For some six centuries before Ptolemy, philosophers and scientists had been debating the supposedly competing credentials of reasoning and of perceptual observation as guides in the quest for truth. Controversy continued into his time and beyond, and was as brisk among harmonic scientists as anywhere else. Earlier commentators had occasionally surveyed the battlefield; but there is little evidence that exponents of the science themselves, after the pioneering years of the fourth century bc, had developed their positions in the light of sober consideration of the merits and deficiencies of the warring camps. They seem, on the whole, simply to have taken up entrenched positions on one side or the other of longstanding barricades, and to have dismissed alternative positions out of hand.

Ptolemy is an important exception. He shows himself to be well informed about the debate, and he offers sharp criticisms of extreme views on either side. His own position is designed to incorporate promising insights from any doctrinal repertoire, while avoiding the faults they had previously carried with them, and to fuse them into a new methodological amalgam, more balanced and more adequate to its task. The business of the present chapter is to examine the statements he makes on these issues in the opening pages of the Harmonics. Here he is not reviewing the postures of his predecessors, though as we shall see immediately, they are implicitly under fire right from the start.

When we set out to use our ears to assess harmonic relations,

It is some measure of the importance Ptolemy attaches to the use of such instruments in harmonics that he devotes nearly six whole chapters, and substantial parts of two more, to descriptions of their design and discussions of their properties. Issues to do with the procedures by which propositions are to be submitted to perceptual tests by means of these instruments are examined, sometimes at length, in at least a dozen other passages.

The questions to be considered in this chapter overlap with those of Chapter 10, but we shall take a slightly different angle of approach. There we based our discussion on what Ptolemy says about his instruments; here we shall concentrate on his account of what we are to do with them when we have got them. Broadly speaking, his comments fall under three headings. Some are concerned with the preparation of the data, that is, with the way in which propositions to be assessed must be expressed, if their content is to be ‘displayed to perception’ on an instrument's strings. Others relate to the procedures by which attunements are to be set up in practice on the instruments themselves, and by which the constructions of reason are actually to be made accessible to perceptual judgement. Finally there are passages that make statements or carry implications about the criteria according to which such judgements can be made. The thesis that they are made ‘by perception’ or ‘by the ear’ is altogether too vague, and we must see what efforts Ptolemy makes to sharpen it up.

The relevant passages are of course not separated out under these headings in the text. In reviewing them there are several important questions that we shall be seeking to answer. More or less tentative answers to some of them have been proposed in Chapter 10, and in these cases we shall be looking for further evidence that might bear on these provisional conclusions.

‘It would not be right to attribute these errors to the power of reason, but to those who ground reason in faulty hupotheseis’ (tois mē deontōs auton hupotithemenois, 15.3–4). This statement stands as a preface to Ptolemy's own account of the concords in 1.7. The Pythagoreans' errors have been shown up principally by recourse to the evidence of perception. That strategy implies that the test of perception is to be trusted; but the errors it reveals should not deter us from the quest for rational principles. Hence where propositions derived from supposedly rational hupotheseis are at odds with the perceptual data, neither reason as such nor the senses should be blamed for the conflict, nor should either be dismissed as unreliable in its own sphere of competence. The proper conclusion is that the hupotheseis have been misconceived or wrongly applied.

Ptolemy's exposition of the correct principles begins with a three-fold classification of musical intervals. ‘Preeminent in excellence is the class of homophones, second that of concords, and third that of the melodic. For the octave and the double octave plainly differ from the other concords as do the latter from the melodic, so that it would be more appropriate for them to be called “homophones”.

Ptolemy's reputation as one of the outstanding scientists of the ancient world rests mainly on his epoch-making treatise in astronomy, the Mathematike Syntaxis (usually known as the Almagest). Modern students of the ancient sciences know him also for his writings on optics, geography and astrology; but with a few honourable exceptions they have shown rather little interest in his Harmonics. Those who have examined it in detail have not, for the most part, been historians of science. Either they have been concerned less with the text in its original setting than with its afterlife in Renaissance musicology, or else they are dedicated (even fanatical) specialists in ancient musical theory; and we are rather few. But even if the subject it addresses continues to languish (as it should not) in a cobwebby corner of our gallery of the Greek sciences, the Harmonics itself deserves much wider attention.

It is in the first place a work of real intellectual distinction, and its skilfully mustered arguments, despite their technical intricacies, are presented with a flair and panache that should commend it to any connoisseur of scientific writing. Secondly, it is quite unusually explicit and self-conscious about its own methodology and procedures. In this respect it has a good deal more to offer than the Syntaxis, whose overt reflections on the general features of the science are relatively brief and less directly methodological, and play a notably less prominent role in the development of the subsequent argument.

At the end of 1.7 Ptolemy restates the conclusions he has derived from his hupotheseis, and explains what his next step will be.

There is nothing new about his results, of course. What is important is that they have been shown to follow from hupotheseis that are both acceptable to reason and capable of being intelligibly represented as precise, mathematical counterparts of the relevant perceptual impressions. As the final sentence indicates, 1.8 will describe the ways in which the results can be made to display their credentials before the court of perception.

By the end of Book 1 Ptolemy has completed his analysis of divisions of the tetrachord; but he takes one further step before moving on to a new topic. 11.1 is occupied by an account of an alternative method of confirming the patterns of ratios attributed to the attunements of practical musicians in 1.16. Here Ptolemy reverses his former procedure. Instead of first arguing to the values of the ratios on ‘rational’ grounds and then confirming the results by ear, he now begins by constructing the attunements by ear on the strings of an eight-stringed instrument, and then argues that the ratios of intervals constructed in this way must indeed have the values he has assigned to them. From here Ptolemy is led on to a discussion of certain other instruments that can be used for the same purpose; this occupies 11.2. We shall review the contents of these two passages, among others, in Chapters 10 and 11.

