The Euclidean algorithm (which comes down to us from Euclid’s Elements) computes the greatest common divisor of two given integers. It is shown here that the structure of the Euclidean algorithm may be used to generate, very efficiently, a large family of rhythms used as timelines (ostinatos), in sub-Saharan African music in particular, and world music in general. These rhythms, here dubbed Euclidean rhythms, have the property that their onset patterns are distributed as evenly as possible. Euclidean rhythms also find application in nuclear physics accelerators and in computer science, and are closely related to several families of words and sequences of interest in the study of the combinatorics of words, such as Euclidean strings, to which the Euclidean rhythms are compared.