Why Do Durations in Musical Rhythms Conform to Small Integer Ratios?

One curious aspect of human timing is the organization of rhythmic patterns in small integer ratios. Behavioral and neural research has shown that adjacent time intervals in rhythms tend to be perceived and reproduced as approximate fractions of small numbers (e.g. 3/2). Recent work on iterated learning and reproduction further supports this: given a randomly timed drum pattern to reproduce, participants subconsciously transform it toward small integer ratios. The mechanisms accounting for this ‘attractor’ phenomenon are little understood, but might be explained by combining two theoretical frameworks from psychophysics. The scalar expectancy theory describes time interval perception and reproduction using Weber’s law: just detectable durational differences equal a constant fraction of the reference duration. The notion of categorical perception emphasises the tendency to perceive time intervals in categories, i.e. “short” vs. “long”. In this piece, we put forward the hypothesis that the integer-ratio bias in rhythm perception and production might arise from the interaction of the scalar property of timing with the categorical perception of time intervals, and that neurally it can plausibly be related to oscillatory activity. We support our integrative approach with mathematical derivations to formalise assumptions and provide testable predictions. We present equations to calculate predicted durational ratios by: (i) specifying one (of K) category of ratios, (ii) assuming a scalar timing constant, and (iii) parameterising the relationship between durational categories. Our derivations provide the basis for future computational, behavioural and neurophysiological work to test our model.