal Coherence on the Tonnetz Across Classical and Popular Repertoires

Authors: Weilun Xu, Edward Hall, Martin Rohrmeier

Organizations: School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, Switzerland · Digital and Cognitive Musicology Lab, École Polytechnique Fédérale de Lausanne, Switzerland

Abstract

How do different musical traditions achieve tonal coherence? Most computational measures to date have analysed tonal coherence in terms of a single dimension, whereas a multi-dimensional analyses have not been sufficiently explored. We propose a new model drawing on the concept of the Tonnetz -- we define two partially independent measures: tonal focus, the concentration of pitch content near a tonal center; and tonal connection, the degree to which pitch content reflects structured intervallic pathways back to that center. Analyzing over 2,800 pieces from Western classical and popular traditions, we find that these traditions occupy overlapping yet distinguishable regions of the two-dimensional space. Popular music shows higher tonal focus, while classical music exhibits higher tonal connection. Our complementary measures ground the differences between different tonal styles in quantitative evidence, and offer interpretable dimensions for computational music analysis and controllable generation.

Figures & tables

Figure 1: Two dimensions of tonal organization. (a) Tonal focus (gravitational centering) along the line-of-fifths: Pitch content is concentrated within a small diatonic region around the tonic, showing strong gravitational pull but limited exploration. (b) Tonal connection (structured exploration) across the Tonnetz lattice: Pitch content systematically explores distant tonal regions through structured intervallic pathways, maintaining coherence despite potentially lower concentration near the tonic.

Figure 2: Two dimensions of tonal organization. (a) Tonal focus ( $k$ : window half-width on the line-of-fifths): popular music concentrates more pitch content near the tonic. (b) Tonal connection ( $\lambda$ : TDM path-length parameter): classical music shows higher values with notably lower variance. (c) Tonal archetypes : the two-dimensional space reveals four characteristic regions defined by the combined median of each dimension. All effect sizes $d$ are Cohen’s $d$ (pooled SD); ${}^{*}|d|{\geq}0.2$ , ${}^{**}{\geq}0.5$ , ${}^{***}{\geq}0.8$ .

Figure 3: Characterizing the two dimensions. (a) Mean weight profiles across the six TDM intervals with $\pm$ 1 SD error bars. Classical music allocates substantially more weight to perfect fifths (fifth dominance $d=0.31$ ; weight kurtosis $d=0.63$ ). (b) Ridge plots of tonal focus distributions from $k=2$ to $k=7$ . At $k=3$ (primary threshold), classical music is unimodal while popular music is broader and bimodal; both converge toward 1.0 at large $k$ .

Figure 4: Historical trajectory in the two-dimensional space. Each point represents a composer’s average position, colored by stylistic era; dashed contours show KDE density regions for each era. The arrow traces the centroid path from Baroque through the early 20th century, revealing a progressive expansion from tight, high-focus clustering toward greater dispersion in both dimensions. The small disconnected Romantic contour reflects the density tail of low-focus outliers such as Wagner and Liszt.

Music Information Retrieval

Music Information Retrieval develops computational methods for finding, classifying, describing, and analyzing music from audio, scores, and metadata.

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A Multidimensional Model for Quantifying Tonal Strength: A Continuous Framework of Analyzing Tonal Evolution Beyond Tonal-Atonal Binary Classification

Yuliang Li, Nan Nan, Meilian Gu, Xiaohong Guan

Frontier Institute of Science and Technology, Xi'an Jiaotong University, Xi'an, China · Interdisciplinary Research Center of Frontier Science and Technology, Xi'an Jiaotong University, Xi'an, China · MOE KLINNS Lab, Faculty of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, China+2

Computational Musicology Music Information Retrieval

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Tonality is a fundamental organizing principle of Western music. However, the gradual transition from common-practice tonality to atonality around the turn of the twentieth century cannot be captured by binary labels. To characterize intermediate tonal states and trace this evolution, we propose a multidimensional framework for quantifying tonal strength, grounded in tonal theory and mechanisms underlying tonal dissolution. This framework operationalizes the global statistical regularity, temporal coherence, and local perceptual alignment of tonal organization through three complementary features: pitch-class distribution uniformity, temporal stability of tonal centers, and local tonal clarity, respectively. To validate the model, we introduce two new datasets: a binary dataset comprising 196 tonal and atonal pieces, and a historical MIDI corpus of 1,561 compositions by 21 composers spanning the Baroque era to the early twentieth century. The proposed model achieves high discriminative performance on the binary classification task (F_1=0.985), and quantitatively reveals a continuous historical trajectory from established tonality through progressive weakening to eventual dissolution. Computational analysis of representative composers further confirms significant declines in tonal strength throughout the careers of late-Romantic and modernist figures. Additionally, analysis of EMOPIA dataset reveals a weak but statistically significant positive correlation between higher tonal strength values and positive emotional valence. This work establishes a unified quantitative model for analyzing tonal evolution and provides a continuous measure for future behavioral and neurophysiological investigations of the relationships among tonal organization, musical emotion, and cognition.

Abstract

Music Information Retrieval

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