A Circular Plot for Rhythm Visualization and Analysis

[1] This report presents a graphing method designed to aid the study of rhythm and expressive timing in beat-based music.(1) I show how the polar coordinate system can be used to describe and analyze different features of expressive timing. Numerous studies have shown that expressive timing lies at the core of rhythm production. The evidence-reviewed by Clarke (1999)-confirms that musicians often place attack points along a continuum of beat subdivision values rather than on the predetermined slots afforded by metrical grid spacing, thus imbuing the performance with expressive depth. This departure from a clear-cut temporal lattice is inadequately, if at all, represented by standard music notation. Like the "piano roll" representation used by MIDI sequencers, standard music notation is a kind of visualization tool whose two "axes" capture two paramount features of music: time horizontally and pitch vertically. The need to represent specific aspects of music in a detailed way has led to the design of various visualization methods, as I discuss next.Visualization and Circles[2] Certain properties of musical structure can be depicted using rhythmograms (Todd, 1994), self-similarity squares (Foote & Cooper, 2001), or hierarchic trees (Lerdahl & Jackendoff, 1983). Timbre can be viewed with various forms of spectrograms, hemiolas with ski-hill graphs (Cohn, 2001), harmony with quotient topologies (Tymoczko, 2006), and the pitch chroma cycle with helices (Shepard, 1983). Geometrical thinking has also served the composition process, as evidenced by the hand-drawn schematics of Reynolds (2004) and Wishart (1996), to name just two recent examples.[3] Visualizations also play a role in the realm of rhythm and expressive timing. Even though microtiming information is sometimes displayed with numerical tables, visualization strategies are often used to communicate information on a more perceptually intuitive level. Desain & Honing (2003) devised a triangular chronotopic map that plots the temporal nuances and categorical boundaries of three-note rhythms. An important and appealing feature of their graphing method is that every point in the map represents a unique rhythmic pattern. Dots that are near each other in the graph denote similar sounding rhythms, forming "clumps" that represent distinct perceptual categories. However, the map is restricted to rhythms that consist of three durations only and is therefore of limited use in most real-world musical contexts. For longer rhythms, note-for-note expressive timing data are often visualized with an XY graph where evenly partitioned time units (such as notes or measures) demarcate score position along the abscissa; the ordinate usually plots tempo, interonset interval, or deviation from a metronomic subdivision. This type of design has proved helpful in different musical contexts including jazz (e.g., Benadon, 2006; Collier & Collier, 2002) and Western "classical" music (e.g., Friberg & Sundberg, 1999; Palmer, 1996; Repp, 2002), but its linear left-to-right orientation tends to conceal the recursive nature of beat-based patterns.[4] Circles enjoy a privileged status in the visualization of musical time. They have been tapped by music theorists, ethnomusicologists, and computer scientists to represent cyclical aspects of rhythm. London (2004, p. 64) visualizes meter by placing "peaks of attentional energy" (beats and subdivisions) along a circle's circumference. Time also flows around a circle in Becker's (1980, p. 107) representation of Javanese gamelan gongan (structural units of time marked by a gong), which are "cyclical rather than linear," and in Anku's (2000) representation of African rhythms. Locke (1996, p. 90) and Collins (2004, p. 59) also use circles to characterize African rhythms, taking the circular concept one step further by employing concentric circles that describe the stratification of polyrhythm. In the work of Toussaint (2005) and McLachlan (2000), the circle facilitates mathematical explanations of rhythm such as maximal evenness and similarity measures. …