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

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: \emph{tonal focus}, the concentration of pitch content near a tonal center; and \emph{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 (4)

• 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.