The transition to a new phase of the investigation is clearly signalled at the beginning of 11.3. ‘Let that be our outline of what is scientifically understood (ta theōroumena) concerning the concordant and melodic relations between notes that are established in conformity with the lengths of string plucked, the homophones being included along with the concords. The next topic for discussion after these is that dealing with the systēmata’ (49.4–7).

Since the opening of 11.3 declares that the programme of harmonic science has now been completed, the rest of the work really falls outside the scope of this methodological study. I shall consider one part of it, the introductory section, in a little detail, and offer only a brief sketch of the contents of the rest. This will be enough, I think, to give us some purchase on Ptolemy's conception of the place of harmonics among the sciences, and its role in equipping us to interpret and engage with the universe we inhabit. We shall then be in a better position to understand why a scientist of Ptolemy's stature should have found this apparently small and insignificant corner of the Greek intellectual tradition worth the meticulous attention he has given it.

After drawing a line under the completed business of the science of harmonics, 111.3 proceeds as follows.

‘Since, then, not even these people have divided the primary genera of the tetrachords in a way that agrees with perception, let us ourselves try, here as well, to preserve what is consistent both with our hupotheseis concerning melodic relations and with the appearances, in accordance with those conceptions of the divisions that are primary and natural’ (33.1–5). So begins 1.15. From a technical point of view this long chapter is the core of the Harmonics, and the analyses that it contains provide the basis for all Ptolemy's later constructions. It will be as well to remark at the outset, however, that they are not his last word on the division of tetrachords. His object is to identify the rational credentials of systems that perception will recognise as perfectly formed. The divisions derived here are perfectly formed from a rational perspective, and in a certain sense from that of perception too; yet it turns out that few of them are acceptable in musical practice precisely as 1.15 describes them. The relation between theoretical perfection and aesthetic acceptability is more complex than has so far emerged. Ptolemy is probably alluding to distinctions of this sort when he describes the conceptions developed here as ‘primary and natural’. They constitute in some way both the mathematical and the aesthetic foundations of the systems used in practical music-making, without being quite identical with them.

Ptolemy mentions by name rather few of his predecessors. When he does, it is seldom to record his debts, though some of them emerge clearly enough, as we shall see. His first major topic is the musical concords; and he sets out the approaches of two schools of thought on this matter in some detail, mainly to criticise them. The strategy is designed to throw into sharper relief his own procedures and their merits, and his treatment of the issues is linked very closely to his criticisms of theirs. But the roles of the two critiques in his wider enterprise are different. Only one of them will be discussed in this chapter. (For the other, see Chapter 6.)

In considering what Ptolemy says about these earlier theorists, one of my aims is similar to his own. A study of his criticisms will clarify the challenges which his own procedures must meet, and will provide a yardstick by which we can judge their success from his point of view. But at the same time I shall draw attention to ways in which some of Ptolemy's own views turn out to be developments or refinements of ones he criticises, though he is never quite explicit in acknowledging the fact. His borrowings are worth mentioning not merely to elucidate his intellectual biography, or in the spirit of Porphyry's Commentary, to convict Ptolemy of surreptitious plagiarism.

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.

Readers intent on the issues that are central to Ptolemy's methodology can be forgiven for ignoring some of the present chapter's more detailed ruminations in favour of their own, less pernickety reading of 1.3. My remarks in this chapter amount to a partial commentary on that stretch of text, and on little else. As I remarked earlier, the phase of the investigation conducted here, which I labelled as Stage (i), is only a preliminary. It is designed to establish the proposition that pitch is a quantitative attribute of sound, and to identify the causal factors responsible for its variations. Ptolemy treats these questions as closely interconnected. The proposition about pitch cannot be established without a study of the causes; and in practice the two issues are pursued simultaneously. Both have important roles in the sequel. The first will legitimise Ptolemy's policy of expressing pitch relations as ratios of numbers, in accordance with their mathematical forms rather than with the corresponding pathē. The second will serve as a basis for correlating the one mode of description with the other; it will also provide an account of the principles underlying the construction of experimental instruments, and the groundwork for an understanding of their use. The main purposes of my project would be served well enough by a bare sketch of the arguments in this passage.

Nevertheless, the details are of some interest, and I shall spend a little time on them.

The Discourse on the Method and the three essays that were published with it, the Dioptrics, the Meteors, and the Geometry, make up a very curious book. The very title page emphasizes the preliminary discourse, and that discourse, the Discourse on the Method, emphasizes method, the importance that method had for Descartes in making the discoveries he made, the importance that the method Descartes claims to have found will have for the progress of the sciences and for the benefit of humankind as a whole. Descartes is not, of course, telling us that we are obligated to follow his method; the Discourse is, after all, proposed “as a story, or, if you prefer, as a fable” (AT VI 4). But Descartes expects that we will all see the light, the light of reason, of course, and follow his example. It is curious, then, that Descartes gives the reader only brief hints of what that method is, four brief, vague, and unimpressive rules that, taken by themselves, would hardly seem to justify Descartes' enthusiasm, not to mention a whole discourse in their honor. Furthermore, explicit methodological concerns are hardly in evidence in the Dioptrics, the Meteors, and the Geometry, which are, Descartes claims, “essays in this method,” as he identifies them on his title page. Indeed, one is hard pressed to find much evidence of the method at all after 1637, either explicit discussions of the method or explicit applications of the method in any of Descartes' writings, published or unpublished. Very curious.

These observations raise quite a number of questions about the development of Descartes' thought and the state of his program as of 1637.