Kid Algebra: Radiohead's Euclidean and Maximally Even Rhythms

KID ALGEBRA: RADIOHEAD’S EUCLIDEAN AND MAXIMALLY EVEN RHYTHMS BRAD OSBORN HE BRITISH ROCK GROUP Radiohead has carved out a unique place in the post-millennial rock milieu by tempering their highly experimental idiolect with structures more commonly heard in Top Forty rock styles.1 In what I describe as a Goldilocks principle, much of their music after OK Computer (1997) inhabits a space between banal convention and sheer experimentation—a dichotomy which I have elsewhere dubbed the ‘Spears–Stockhausen Continuum.’2 In the timbral domain, the band often introduces sounds rather foreign to rock music such as the ondes Martenot and highly processed lead vocals within textures otherwise dominated by guitar, bass, and drums (e.g., ‘The National Anthem,’ 2000), and song forms that begin with paradigmatic verse–chorus structures often end with new material instead of a recapitulated chorus (e.g., ‘All I Need,’ 2007). In this T 82 Perspectives of New Music article I will demonstrate a particular rhythmic manifestation of this Goldilocks principle known as Euclidean rhythms. Euclidean rhythms inhabit a space between two rhythmic extremes, namely binary metrical structures with regular beat divisions and irregular, unpredictable groupings at multiple levels of structure. After establishing a mathematical model for understanding these rhythms, I will identify and analyze several examples from Radiohead’s post-millennial catalog. Throughout the article, additional consideration will be devoted to further ramifications for the formalization of rhythm in this way, as well as how hearing rhythm in this way may be linked to interpreting the lyrical content of Radiohead’s music. After doing so, I will suggest a prescriptive model for hearing these rhythms, and will then conclude with some remarks on how Radiohead’s rhythmic practices may relate to larger concerns such as style and genre. EUCLIDEAN RHYTHMS: SOME MATHEMATICAL PRELIMINARIES A familiar example from Radiohead’s recent song ‘Codex’ (2011, see Example 1) will serve to illustrate the mathematics involved in the rest of this article. Stylistically competent rock listeners will entrain to a sixteen-semiquaver subdivision in each measure. That is to say, the lowest common denominator that encompasses all rhythmic onsets in the voice, piano, and quiet kick drum involves sixteen evenly spaced time points per notated measure.3 Observing the underlying piano rhythm in this section (notated in full in measures 1, 3, and 5–7 of example 1—the remaining measures are rhythmic subsets of this same riff), notice that there are five onsets per measure, meaning that there are five onsets played for every sixteen possible time points. But, these are not just any five onsets within that space of sixteen. In fact, the particular spacing of those five onsets represents a maximally even spacing of five onsets within a grid of sixteen evenly spaced time points. That is to say, although one cannot divide sixteen by five evenly, the rhythm we hear in the piano is the closest-to-even distribution possible of sixteen units into five sets (4+3+3+3+3). There exists only one such possible distribution given k onsets over n evenly spaced time points, and this is the rhythm that I will henceforth refer to as the Euclidean distribution.4 Here it will be useful to establish a notation system that will be used consistently throughout the rest of the article. For any abstract Euclidean distribution, the unordered set will be notated within curly Kid Algebra: Radiohead’s Euclidean and Maximally Even Rhythms 83 braces with the smallest inter-onset intervals packed to the left. Specific ordered presentations of that rhythm will be enclosed within angle brackets, placing the inter-onset intervals in order beginning on the perceptual downbeat. ‘Codex’ could then be related abstractly to other Euclidean distributions of five in sixteen by notating the unordered multi-set of inter-onset intervals {3,3,3,3,4} (notated in order from lowest to highest values), and, of the five possible rotations of that set, the specific ordering heard here is . At other times in this article, I will find it necessary to notate numbered onsets in a string of n evenly spaced time points (numbered 0 through n–1), so that the attack points heard in the ‘Codex...