SIAM News Blog

Improving Musical Synchronization with Mathematical Modeling

By Lina Sorg

Music functions as a universal connector that pervades most cultures. More specifically, rhythm and synchronization – both within and beyond the realm of music – are forms of communication that stimulate brain activity.

In a recently-published paper in the SIAM Journal on Applied Mathematics, authors Donald Drew, Kevin Dolch, and Maury Castro propose a stochastic differential equation model that simulates how musical performers in a large ensemble sustain tempo and phase while responding to a conductor, other musicians, and additional distractions modeled as “noise.” In an ideal situation, musicians would be able to perfectly coordinate the rate of change at which pitch and relative loudness occur while simultaneously ignoring noise and the distractions of the other musicians. However, the authors recognize that the aforementioned stimuli cause execution errors from each individual.

The authors assume that individual performers preserve an internal tempo when responding to the conductor, who offers the correct rhythm sequence. Their phase correction model assumes that the correction of a rate of error is contingent on the ratio of tempo variation to a performer’s ability to resist noise distraction and concentrate solely on the conductor. The correction model is based on deliberate responses of the human brain when determining tempo and phase, rather than assuming error correction based on biochemical oscillators, as in other models.

The authors acknowledge that musical performances involve a certain amount of individual tempo variations to achieve a sense of artistry. But their proposed models offer a means by which to manage tempo discrepancies, improve synchronization, and thus enhance the overall quality of performed music.

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Source Article: A Model for Tempo Synchronization in Music Performance. SIAM Journal on Applied Math, 75(6), 2540-2561. (Online publish date: December 1, 2015).

About the authors: Donald Drew is now retired; before retiring he was the Chair of Mathematical Sciences and Professor of Mechanical, Aerospace, and Nuclear Engineering at Rensselaer Polytechnic Institute; Kevin Dolch studied at Rensselaer Polytechnic Institute; Maury Castro is the Organist & Choirmaster and Director of Communications at St. Christopher’s Episcopal Church.

About SIAM: The Society for Industrial and Applied Mathematics (SIAM), headquartered in Philadelphia, Pennsylvania, is an international society of more than 14,000 individual, academic and corporate members from 85 countries. SIAM helps build cooperation between mathematics and the worlds of science and technology to solve real-world problems through publications, conferences, and communities like chapters, sections and activity groups. Learn more at

   Lina Sorg is the associate editor of SIAM News. 

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