Rhythmic segment analysis: Conceptualizing, visualizing, and measuring rhythmic data

Abstract

This paper develops a framework for conceptualizing, visualizing, and measuring regularities in rhythmic data. I propose to think about rhythmic data in terms of interval segments: fixed-length groups of consecutive intervals, which can be decomposed into a duration and a pattern (the ratios between the intervals). This simple conceptual framework unifies three rhythmic visualization methods and yields a fourth: the pattern-duration plot. When paired with a cluster transition network, it intuitively reveals regularities in both synthetic and real-world rhythmic data. Moreover, the framework generalizes two common measures of rhythmic structure: rhythm ratios and the normalized pairwise variability index (nPVI). In particular, nPVI can be reconstructed as the average distance from isochrony, and I propose a more general measure of anisochrony to replace it. Finally, the novel concept of quantality may shed light on wider debates regarding small-integer-ratio rhythms.

Figures (7)

• Figure 1: Rhythmic segment analysis studies fixed-length segments of a sequence of durations: interval pairs or triplets for example. Every segment has a rhythmic pattern: the ratios between its intervals (A). Length-$n$ patterns are points in a $(n-1)$-dimensional simplex. Panel B shows the space of length-2 patterns: the rhythm line. Values along this line are known as rhythm ratios$r$ with isochrony corresponding to $r=0.5$. Normalization maps segments to their patterns, as illustrated for segments $D$ and $E$ with patterns $d$ and $e$. Panel C shows the space of length-3 patterns, the rhythm triangle. The white crosshairs shows how to read the coordinates of patterns inside the triangle: pattern $f$ has relative durations $(.25, .25, .5)$. Both panels show how the segment distance is measured along the gridlines of the ambient space, while the pattern distance is measured along the gridlines of the simplex.
• Figure 2: Pattern-duration space. Segment can be represented either in segment space (A) or in pattern-duration space (B), here by plotting the rhythm ratio horizontally and the duration vertically. This explicitly isolates tempo (vertically) from the rhythmic content (horizontally). Moving between the two descriptions is a coordinate transform similar to polar coordinates (C): the pattern $r$ determines the direction, the duration $d$ the distance from the origin. A square of segments (black dots) is plotted in both spaces to illustrate the transformation, along with lines of constant pattern, duration, and interval. For example, all points on a dashed blue line represent the same pattern, and lines through the origin in phase space become vertical lines in pattern-duration space. Labels in (B) show the coordinates of the segments in segment space.
• Figure 3: Quantality. A rhythmic dataset is quantal when intervals tend to be integer multiples of a fixed quantum $q$ (row 3). Intervals between events that approximately fall on a grid (row 1) are an example of quantal data. Intervals in a quantal dataset are approximately discrete, and since they are all multiples of a quantum, must be related by integer ratios (row 2). These three phenomena are illustrated for two types of random data: noisy multiples of a quantum, with the multiples drawn from either (A) a uniform distribution or (B) a geometric distribution, which favour short-intervals.
• Figure 4: Four visualization methods. Columns show three types of synthetic data: (1) uniform intervals, (2) geometric quantal intervals, and (3) a noisy, repeated rhythm. (A) Raster plots sort segments vertically by their duration; the shorter interval of each segment is plotted to the left, the longer one to the right. (B) Phase plots plot the first against the second interval. (C) Pattern-duration plots show the rhythm ratio horizontally and the duration. (D) A ratio plot is a marginal density plot of a pattern-duration plot. See the main text for details.
• Figure 5: A cluster transition network. (A) Noisy repetitions of a fixed rhythm are shown as a pattern-duration plot with marginal density plots. This dataset is quantal and so durations (vertical axis) have been expressed as multiples of the quantum $q=0.5$s. Colors indicate clusters, and a cluster transition network shows transitions between clusters, with the thicker lines indicating more frequent transitions. Gray lines in the background show individual segment transitions. All possible segments are marked and annotated: $2 : 3$, for example, indicates the segment $(1.0, 1.5)$ consisting of two and three quanta. (B) Reading only the second of the annotated numbers, gives the rhythm produced by a path through the network. This is illustrated for the final bar of the repeated rhythm shown in (C).