The arrangement of this book is only partly chronological, as I have said. Still, one would like to begin at the beginning. But it is hard to know what would count as a solid historical starting point. New intellectual enterprises do not spring into being in a single bound, fully armed and fully recognisable; typically they trickle together as a confluence of tribu- taries which are themselves side-shoots of other, pre-existing traditions of thought, none of which ‘is’ this new thing, but whose developing com- bination can be seen, in retrospect, gradually to have become it. In any case, such information as we have about the earliest explorations of terri- tory eventually claimed by this science is vague, unreliable and hopelessly entangled with legends and misconceptions that grew up in later times. Pythagoras, of course, is often identified as the fountain-head, and if that were true it would take us back to the late sixth century. There are also reports that his younger contemporary, the poet and composer Lasus of Hermione, wrote the first ‘treatise’ (logos) about music,1 and allusions in
two other writers seem to suggest that Lasus’ work had some connection with issues in harmonics. But one of these latter references is too enigmatic to be interpreted safely;2 in the second, which hints at connections with
Pythagorean speculations in acoustics, there is a serious gap in the surviving text, and we do not know precisely what the author was asserting.3If Lasus
did indeed write a work of the sort with which he is credited, it may not have survived for long after its author’s lifetime; even Aristoxenus, who apparently had some inkling of its contents, may not have known it as a whole or at first hand. We are in no position to reconstruct Lasus’ ideas or
1 See Martianus Capella 9.936, and the entry under Lasus’ name in the Suda. The latter gives no details;
the former assigns to Lasus a set of distinctions between branches of musical theory which certainly originated in later times. See Privitera1965: 37–46.
2 Aristox. El. harm 3.21–4, which indicates that Lasus offered a rather curious view about the nature
of a musical note. The passage does at least do something to confirm that a musical treatise by Lasus existed, and we shall revisit it briefly in Chapter6.
3 Theo Smyrn. 59.7–12.
to identify the relation, if any, in which his logos stood to the writings of the later theorists.4
The familiar figure of Pythagoras, mystic, mathematician, philosopher and scientist, is almost entirely a construct of the Pythagorean revival of Roman times, though its creators found some of the material from which it was built in writings of the fourth century. Very little of the colourful information retailed by Nicomachus, Porphyry, Iamblichus and others can be taken at face value, and the amount of reliable evidence they offer to link Pythagoras with harmonic science is vanishingly small. Much the same is true of the accounts we possess of the work of Pythagoras’ followers down to the late fifth century, at least in point of detail. What we know, rea- sonably securely, about Pythagorean harmonics around 400 bc certainly presupposes an earlier tradition, which may go back to Pythagoras himself. But we are deceiving ourselves if we think that we can pin down its con- tents with any precision, let alone attribute specific ideas with justifiable confidence to particular individuals and dates.5
In the light of these dismal reflections I shall abandon the search for a historical beginning. I shall take as my starting point remarks made about their predecessors by writers of the fourth century, which seem to refer to ideas and activities current in the later years of the fifth. It may be possible, on the basis of a study of these reports, to work our way with due caution a little further into the past. But there is a general issue that needs to be faced first. In the rest of this short chapter I shall consider its broad outlines; more intricate details will be examined in later parts of the book.
A melody, a scale or an attunement is a complex of pitched sounds, or notes, whose pitches stand to one another in particular relations. It is in virtue of these relations that they are melodies, scales or attunements; and it is these relations that must form the principal focus of any attempt to analyse them. Students of these matters must therefore be able to conceptualise relations between pitches in a way that allows them to compare one relation with another, and to map groups of such relations into intelligible patterns; and they need linguistic resources – and perhaps others too, as we shall see – which will allow them to express, with a high degree of precision, the way in which one relation differs from another, and to specify the different
4For some fascinating discussion and suggestions see Porter (2007).
