Western classical music has undergone significant evolution, with notable differences in tonality across centuries. Music theorists have offered various explanations for this development, often focusing on changes in the relationships between notes. Fétis's work, for instance, highlights a transition from *tonalité ancienne* to *tonalité moderne*, exemplified by a specific passage in Monteverdi's music, though this claim has been debated. The field of musical corpus studies offers a new perspective by applying computational and statistical methods to large datasets of digitized music, aiming to bridge qualitative-historical and quantitative-empirical approaches. This paper contributes to this endeavor by using computational modeling for a data-driven analysis of historical changes in tonality, focusing specifically on the historical development of modes.

Prior empirical research on tonality often uses pitch-distribution models, reflecting both musical corpora and psychological and neuroscientific findings on mental representations of pitch-class distributions. Studies have investigated diachronic developments using various statistical methods on pitch-class distributions, such as bigram analysis, dominant-seventh chord ratios, and Fourier transforms. These studies have revealed trends such as increasing dominant-seventh chord usage, expansion of tonal material, and shifts in chord transition directionality. Although several studies touched upon modes, few focused exclusively on their historical evolution, often relying on a priori assumptions of major and minor modes. A notable exception is Albrecht et al. (2014), whose methodology shares similarities with the current study but differs crucially in its a priori assumption of the existence and characteristics of major and minor modes. The current study directly addresses this limitation by inferring both the number and characteristics of modes from the data itself.

This study employed a dataset of approximately 13,402 pieces of Western classical music in MIDI format, sourced from ClassicalArchives and other scholarly resources. The data was preprocessed to extract pitch-class counts weighted by duration. The composition years were used to divide the data into five historical periods: pre-1649 (Renaissance), 1650-1758 (Baroque), 1759-1817 (Classical), 1818-1856 (Early Romantic), and post-1856 (Late Romantic). Two computational models were used. The first model is a geometric model that uses unsupervised learning to determine the optimal number of modes in each period by considering the pieces as clusters in a non-Euclidean space. This model helps to determine the number of distinct clusters. The second is a cognitively plausible Bayesian model that represents modes as Dirichlet distributions. This model was used to infer the characteristics of each mode, such as their associated pitch-class distributions. The Bayesian model uses a Gibbs sampling method to obtain a posterior distribution for each piece's root and mode. The maximum-a-posteriori estimate determines the classification of each piece. Dimensionality reduction via t-distributed stochastic neighbor embedding (t-SNE) was employed for visualization of the mode space. Silhouette scores were used to evaluate the appropriateness of different numbers of clusters for each period. Mode clarity, defined as the proportion of correctly predicted modes based on metadata, was calculated to further evaluate the models.

The geometric model, informed by silhouette scores, revealed that four modes were most plausible for the Renaissance, two modes (consistent with major and minor) for the Baroque and Classical periods, and no clear separation into distinct modes for the 19th century. The Bayesian model confirmed these findings, showing a strong correspondence between the two-dimensional t-SNE representation and the unsupervised mode classification. Mode clarity was highest in the Classical period and significantly higher for the Baroque and Classical periods compared to the Romantic periods, confirming the clear distinction between major and minor modes in the Common-Practice period. Analysis of the Renaissance period, assuming four modes, revealed pitch-class distributions resembling music-theoretical descriptions of various Renaissance modes. Violin plots illustrated the pitch-class distributions for each cluster. The Common-Practice period showed distinct major and minor mode distributions with clear thresholds separating in-scale and out-of-scale notes. A comparison of mode templates generated by the Bayesian model with those from previous research (psychological experiments and corpus statistics) revealed similarities, particularly in the consistent identification of major and minor thirds as distinctive pitch classes. Radar plots were used to better visualize the circular nature of the pitch space and highlight the consistent symmetry between major and minor modes across the datasets.

The findings strongly support cognitive theories of mode inference through statistical learning. The success of the models in identifying the number and characteristics of modes in different historical periods without a priori assumptions demonstrates that basic pitch statistics are sufficient to reveal significant patterns in tonal evolution. The clear distinction between major and minor modes in the Baroque and Classical periods confirms music-theoretical concepts of the Common-Practice period. The lack of clear modal separation in the 19th century reflects the significant changes in tonality during that period, including the use of novel scales and harmonic relations. The results suggest that the concept of a single global mode is less suitable for 19th-century music due to frequent local modulations.

This study provides a data-driven approach to investigating the historical evolution of tonality. The proposed Bayesian and geometric models offer a novel method for inferring the number and characteristics of modes without pre-existing assumptions. Future research could explore more nuanced models (e.g., hierarchical models) to analyze tonality at multiple levels of musical composition, incorporate algebraic models of tonal spaces, and expand the dataset to encompass a wider range of musical styles and cultures.

The study's reliance on the ClassicalArchives dataset, which is crowd-sourced and may contain biases in terms of composer representation and historical balance, is a potential limitation. The assumption of transpositional invariance, while common in music analysis, may not be entirely accurate for all historical periods, particularly the Renaissance. The accuracy of key labels in the metadata is also a factor that might influence the interpretation of results.