Exploratory interpretations of power law relationships in Debussy’s Syrinx
Authors: Scott Douglas , Ludick Zani 
Published: February 2022

Background in music analysis. Music analysis often relies on pre-formed concepts such as “binary form”, key centers, or rhythmic tropes to provide templates for comparison between works. To the extent that such tools are based primarily on semantic level considerations, they can present obstacles to translation between different 4E domains of embodied, embedded, extended, and enactive cognition. Power law analysis, representing a context neutral information theoretic approach which is amenable to task specific weighting, can help bridge the gap.

Background in applied mathematics. In this paper we investigate how deviations from the near universal phenomenon of logarithmic perception correspond to psychologically tractable interpretations of a musical score. We show that variations in the slope of a straight-line log-log regression and its residuals, together with variation in structural features produced by varying counting methods and levels of analysis correspond to academic and expert assessments in various ways. We then show how wavelet analysis can be used to smooth over level of analysis problems.

Aims. We seek to explore how violations of simple power law descriptions (Zipf’s law) relate to structural features of Claude Debussy’s Syrinx for solo flute as in the form of a variety of analyses and performances.

Main contribution. Power laws are an important implicit feature of musical notation through such characteristics such as octave equivalence and beat hierarchies, but are rarely explicitly employed as higher-level analytical tools directly, despite the fact that the scale-invariance property and empirical observation suggests that they can be useful at that level too. The simple historical reason is that generating such descriptions, while conceptually straightforward, is quite computationally intensive. Now that computation is readily available it has become possible to extend power law descriptions into these higher levels of analysis with relative ease. Our contribution is to explore some general techniques that can be expanded upon and refined as a broader research program with substantial interdisciplinary implications for the study of perception and cognition generally, since these phenomena obey similar laws.

Implications. The present study is correlational in nature, so the direct implications are limited for either music analysis, which provides the interpretive data, or power law representations, which follows well-established practices. However, having established that a set of relatively approachable mathematical operations can readily account for key music-theoretical structural features, the door is opened to a systematic application of the principles involved. This provides an opportunity to state music theories in ways that are precise, falsifiable and directly relatable to modes of description employed in other sciences. Furthermore, because these procedures have broad relevance to many domains outside of music theory, it allows for insights gained from studying musical structure to be applied directly to other, ostensibly unrelated problems.