During the 1970s and 80s, it was my regular habit to take Philosophy undergraduates at the University of Warwick on a guided tour around a selection of Platonic and Aristotelian texts; and I generally found myself placing issues about the nature of knowledge, and about the procedures by which it may be pursued, firmly at the centre of our agenda. I became more and more fascinated, in the course of this annual pilgrimage through Meno, Phaedo, Republic, Theaetetus,Posterior Analytics, Physics and Nicomachean Ethics, by their intricate negotiations between what we would call ‘rationalist’ and ‘empiricist’ conceptions of the route towards knowledge in a variety of different fields of enquiry. In 1976 the University of Warwick allowed me to accept an invitation to spend two years teaching in the Faculty of Classics at Cambridge; and it was there, with my mind full of these matters, that I first stumbled, largely by accident, into the thickets of the Greek musical sciences. As I worked backwards from Aristoxenus to Plato and the early Pythagoreans, and then forwards into later antiquity, I discovered that the surviving texts of that unfamiliar tradition can be read as a record of continual controversy, not so much over musicological details as over the general character of the understanding sought by scientists in this field, the methods by which it is to be pursued and secured, and the relations that hold between the propositions of this science and those belonging to other domains.