Rhythm Complexity

A

Clapping Music be informed in terms of rhythm complexity? Has the evolution of the

popular rhythms of the world favored an increase in their complexity? Can the difficulty of learning to perform a rhythm be predicted with a simple and elegant mathematical for- mula? How similar is the rhythmic oddity property prevalent in the Aka Pygmy music to the Western concept of syncopation? How powerful is rhythm complexity as a feature for music genre classification and music information retrieval? An introduction to the search for answers to these questions is the focus of this chapter.

Rhythm is arguably the most fundamental aspect of music,* _{and complexity is one of its}

most salient features.† _{Musicologists routinely comment on the complexity of rhythm pres-}

ent in music from different cultures. In his analysis of African rhythmic systems, Simha Arom writes that they are “the most complex of all those which are known all over the world.”‡ _{According to the Reverend Arthur Morris Jones: “No European musician could}

clap and sing any but the more simple examples of African music.Ӥ _{Yet, the formal inves-}

tigation of the complexity of rhythm has been largely overlooked in the literature. A musi- cal concept closely related to rhythm complexity is syncopation, a topic already explored in Chapter 13. However, as we saw there, formal definitions of syncopation are lacking. A  typical definition of syncopation is the one found in the Collins English Dictionary:

* _{Although most musicologists argue for the supremacy of rhythm over other features of music, the thesis is not without}

its detractors. The composer Olivier Messiaen, for example, writes: “The melody is the point of departure. May it remain sovereign! And whatever may be the complexities of our rhythms and our harmonies, they shall not draw it along in their wake, but, on the contrary, shall obey it as faithful servants.” See Messiaen, O. (1956), p. 13.

† _{Gabrielson, A. (1973a,b) used factor analysis and multidimensional scaling to uncover 15 perceptual features of rhythm,}

and complexity stood out among them. Conley, J. K. (1981), p. 69, experimented with 10 physical features of music com- plexity calculated from Beethoven’s Eroica Variations, Op. 35, and found that the rate of rhythmic activity (in terms of the number of rhythmic events) was the most powerful measure of complexity. Interestingly, Wang, H-M., Lin, S.-H., Huang, Y.-C., Chen, I.-C., Chou, L.-C., Lai, Y.-L., Chen, Y.-F., Huang, S.-C., and Jan, M.-Y. (2009) showed that the com- plexity of rhythms can modify the interbeat duration patterns of the heart of the listener.

‡ _{Arom, S. (1984), p. 51.} § _{Jones, A. M. (1949), p. 295.}

108    ◾   The Geometry of Musical Rhythm

“The displacement of the usual rhythmical accent away from a strong beat onto a weak beat.” A mathematician would not only demand formal definitions of “strong” and “weak” beats, but would be baffled by how to interpret the term “usual.” On the other hand, many formal (mathematical) definitions of complexity do exist, most from domains other than music, but some from music itself. A typical example of the former is the Lempel–Ziv com- plexity of a binary sequence,* _{and two representatives of the latter are the rhythmic oddity}

property† _{and the off-beatness discussed in the previous chapter.}‡

Concerning the rhythms of India, the journalist and producer Joachim-Ernst Berendt writes: “It is necessary . . . to say a few words about the mysteries of Indian music. Its talas, its rhythmic sequences—incomprehensible for Western listeners—can be as long as 108 beats; yet the Indian ear is constantly aware of where the sam falls.”§ _{Kofi Agawu reviews}

a plethora of published claims about the purported complexity of African rhythms.¶

Comparing African and Indian music with European music, Benjamin I. Gilman writes: “Hindu and African music is notably distinguished from our own by the greater complica- tion of its rhythms. This often defies notation.”** _{Concerning the measurement of rhythmic}

complexity, Martin Clayton writes: “I can think of no objective criteria for judging the relative complexity or sophistication of rhythm in, for example, Indian rag music, Western tonal art music, and that of African drum ensembles.”††

The concept of complexity is extremely fluid. Its definition depends to a great extent on the context‡‡ _{and the purpose to which it is put.}§§ _{In an information theory context, a metro-}

nomic pulsation is least complex, and random noise is most complex. However, in a musi- cal context, random noise is not complex at all. The most complex musical rhythms exhibit a degree of complexity that lies somewhere between complete order and complete disor- der.¶¶ _{However, determining the exact point within this continuum is easier said than done.}

As a consequence, numerous definitions of complexity have been proposed. Complexity is also multidimensional, and there are many ways of measuring and combining these dimensions.*** _{Ilya Schmulevitch and Dirk-Jan Povel distinguish between three broad cate-}

gories of complexity measures for musical rhythms: hierarchical, dynamic, and generative.†††

Hierarchical measures refer to structure at several levels simultaneously, dynamic measures refer to the nonstationarity of the input over time, and generative measures depend on the

* _{Lempel, A. and Ziv, J. (1976). See Coons, E. and Kraehenbuehl, D. (1958) for some early work on the application of}

information theory to the analysis of musical structure.

† _{Chemillier, M. (2002), Chemillier, M. and Truchet, C. (2003).} ‡ _{Toussaint, G. T. (2005b).}

§ _{Berendt, J.-E. (1987), p. 202.} ¶ _{Agawu, K. (1995).} ** _{Gilman, B. I. (1909), p. 534.} †† _{Clayton, M. (2000), p. 6.}

‡‡ _{Repp, B. H., Windsor, W. L., and Desain, P. (2002). Toussaint, G. T. (1978) provides a tutorial survey on the dependence}

between the perception and recognition of patterns in spatial (visual) and temporary (auditory) modalities, and the context in which those patterns are perceived.

§§ _{Wolpert, D. H. and Macready, W. (2007). See Crofts, A. R. (2007), p. 25, for the relevance of complexity to evolution.} ¶¶ _{Eglash, R. (2005), p. 154.}

*** _{Sioros, G. and Guedes, C. (2011), p. 385, propose a complexity measure that combines the density of events in a rhythm}

with its syncopation by means of the formula: complexity = {density2 _{+ syncopation}2_}1/2_{. See also Essens, P. (1995).} ††† _{Shmulevich, I. and Povel, D.-J. (1998, 2000a,b).}

Rhythm Complexity    ◾   109  

amount of effort required to generate the rhythms. Rhythm complexity may also be mea- sured with respect to perception and performance (also called production*_{). Furthermore,}

these complexities depend on additional factors such as tempo and the underlying meter.†

Experiments by D. J. Povel demonstrated that “changing the tempo of temporal sequences may cause dramatic changes in the perceived rhythmical characteristics.”‡ _{In some con-}

texts such as music transcription, it is desirable to determine the notation of a rhythm that minimizes the performance complexity while affecting the perceptual complexity as little as possible.§ _{In this chapter, several definitions and measures of rhythm complexity}

are compared.¶ _{Some of these are better than others, and the reader may wonder: why not}

just describe the best one? The answer is that there is no best. The usefulness of a measure depends on its intended application. Furthermore, it is hoped that some readers may be inspired by these concepts to invent new measures that may perhaps combine features of the measures described here. It has been my experience during many years of teaching at universities that just presenting the best correct algorithms is not necessarily the best way to teach. It is sometimes better to teach incorrect algorithms first. Even better is to teach incorrect algorithms that students believe to be correct. Then, after seeing counter- examples, the students have the opportunity to learn the reasons for their failure, and to attempt to fix them. Following such an experience, students have greater appreciation for the correct solutions. The best learning does not always happen when knowledge is served on a plate, but when the learner has to construct that knowledge.