From the Cover: Synchronization in human musical rhythms and mutually interacting complex systems

Though the music produced by an ensemble is influenced by multiple factors, including musical genre, musician skill, and individual interpretation, rhythmic synchronization is at the foundation of musical interaction. Here, we study the statistical nature of the mutual interaction between two humans synchronizing rhythms. We find that the interbeat intervals of both laypeople and professional musicians exhibit scale-free (power law) cross-correlations. Surprisingly, the next beat to be played by one person is dependent on the entire history of the other person’s interbeat intervals on timescales up to several minutes. To understand this finding, we propose a general stochastic model for mutually interacting complex systems, which suggests a physiologically motivated explanation for the occurrence of scale-free cross-correlations. We show that the observed long-term memory phenomenon in rhythmic synchronization can be imitated by fractal coupling of separately recorded or synthesized audio tracks and thus applied in electronic music. Though this study provides an understanding of fundamental characteristics of timing and synchronization at the interbrain level, the mutually interacting complex systems model may also be applied to study the dynamics of other complex systems where scale-free cross-correlations have been observed, including econophysics, physiological time series, and collective behavior of animal flocks.In his book Musicophilia, neurologist Oliver Sacks writes: “In all societies, a primary function of music is collective and communal, to bring and bind people together. People sing together and dance together in every culture, and one can imagine them having done so around the first fires, a hundred thousand years ago” (1). Sacks adds, “In such a situation, there seems to be an actual binding of nervous systems accomplished by rhythm” (2). These thoughts lead to the question: Is there any underlying and quantifiable structure to the subjective experience of “musical binding”? Here, we examine the statistical nature of musical binding (also referred to as musical coupling) when two humans play rhythms in synchrony.Every beat a single (noninteracting) layperson or musician plays is accompanied by small temporal deviations from the exact beat pattern, i.e., even a trained musician will hit a drum beat slightly ahead or behind the metronome (with a SD of typically 5–15 ms). Interestingly, these deviations are statistically dependent and exhibit long-range correlations (LRC) (3, 4). Listeners significantly prefer music mirroring long-range correlated temporal deviations over uncorrelated (white noise) fluctuations (5, 6). LRC are also inherent in the reproduction of both spatial and temporal intervals of single subjects (4, 7–9) and in musical compositions, such as pitch fluctuations (a simple example of pitch fluctuations is a melody) (10, 11) and note lengths (12). The observation of power law correlations in fluctuations of pitch and note length in compositions reflects a hierarchical, self-similar structure in these compositions.In this article, we examine rhythmic synchronization, which is at the foundation of musical interaction, from orchestral play to audience hand-clapping (13). More specifically, we show that the interbeat intervals (IBIs) of two subjects synchronizing musical rhythms exhibit long-range cross-correlations (LRCCs), which appears to be a general phenomenon given that these LRCC were found both in professional musicians and in laypeople.The observation of LRCCs may point to characteristics of criticality in the dynamics of the considered complex system. LRCCs are characterized by a power law decay of the cross-correlation function and indicate that the two time series of IBIs form a self-similar (fractal) structure. Here, self-similarity implies that trends in the IBIs are likely to repeat on different timescales, i.e., patterns of IBI fluctuations of one musician tend to reproduce in a statistically similar way at a later time—even in the other musician’s play. A variety of complex systems exhibit LRCCs; examples include price fluctuations of the New York Stock Exchange (where the LRCCs become more pronounced during economic crises) (14–16), heartbeat and EEG fluctuations (15, 17), particles in a Lorentz channel (18), the binding affinity of proteins to DNA (15), schools of fish (19), and the collective response of starling flocks (20, 21). The origin of collective dynamics and LRCCs based on local interactions often appears elusive (20), and is the focus of current research (19, 21). Of particular interest are the rules of interactions of the individuals in a crowd (22, 23) and transitions to synchronized behavior (16, 24). We introduce a stochastic model for mutually interacting complex systems (MICS) that generates LRCCs and provides a physiologically motivated explanation for the surprising presence of long-term memory in the cross-correlations of musical performances.Interbrain synchronization has received growing attention recently, including studies of interpersonal synchronization (see ref. 4 for an overview), coordination of speech rhythm (25), social interactions (26), cortical phase synchronization while playing guitar in duets (27, 28), and improvisation in classical music performances (29).Notably, the differences between the beats of two musicians are on the order of only a few milliseconds, not much larger than the typical duration of a single action potential (∼1 ms). The neurophysical mechanisms of timing in the millisecond range are still widely open (30, 31). EEG oscillatory patterns are associated with error prediction during music performance (32). Fine motor skills, such as finger-tapping rhythm and rate, are used to establish an early diagnosis of Huntington disease (33). The neurological capacity to synchronize with a beat may offer therapeutic applications for Parkinson disease, but the mechanisms are unknown to date (34). This study offers a statistical framework that may help to understand these mechanisms.