Computational modeling of interval distributions in tonal space reveals paradigmatic stylistic changes in Western music history

The paper investigates how stylistic changes in Western art music manifest in tonal interval structures over approximately 600 years. Building on music-theoretical conceptions of tonal space (e.g., the Tonnetz), the authors ask: (1) whether there is a historical trend in the exploration of tonal space, and (2) how the relative importance of the primary intervals (perfect fifths, major thirds, minor thirds; each in ascending and descending directions) varies over time. The work situates itself within computational musicology, extending prior corpus-based findings that suggest increasing harmonic complexity, decreasing diatonicity, and evolving tonal practices across periods.

The study reviews computational and theoretical research on diachronic stylistic change: corpus analyses of chord transitions, intervals, and tonal complexity (Weiß et al., 2019); statistical models showing increasing dissonance (Nakamura & Kaneko, 2019); topic modeling of pitch-class distributions showing stable diatonic-key-like topics (Moss & Rohrmeier, 2021); Bayesian modeling of musical modes revealing changes in mode number/shape (Harasim et al., 2021); expansion of tonal material along the line of fifths (Moss et al., 2023). Fourier analyses indicate decreasing diatonicity and period/composer-specific tonal signatures (Yust, 2019; Viaccoz et al., 2022). Entropy-based harmonic complexity increases over centuries (Nardelli et al., 2022). Time-series analyses highlight irreversibility and increasing novelty at period transitions (González-Espinoza et al., 2020; González-Espinoza & Plotkin, 2023). Within-composer and genre studies also show stylistic evolution (e.g., Debussy: Laneve et al., 2023; Brazilian Choro: Moss et al., 2020b). Historically, tonal spaces have been theorized via Euler’s diagram and the Tonnetz (Hauptmann, Hostinský, Riemann, etc.), typically privileging perfect fifths and thirds as primary relations. The present study builds on this lineage via the Tonal Diffusion Model (Lieck et al., 2020).

Corpus: The Tonal Pitch-Class Counts Corpus (TP3C; 2,012 pieces by 75 composers, spanning ~600 years) aggregates tonal pitch-class counts from diverse symbolic sources converted to MusicXML. Each piece is represented as a 35-dimensional vector of tonal pitch classes ordered along the line of fifths (Fbb to B##). The corpus is historically imbalanced and reflects canon and encoding biases; results are interpreted with these caveats in mind. Model: The Tonal Diffusion Model (TDM) treats each piece as a bag of tones generated from a single tonal center via diffusion along the Tonnetz using primary intervals: ± perfect fifth (±P5), ± major third (±M3), ± minor third (±m3). The model parameters per piece are: (a) tonal center c, (b) interval weight distribution w over the six directed primary intervals, and (c) diffusion parameter λ controlling path-length distributions (preference for shorter vs longer paths). For each observed tone, the model marginalizes over all possible diffusion paths from c with dynamic programming, weighted by w and governed by λ. Parameters (c, w, λ) are inferred to maximize the likelihood of the observed pitch-class distribution under uniform priors. Smoothing and uncertainty: Diachronic trends are estimated using LOWESS (statsmodels), applied to 250 bootstrap resamples of the corpus to visualize uncertainty. The fraction parameter is chosen to yield reasonably smooth curves; raw data points are displayed alongside to show distribution density. Operationalizations: exploration of tonal space is measured by λ; relative importance of primary intervals is measured by the inferred weights of ±P5, ±M3, ±m3 across time.

- Diffusion strength (λ): LOWESS curves across 250 bootstrap samples show low variance and values confined to ~0.8–0.9, indicating modest overall change in average tonal spread around the center but a clear increasing historical trend. From the late 14th to the late 17th century, λ rises nearly linearly; the 18th and 19th centuries exhibit local mid-century maxima while maintaining the overall upward trajectory. Diffusion strength correlates positively with the line-of-fifths width (Pearson r = 0.254); on average, a 0.1 increase in λ corresponds to an increase of ~1.54 fifths in span. - Primary interval weights: From the late 14th to late 17th century, ±P5 overwhelmingly dominate, with near-zero weights for ±M3 and ±m3. During this period, the model explains observed third relations primarily as sequences of fifths rather than direct third steps. Within this window, ascending P5 gradually increases at the expense of descending P5, suggesting a growing tendency to relate tones via ascending-fifths paths from a flatter tonal center to sharper pitch material. - 18th century shift: A marked reversal occurs, with descending P5 rising significantly (and ascending P5 declining), yielding a more balanced proportion of authentic/plagal motions. - 19th century thirds surge: Weights for both major and minor thirds rise sharply (especially +M3, −m3, and −M3, with a smaller rise for −M3), sufficiently impacting the relative prominence of P5. Third-based explanations peak around the mid-19th century before decreasing toward the late 19th century. Absolute third weights remain smaller than fifths, but their relative growth in the 19th century is substantial compared to earlier periods. - Among thirds, major thirds (±M3) exhibit the weakest absolute weights overall, contrasting with some historical theoretical expectations, though theoretical importance and empirical frequency need not coincide.

The increasing λ over centuries addresses the first research question by evidencing a long-term expansion in the exploration of tonal space: tones tend to be related to the tonal center via longer, more complex paths on the Tonnetz, consistent with historical increases in chromaticism and enharmonic practice. Regarding the second question, the results show that perfect fifths dominate interval relations for several centuries, with a nuanced directional asymmetry (rising +P5 until the late 17th century, then a strong 18th-century rebound of −P5). The 19th-century surge in third-based relations indicates a stylistic shift wherein direct third connections become more prevalent in explaining pitch-class distributions, aligning with the emergence of extended tonality and corroborating prior computational findings on decreased diatonicity and increased harmonic complexity. These findings demonstrate how a computationally grounded model tied to music-theoretic structures can trace diachronic stylistic dynamics in large corpora and complement example-based historiography.

The study shows: (1) a clear long-term increase in the diffusion strength λ, indicating progressively complex tonal interval relations; (2) centuries-long dominance of perfect fifths in structuring tonal relations; and (3) a pronounced 19th-century rise in both major and minor third relations coinciding with extended tonality. Methodologically, the work illustrates how formal, interpretable computational models rooted in tonal theory can elucidate historical trends from large corpora of pitch-class counts. Future research directions include: extending models to richer musical representations (rhythm, harmony, voice-leading), relaxing single-tonal-center assumptions, modeling transmission mechanisms and causality within hierarchical generative frameworks, expanding corpora beyond canonical repertoires, and extending analyses into the 20th century and beyond.

- Observational, correlational design: trends cannot establish causality for stylistic changes. - Corpus biases and imbalance: TP3C is not a representative, uniformly sampled dataset; it reflects canon formation and availability of encodings. - Model assumptions: single tonal center per piece; bag-of-tones independence; interval relations constrained to the six primary intervals; reliance on pitch-class counts only (no temporal/voice-leading context). - Smoothing and edge effects: LOWESS trends near corpus boundaries (late periods) are sensitive to data sparsity, potentially affecting late-19th/early-20th-century inferences. - Discrepancies between theoretical importance and empirical frequency: frequency-based measures may not align with axiomatic or psychoacoustic assessments of interval salience.