With the work of Aristoxenus this phase of the empirical tradition of harmonic theory has run its course. We now turn the clock back a hundred years or so to consider our earliest evidence about the mathematical form of the discipline, which originated with the Pythagoreans of the fifth century or conceivably with Pythagoras himself in the sixth. Evidence from later writers gives us a fair general picture of the approach to harmonics which they adopted and whose outlines I sketched in Chapter 1; I shall not repeat the programmatic points I made there. But on matters of detail and on the work of individual Pythagoreans of the early period the late sources are often unreliable. For the most part I follow Burkert in treating Aristotle as the most authoritative of our sources on the subject, together with a few fragments from the work of the late fifth-century Pythagorean Philolaus which most modern commentators take to be genuine. If they are, Philolaus is the only Pythagorean harmonic theorist before the fourth century of whose work we have solid and significant details.

The prize exhibit is a short passage from Philolaus' essay On Nature quoted by Nicomachus and Stobaeus. In modern editions it is sometimes printed as a continuation of Philolaus fragment 6, sometimes as a separate item; I shall refer to it as frag. 6a. It has been much discussed, but I have a serious purpose in re-examining it at some length here.

There is a long Western tradition, one no longer much in fashion, in which the self is understood as being essentially double. On the one hand, each human being lives in and through the body, whereby his mind is filled with perceptions and desires that connect him to the surrounding world. On the other hand, many philosophers have thought that beyond perceptions and experiences there must be something or someone to whom the perceptions belong and who also unifies them; that is, that there is a unified centre of consciousness or rationality. This idea gets one of its expressions in the Kantian division between empirical self and transcendental subject. The two levels or natures of humanity which this kind of thinking entails are separated by having different functions, sometimes, as in Plato, even their own proper desires, and depending on the point of view of the inquiry, both have a claim to be a – or the – self. The division into a transcendental or higher, and an empirical or lower, self is connected with but not identical nor reducible to that of soul and body. For example, a dualist as well as a hylomorphist – someone who believes that the soul is not separable from the body but the form of the living body – both see human nature as essentially double, and both can argue that there is only one self, choosing either the composite or one of the parts that make up the composite as the self.

Up to the end of the fourth century, all our direct evidence about mathematical harmonics comes in the form of scraps. There are the fragments of Philolaus and Archytas, together with a few brief later reports on which we can reasonably rely; there are Plato's comments in the Republic and his psycho-musical construction in the Timaeus; there is a scattering of allusions and discussions in Aristotle and a couple of acid comments in Aristoxenus. Apart from the critique mounted by Theophrastus, which we shall consider in Chapter 15, there is very little else.

In earlier chapters I have tried to extract as much enlightenment from these bits and pieces as they can yield, and the amount is not negligible. But in some respects the absence of any complete treatise in the field leaves serious gaps in our knowledge. Quite apart from the loss of theories and arguments, we simply do not know what a ‘complete treatise’ of that sort would have looked like. We have nothing that unambiguously reveals the aims of such a work, the list of items that we might formulate as its table of contents, the way in which its propositions were combined with one another and integrated into a systematic whole (if indeed they were), or the style of presentation it adopted.

We have now reviewed virtually all the significant data we have about Aristoxenus' predecessors in the empirical tradition, and have put in place the Aristotelian ideas about scientific method which form an essential background to his own work in harmonics. They will figure extensively in Chapters 6–8. Most of his surviving reflections on the subject are contained in the text we know as the Elementa harmonica. The present chapter is concerned mainly with the structure of this work as we now have it, and we shall tackle the much-debated question whether the whole of the surviving text originally belonged to the same treatise, and if it did not, how the relations between its parts are to be understood. These are issues which any serious student of Aristoxenus must address; but they are quite intricate and involved, and I cannot pretend that this chapter is easy reading. Some readers may prefer to cut to the chase, and after glancing at my preliminary paragraphs on Aristoxenus' life and writings, to jump to the discussion of the substance of his theories which begins in Chapter 6. If so I am happy to forgive them; they may perhaps be motivated to come back to the present chapter at a later stage.

ARISTOXENUS' LIFE AND WRITINGS

Aristoxenus was born in Taras, a Greek city in south-east Italy (Tarentum to the Romans, modern Taranto); the date of his birth is uncertain, but can be no later than about 365 bc.

