Probability distributions over external states (priors) are essential to the interpretation of sensory signals. Priors for cultural artifacts such as music and language remain largely uncharacterized, but likely constrain cultural transmission, because only those signals with high probability under the prior can be reliably reproduced and communicated. We developed a method to estimate priors for simple rhythms via iterated reproduction of random temporal sequences. Listeners were asked to reproduce random 'seed' rhythms; their reproductions were fed back as the stimulus and over time became dominated by internal biases, such that the prior could be estimated by applying the procedure multiple times. We validated that the measured prior was consistent across the modality of reproduction and that it correctly predicted perceptual discrimination. We then measured listeners' priors over the entire space of two- and three-interval rhythms. Priors in US participants showed peaks at rhythms with simple integer ratios and were similar for musicians and non-musicians. An analogous procedure produced qualitatively different results for spoken phrases, indicating some specificity to music. Priors measured in members of a native Amazonian society were distinct from those in US participants but also featured integer ratio peaks. The results do not preclude biological constraints favoring integer ratios, but they suggest that priors on musical rhythm are substantially modulated by experience and may simply reflect the empirical distribution of rhythm that listeners encounter. The proposed method can efficiently map out a high-resolution view of biases that shape transmission and stability of simple reproducible patterns within a culture.