A Sintaxe Harmónica Atonal no Espaço de Fourier: Para Uma Representação Esférica das Fases

Abstract

Harmonic syntax archetypes and the function they play within the temporal structure of Western tonal music is vastly systematized across music theory treatises as rule-based principles. E.g., the function a chord plays within tonal structures is typically reduced to a threefold categorization (tonic, subdominant, and dominant), whose temporal alignment implies meaning. In post-tonal music since the early 20th century, this phenomenon is much less systematized and understood, evolving even in recent years, to the existence of almost as many theories and compositional systems as the number of composers or individuals works. In this context, we advance some methods and tools within computational musicology, which aims to unpack typical syntactic (i.e. temporal) patterns in atonal harmonic structures. To this end, we follow recent methods by Yust on the adoption of the phase information from the discrete Fourier transform (DFT) of pitch class sets. Our contribution extends previous research by proposing a three-dimensional spherical representation resulting from the phase information of two Z_12 magnitude-salient coefficients to plot the temporal dimensions of pitch class sets within an atonal music piece. Resulting spatial mappings automatically identify interval-content clusters and typical harmonic patterns across them. Finally, to test this three dimensional representation, we use it to analyze the "Sechs kleine klavierstück", Op.19 by Arnold Schoenberg and as an assistant to the composing process, suggesting movement patterns in the harmonic material presented by the composer. In summary, we intend with this study to: (1) plot the harmonic sets of Arnold Schoenberg's Op. 19 in a three-dimensional space according to its interval content; (2) group such sets into different clusters; (3) identify the existence of standardized movements between these clusters; (4) compare the results of the representation with previous analysis and (5) assess the utility of the system in the composer's creative process.