The third book of the El. harm. consists almost entirely of a set of twenty-three demonstrative proofs of propositions about melodic sequences (62.34–74.8). Before them come four preliminary arguments establishing points on which the proofs will rely (58.14–62.33); and they are followed by the beginning of an argument designed to show that there are three and only three melodically acceptable ways of arranging the constituent intervals of a perfect fourth (74.9–25). At this point our manuscripts break off.

Here and there in the course of his exposition Aristoxenus pauses to examine a methodological issue or to clarify theses which his hearers, so he says, have previously found obscure. The bulk of the text, however, is devoted to the logical derivation of rules about melodic sequences, one at a time, from axiomatic principles. Though the arguments are set out with various degrees of formality, all are presented within the conventions familiar to us from Greek mathematical works, notably from Euclid's Elements. A proposition is stated; there follows an argument deriving it directly or indirectly from the agreed axioms; finally, in many cases but not all, the proposition is restated as an established conclusion. The unadorned rigour of the mathematical treatises is reflected also in Aristoxenus' style of writing in this book; it is severe and concentrated, admitting few of the rhetorical flourishes and none of the barbed allusions to other theorists which enliven Books i and ii.

The study of Pythagorean thought, including harmonics, is firmly entrenched in the agendas of several scholarly disciplines, and the writings devoted to it in the last two centuries alone would fill a respectable library. The other style of harmonic analysis mentioned in the Republic, by contrast, has rarely been examined except – briefly – in commentaries on the Republic itself, and by a handful of specialists in ancient musical theory. The force of its exponents' claim to more generous treatment should not be exaggerated; they did not, like the Pythagoreans, pioneer a system of ideas which played a major part in the formation of scientific, philosophical and popular thought for over two millennia. But they deserve more than the occasional learned footnote. Their work broke new ground in the study of music; it casts valuable light on the history of the empirical sciences in the period before Aristotle; and it adds, in small but significant ways, to our knowledge of the environment and the practices of Greek culture between about 450 and 350 bc. In the present chapter I shall be exploring such evidence as we have about the character and content of their contributions to musical knowledge; the discussion will inevitably involve some technicalities, though none of them are alarmingly abstruse.

Archytas of Tarentum, according to Aristoxenus, was the ‘last of the Pythagoreans’ in the continuous tradition stretching back to the founder. The precise dates of his birth and death are not known, but his life seems to have spanned almost exactly the same period as Plato's (427–347 bc), and if the seventh Platonic letter is genuine they were personal friends. Certainly there are close connections between their writings on harmonics, though the evidence about the relation is not always easy to interpret.

Archytas was by all accounts a remarkable man, distinguished simultaneously as a philosopher, a mathematician, an inventor of ingenious gadgets, a statesman and a military commander, and admired also for his personal qualities, his kindness, resourcefulness, self-control and affection for children. He counts as a heroic figure in the early history of mathematical harmonics, which he raised to new levels of conceptual and technical sophistication and channelled in unprecedented directions. Only a few fragments of his writings survive, along with reports in various (and variously reliable) later sources, but they are enough to allow us to reconstruct a coherent general outline of his approaches and ideas, and to piece certain parts of his work together in some detail.

Half a millennium later, in the most accomplished of all Greek essays in the mathematical style of harmonics, Ptolemy speaks of Archytas with evident admiration.

Harmonic theory is mentioned by various writers of the period between about 300 bc and the beginning of the Christian era, most of them (Philodemus, Vitruvius and Dionysius of Halicarnassus, for example) in its latter years, but nothing they say suggests that original works in the discipline were still being produced. The fact probably reflects more than an accidental gap in our evidence; there is little to encourage the thought that harmonics continued to flourish during those centuries in the hands of theorists whose writings have been lost. The authors of treatises from the Roman imperial period refer quite often to their predecessors; but some of those they mention are their own near-contemporaries, and nearly all the others belong to the fifth or fourth centuries bc, almost none to the third, second or first. The later theorists seem to have thought of themselves, in fact, as the immediate successors of Philolaus, Archytas, Plato, Aristoxenus and ‘Euclid’, as if the intervening generations had said nothing at all on the subject, or at any rate nothing interesting and new.

Writings devoted wholly or in large part to harmonics reappear early in the first century ad, and continue, if not in a torrent then at least in a steady trickle, right down to Boethius in the early sixth. Through Boethius' Latin reformulation, a version of Greek mathematical harmonic theory passed into the medieval tradition.

