The diachronic development of Debussy's musical style: a corpus study with Discrete Fourier Transform

The late 19th and early 20th centuries witnessed significant stylistic shifts in Western classical music, moving away from the common practice period's diatonic tonality towards extended tonality, impressionism, and eventually atonality. Claude Debussy (1862-1918) stands as a pivotal figure in this transformation, renowned for his innovative and original style. While his music retains elements of diatonic tonality (tonic-dominant polarity), it significantly weakens goal-directed tonal harmony, prioritizing coloristic and textural aspects. Debussy also employs a broader harmonic palette, blending diatonic chords with octatonic, whole-tone, and chromatic structures, often shifting between multiple pitch-class sets within a single piece. This study aims to quantitatively investigate the diachronic evolution of these stylistic features in Debussy's solo piano works, focusing on three key aspects: the use of symmetric pitch-class materials (leading to tonal ambiguity), the coexistence of multiple pitch-class structures within a single piece (fragmentation), and the overall coherence of the piece (global prototypicality). The research hypothesizes that Debussy's mature style exhibits increased tonal ambiguity and fragmentation but maintains a degree of coherence.

Existing music-analytical approaches suggest a stylistic evolution in Debussy's music, with whole-tone, chromatic, and octatonic procedures becoming more prominent in later works. This evolution is characterized by weakened directionality, ambiguous tonal organization, and an equalization of the twelve tones. However, a comprehensive quantitative characterization of these diachronic trends is lacking. Previous corpus studies have shown evidence of a decrease in diatonicity over a broader historical span, and the use of DFT has been explored to analyze pitch-class distributions in individual pieces. This study builds upon this existing research by applying a novel, comprehensive DFT-based method to Debussy's complete solo piano oeuvre to test hypotheses regarding tonal ambiguity, fragmentation, and coherence.

This study employs a corpus-based approach using Debussy's complete solo piano works (82 pieces) represented in machine-readable XML format. The Discrete Fourier Transform (DFT) is used to analyze pitch-class vectors (PCVs) derived from the scores. PCVs represent the duration of each pitch class in a given segment of a piece. The DFT decomposes these PCVs into components representing various pitch-class prototypes (chromaticity, tritonicity, augmentedness, octatonicity, diatonicity, whole-toneness). A hierarchical representation, called wavescapes, is employed to analyze the DFT components at different time scales within each piece. A novel summary wavescape is created for each piece, indicating the most salient DFT component for each segment at various hierarchical levels. Three key metrics are defined: 1. **Prevalence:** The proportion of nodes in the summary wavescape where a specific DFT coefficient (representing a pitch-class prototype) has the highest magnitude, weighted by entropy. This quantifies the prominence of each prototype in a piece. 2. **Fragmentation:** The measure-theoretic entropy of the partition of the summary wavescape's nodes into maximal connected regions with identical labels. This captures the variability of dominant pitch-class structures across different segments of a piece. 3. **Global Prototypicality:** The slope of a regression line fitted to the average maximal magnitude of the DFT components at each hierarchical level. This assesses the degree to which larger segments of a piece express a defined pitch-class prototype. Bayesian mixed-effects models were used to assess linear trends in prevalence, fragmentation, and global prototypicality over Debussy's compositional lifespan. The models incorporate the effects of composition year and piece length and their interaction to account for potential confounding factors. A post-hoc analysis using a moment of inertia measure, weighted towards higher hierarchical levels, was also conducted to investigate the robustness of the findings.

The analysis yielded the following key findings: 1. **Decrease in Diatonicity:** Strong evidence was found for a significant decrease in the prevalence of the 5th DFT component (diatonicity) over Debussy's career. This decrease was consistent across different weighting schemes (entropy-weighted prevalence and moment of inertia). This supports the hypothesis of a progressive departure from common practice diatonic structures. 2. **Increase in Octatonicity:** A significant increase in the prevalence of the 4th DFT component (octatonicity) was observed over time. This increase, particularly evident in Debussy's later works, reinforces the importance of octatonic structures in his mature style and suggests a shift towards harmonic functionality expressed in a chromatic context. 3. **Increase in Fragmentation:** A significant increase in the degree of fragmentation was observed throughout Debussy's compositional career, suggesting an increased use of multiple pitch-class structures within individual pieces. This supports the notion of an increasing reliance on juxtaposing disparate pitch materials. 4. **Stable Coherence:** Despite the increase in fragmentation, global prototypicality did not significantly decrease over time. This suggests that the increase in fragmentation did not lead to a loss of overall coherence in Debussy's compositional style, implying the preservation of a generalized notion of harmonic functionality beyond the diatonic system. Furthermore, a positive correlation was found between global prototypicality and fragmentation, suggesting that increased complexity does not necessarily lead to incoherence. The models also indicated that longer pieces tended to exhibit lower diatonic prevalence and higher global prototypicality.

The findings strongly support the hypotheses that Debussy's style progressively moved towards increased tonal ambiguity and fragmentation while maintaining a level of coherence. The decrease in diatonicity and increase in octatonicity suggest a transition away from the inherent directedness and orientation of diatonic scales toward more ambiguous, chromatic structures. The increase in fragmentation indicates a compositional strategy emphasizing the juxtaposition and blending of diverse pitch-class sets. The maintenance of global prototypicality implies that this increased complexity did not result in a loss of overall coherence; rather, Debussy likely developed a novel form of organizational coherence within his evolving harmonic language. These findings resonate with musicological interpretations of Debussy's style, highlighting its deliberate organization despite its apparent fragmentation and ambiguity. The significant increase in octatonic structures may be interpreted within the framework of extended tonality theories, suggesting a generalization of functional harmony beyond the constraints of the diatonic system.

This study provides robust quantitative evidence for the evolution of Debussy's musical style, confirming and extending existing qualitative analyses. The findings highlight the significant diachronic shifts in diatonicity, octatonicity, fragmentation, and global prototypicality, offering a nuanced understanding of Debussy's departure from common-practice tonality. Future research could explore the interaction of textural and structural features, further differentiate between prototypes associated with the same DFT coefficient, and investigate the perceptual implications of the observed stylistic changes. The methodologies employed in this study can be applied to other musical repertoires to investigate similar stylistic developments in extended-tonal and post-tonal music.

The analysis is based on a specific definition of coherence (global prototypicality), which may not fully capture all aspects of musical organization. The DFT method, while powerful, does not explicitly consider temporal ordering or voice-leading, which may be crucial in certain interpretations of Debussy's music. The analysis of phase information in the DFT coefficients could be further developed to capture specific juxtapositions and contrasting patterns of pitch-class prototypes. Finally, the study focuses on solo piano works, limiting generalizability to other genres and instrumental settings.