Diachronic stylistic changes in Western art music are a central theme in musicology. Music theorists and historians have long identified stylistic characteristics common to pieces from specific composers or eras, using these to distinguish between styles like Baroque and Romantic. Recent years have seen a rise in computational approaches to analyzing large music corpora to investigate large-scale diachronic developments. Studies have examined musical periods based on audio features, chord transitions, intervals, and tonal complexity, often visualizing changes on evolution curves to demonstrate diachronical developments. Other research has employed statistical models to identify trends in dissonant intervals, analyzed tonal material expansion along the line-of-fifths, and used the Discrete Fourier Transform (DFT) to identify tonal characteristics. While some studies have noted the irreversibility of musical time series, pointing to rich deep structures, others have focused on specific eras, like the Classical period, providing a more nuanced understanding of harmonic complexity and novelty. This study contributes to this growing body of work by specifically focusing on tonal interval distributions and their changes throughout the history of Western music. It uses the Tonal Diffusion Model (TDM), a computational model that formalizes music-theoretic conceptualizations of tonal space, to analyze a large corpus, investigating how changes in the prevalence of primary intervals relate to the history of tonality. The main research questions are: (1) Can we observe a historical trend in the exploration of tonal space? (2) What is the relative importance of the primary intervals, and how do they vary over time?

The paper reviews existing literature on computational musicology and its application to the study of stylistic changes in music. It highlights various approaches, including the use of large-scale corpora, analysis of audio recordings, extraction of musical features, and the application of statistical models like Latent Dirichlet Allocation (LDA). The authors discuss studies focusing on chord transitions, intervals, tonal complexity, and the evolution of musical modes. They also mention studies using the Discrete Fourier Transform (DFT) to analyze pitch-class distributions and the development of dynamical score networks for measuring harmonic complexity. The literature review emphasizes the use of diverse methodologies and the challenges associated with working with historical data, acknowledging issues of data quality, imbalance, and biases in corpus construction.

The study uses the Tonal Pitch-Class Counts Corpus (TP3C), a dataset containing tonal pitch-class counts from 2,012 pieces by 75 composers spanning approximately 600 years. The TP3C, while not a perfectly representative sample, combines data from multiple open resources, reflecting a 'consensus strategy' in digital musicology. The authors acknowledge the inherent imbalances and biases in historical datasets and the effects of these on analysis. The core methodology employs the Tonal Diffusion Model (TDM). The TDM models tonal space based on the Tonnetz, representing pieces as distributions of tonal pitch classes and estimating a tonal center. It models the generation of tones through a diffusion process along primary intervals (perfect fifths, major and minor thirds). The model infers interval weights and a diffusion parameter for each piece. The diffusion parameter indicates the complexity of tonal relations. The interval weights reflect the relative importance of each primary interval. The authors use Locally Weighted Scatterplot Smoothing (LOWESS) with bootstrapping to analyze temporal changes in these parameters, assessing historical trends and uncertainties. The study operationalizes its research questions by linking the diffusion parameter to the exploration of tonal space and the interval weights to the importance of primary intervals.

The analysis reveals a clear trend toward increasing complexity in tonal interval relations over time. The diffusion parameter, indicating the average length of paths on the Tonnetz, shows a generally increasing trend from the 14th to the 19th centuries, suggesting more complex intervallic relationships. This correlates with the findings of another study using fifth width as a measure of tonal spread. The study also finds that perfect fifths overwhelmingly dominate the primary interval distributions from the late 14th to the late 17th centuries. During this period, the model explains minor and major thirds primarily as sequences of fifths rather than direct steps, reflecting the dominance of diatonic structures. A shift occurs in the 18th century, with a noticeable increase in descending perfect fifths. However, the most striking finding is the sharp increase in the relative importance of major and minor thirds during the 19th century, coinciding with the emergence of extended tonality. While the absolute weights of thirds remain lower than those of fifths, their relative increase is significant, indicating a shift in the ways in which the model explains tonal relationships. This aligns with theoretical accounts of 19th-century harmony and supports the findings of other studies showing decreasing diatonic pitch-class set usage.

The findings address the research questions by demonstrating a clear historical trend in the exploration of tonal space, marked by increasing complexity of intervallic relations and a shift in the relative importance of primary intervals. The increase in the diffusion parameter supports the notion of an expanding tonal vocabulary and increasing chromaticism. The dominance of perfect fifths in early periods reflects the centrality of diatonic structures, while the rise of thirds in the 19th century reflects the move toward extended tonality. The results strongly corroborate existing music-theoretical literature and computational studies. The study highlights the complementary nature of corpus studies and manual music analysis, emphasizing the benefits of quantitative approaches for revealing nuanced patterns in historical data.

This study successfully demonstrates the utility of computational modeling for investigating historical stylistic changes in music. The TDM provides an interpretable framework for analyzing large-scale trends in tonal interval usage, revealing a consistent pattern of increasing complexity and a significant shift in the relative importance of primary intervals during the 19th century. Future research could incorporate more sophisticated models, richer data, and explore models of cultural transmission to provide a more complete understanding of the observed changes.

The study acknowledges several limitations. The TP3C, while extensive, is not a fully representative sample of Western art music, and its biases might affect the results. The model's simplicity, focusing solely on pitch-class counts, limits the depth of analysis. While the model's results align with existing literature, it cannot explain the causal mechanisms behind the observed stylistic changes. Finally, the limited data toward the end of the investigated period affects the reliability of the analysis for the latest centuries.