Music has a tempo (or frequency of the underlying beat) that musicians maintain throughout a performance. Musicians maintain this musical tempo on their own or paced by a metronome. Behavioral studies have found that each musician shows a spontaneous rate of movement, called spontaneous motor tempo (SMT), which can be measured when a musician spontaneously plays a simple melody. Data shows that a musician's SMT systematically influences how actions align with the musical tempo. In this study we present a model that captures this phenomenon. To develop our model, we review the results from three musical performance settings that have been previously published: (1) solo musical performance with a pacing metronome tempo that is different from the SMT, (2) solo musical performance without a metronome at a spontaneous tempo that is faster or slower than the SMT, and (3) duet musical performance between musician pairs with matching and mismatching SMTs. In the first setting, the asynchrony between the pacing metronome and the musician's tempo grew as a function of the difference between the metronome tempo and the musician's SMT. In the second setting, musicians drifted away from the initial spontaneous tempo toward the SMT. And in the third setting, the absolute asynchronies between performing musicians were smaller if their SMTs matched compared to when they did not. Based on these previous observations, we hypothesize that, while musicians can perform musical actions at a tempo different from their SMT, the SMT constantly acts as a pulling force. We developed a model to test our hypothesis. The model is an oscillatory dynamical system with Hebbian and elastic tempo learning that simulates music performance. We simulate an individual's SMT with the dynamical system's natural frequency. Hebbian learning lets the system's frequency adapt to match the stimulus frequency. The pulling force is simulated with an elasticity term that pulls the learned frequency toward the system's natural frequency. We used this model to simulate the three music performance settings, replicating behavioral results. Our model also lets us make predictions of musician's performance not yet tested. The present study offers a dynamical explanation of how an individual's SMT affects adaptive synchronization in realistic musical performance.

A musician’s spontaneous rate of movement, called spontaneous motor tempo (SMT), can be measured while spontaneously playing a simple melody. Data shows that the SMT influences the musician’s tempo and synchronization. In this study we present a model that captures these phenomena. We review the results from three previously-published studies: solo musical performance with a pacing metronome tempo that is different from the SMT, solo musical performance without a metronome at a tempo that is faster or slower than the SMT, and duet musical performance between musicians with matching or mismatching SMTs. These studies showed, respectively, that the asynchrony between the pacing metronome and the musician’s tempo grew as a function of the difference between the metronome tempo and the musician’s SMT, musicians drifted away from the initial tempo toward the SMT, and the absolute asynchronies were smaller if musicians had matching SMTs. We hypothesize that the SMT constantly acts as a pulling force affecting musical actions at a tempo different from a musician’s SMT. To test our hypothesis, we developed a model consisting of a non-linear oscillator with Hebbian tempo learning and a pulling force to the model’s spontaneous frequency. While the model’s spontaneous frequency emulates the SMT, elastic Hebbian learning allows for frequency learning to match a stimulus’ frequency. To test our hypothesis, we first fit model parameters to match the data in the first of the three studies and asked whether this same model would explain the data the remaining two studies without further tuning. Results showed that the model’s dynamics allowed it to explain all three experiments with the same set of parameters. Our theory offers a dynamical-systems explanation of how an individual’s SMT affects synchronization in realistic music performance settings, and the model also enables predictions about performance settings not yet tested.