Who am I? This question continues to trouble us. The somewhat cryptic three-word sentence breaks up into various queries: What kind of being am I? What essential features do I share with animals or with other human beings? What differentiates me from them? What or who is this thing now sitting or writing this introduction, thinking these thoughts? Why do I often experience myself as a single subject of experience, or believe myself to be an autonomous agent capable of causing changes in the world, but at others feel a divided and inconsistent creature, or a powerless slave of circumstances? Do those experiences reveal anything about my true nature? What in the midst of all this is particular only to me, as this one individual, this person with these thoughts, likings and personal characteristics?

For a long time, the notion of self was considered to be accompanied by ontological commitments that scholars with a physicalist and scientific world-view could hardly entertain. But there is still no escape from these questions. Even though they may ultimately lead to different kinds of inquiries about, for instance, the essential nature of human beings, personal identity, rational agency, etc., and even if many of these issues would be best approached from a third-person rather than a first-person perspective, they still have one aspect in common: they are all, broadly speaking, reflexive. They are all questions that the inquirer asks about his or her own nature.

In Meibom's edition of 1652, whose pagination modern scholars use as their standard reference-point, Book ii of the El. harm. occupies a little less than twenty-eight pages of thirty-four lines each. No more than about six pages are taken up with statements and elucidations of facts about musical structures, or of principles governing them, and all those pages fall within the ‘overlapping passage’ discussed on pp. above. Virtually the whole of their contents, by contrast with their manner of presentation, is already familiar from Book i (see 44.21–47.7, 50.14–52.32, 53.32–55.2). Of the remainder, there are two pages of preface (on the value of prefaces in clearing away potential misunderstandings, 30.9–32.9); the revised list of the science's parts occupies nearly four pages, the bulk of which is devoted, as in Book i, to comments on the procedural failings of Aristoxenus' predecessors (35.1–38.26); one page of the ‘overlapping passage’ is its study of the ways in which the topic of melodic continuity should be addressed (52.32–53.32); three pages describe procedures for constructing discords by the ‘method of concordance’ and for assessing the size of the perfect fourth (55.3–58.5). All the rest, amounting to nearly twelve pages, takes the form of a series of long digressions, concerned with the concepts that must be brought to bear on the subject if it is to be properly understood, and with the methods by which its theses are to be established and expounded. (These are at 32.10–34.34, 38.27–44.20 and 47.8–50.14; the first two are subdivided into two and three separate ‘digressions’ respectively.)

Theophrastus was born in about 370 bc, succeeded Aristotle as head of the Lyceum in 322 bc and died, full of years, early in the second decade of the third century. Like several other Peripatetics of his generation he wrote copiously; Diogenes Laertius (v.42–50) lists the titles of 224 works, many of them in several books, amounting in all to 232,850 lines and addressing an astonishing range of topics. Later writers refer occasionally to his interest in music, and though he can hardly be regarded as a specialist in the field, the works in Diogenes' catalogue include an On Music in three books, and a Harmonics and an On Musicians in one book each. We know little of their contents. The great majority of his writings are lost, or survive only in fragments quoted by others, and from his publications on music we have only a handful of scraps. Only one of them concerns us here. According to Porphyry, the source who quotes it, it comes from the second book of Theophrastus' On Music. It is by far the longest of the fragments on musical topics, running to 126 lines in Fortenbaugh's edition.

The passage begins with a reference to a kinēma melōidētikon, a ‘movement productive of melody’ or a ‘melody-making movement’ which occurs in the soul.

The only sustained discussion of musical issues in Aristotle's surviving works is in the last book of the Politics. Like the conversations in Book iii of Plato's Republic, to which it is in part a response, it focuses on the value of music in the life of a city and its citizens, and it says nothing about the musical sciences. It alludes several times, however, to the work of unnamed experts in musicology, at least some of whom were Aristotle's contemporaries. This suggests that he had some acquaintance with up-to-date studies by musical specialists, perhaps including their work in harmonics; and references to harmonic science, and to concepts used by its exponents, are scattered here and there in his other writings. Almost all of them are brief. It seems fairly clear that Aristotle made no substantial contributions of his own to the subject, and that it was marginal to his main areas of interest. I shall argue, too, that despite the confidence of his various pronouncements, his grasp on some of its concepts and procedures was a little uncertain.

A study of his remarks pays dividends none the less, for three main reasons. First, slight though they are, they contribute rather more than has generally been recognised to our knowledge of mathematical harmonics in the fourth century, especially in its Pythagorean form.