This article analyses the conditions under which mathematics could enter the field of fourteenth-century music. It distinguishes between descriptive and argumentative uses of mathematics. Jean de Murs’ uses of arithmetic to study musical time is an example of the former, Jean de Boen’s study of the division of the whole tone an example of the latter. It is furthermore explained how the mathematical descriptions appear to bring into agreement two types of constraint, namely the physical characteristics of sound and the (...) aesthetic principles of the medieval discourse about music. Within these constraints, mathematics manages to fulfill different argumentative roles: it has an ontological function when music is seen as a part of the quadrivium; but an explicative function in the framework of the scientia media and, in an more innovative spirit for Jean de Boens, it provides a definition of the possible in the argumentation about the division of the whole tone. (shrink)