5The foundation-stone of all recent scholarship on the early Pythagoreans is Burkert1962, with its
English translation of 1972. Though some of Burkert’s views have been disputed and others are still under debate, the book remains indispensable. For an admirable, brief and recent study see Kahn
2001, whose bibliography gives a useful guide to the modern literature and refers to other, more compendious bibliographical surveys.
characteristics of the patterns they form. In our own language (if we set aside for the present the perspective of scientific acoustics) we speak of notes as ‘higher’ or ‘lower’, as if they were placed as points on a vertical continuum of ‘up’ and ‘down’, and we talk of the relations between them as larger and smaller ‘intervals’, as if they were spatial gaps between these points of pitch, and could be measured and compared like distances along a line. This way of depicting the phenomena is simple and convenient, and to us it seems obvious. But of course we know that it is entirely metaphorical. It is not a direct, objective representation of the facts, but a way of thinking and talking which arises from the contingencies of our own culture and its history. Other peoples have other metaphors, and those to whose language this particular piece of imagery is alien may have no easy access to the mode of thought that it invites.
Just occasionally, a Greek writer speaks of notes as ‘above’ and ‘below’,
an¯o and kat¯o; but the usage is very rare.6Where we would call a note ‘high’,
a Greek would most commonly describe it as oxys; where we would call it ‘low’ it is barys. But oxys and barys do not mean ‘high’ and ‘low’; they mean ‘sharp’ and ‘heavy’. I have argued elsewhere that these designations are not conceptually neutral, amounting to nothing more than another way of labelling the same distinctions of pitch that we make, but that they radically condition the way in which the Greeks experienced and envisaged the phenomena.7I shall not labour the point here. It is obvious, however,
that these terms provide no ready access to a metric within which pitches can be precisely compared and the relations between them pinned down. ‘Sharp’ and ‘heavy’ are not even direct contraries, and there is no metric through which the ‘sharpness’ of one sound can be measured against the ‘heaviness’ of another.
The standard Greek word for ‘pitch’ is tasis, which literally means ‘tension’; and another, rather more erudite way of calling a note ‘high’ or ‘low’ was to describe it as syntonos, ‘tense’, or aneimenos (sometimes chalaros), ‘relaxed’ or ‘slack’. This usage too is in a certain sense metaphorical, though it need be none the worse for that; its source is probably in the observation that an increase in tension raises the pitch of an instrument’s string. At first sight it seems more promising, since degrees of tension can be measured and compared with mathematical exactitude. But that is true only in principle. The Greeks had no reliable way of measuring tension, and the one method to which writers on harmonics refer, that of suspending larger or smaller
6 It occurs at Hippocr. De victu i.18, [Ar.] Problems xix.37, 47, and very occasionally in later sources. 7 See Barker2002b.
weights from strings, introduces unfortunate complications; pitch does not vary directly with the weights of the suspended objects.8 Some relatively
complex mathematics would be required in order to express accurately the relations between the pitches of an attunement by reference to tensions measured in this way. No Greek theorist seems to have understood the difficulties fully;9and no Greek theorist outside Pythagorean legend seems
even to have attempted to measure musical relations by this procedure. Nor did the words used by early musicians to designate particular inter- vals, so far as we can recover them, give any purchase to a system of mea- surement. What we call the perfect fourth, for instance, was called syllab¯e, ‘grasp’, or dia tessar¯on, ‘through four [strings]’. The former term seems to refer to the span of strings that the fingers can comfortably grasp on an instrument such as the lyre, and the second merely expresses the number of strings one passes across, in a regular form of attunement, in order to complete this interval. Neither tells us, in any relevant sense, ‘how big’ the interval is. The terms for the perfect fifth and for the octave are no more helpful. The former is di’ oxei¯on, ‘through the high [strings]’, or dia
pente, ‘through five’; the octave is harmonia (‘attunement’, indicating that
the normal compass of an attunement was an octave), or dia pas¯on, ‘through all’. None of this terminology has anything to do with the measurement of the relations between the pitches of notes standing, as we might put it, a fourth, a fifth or an octave ‘apart’ from one another.
How, then, are the relations between pitches to be identified objectively and compared with scientific precision? The issue, as my previous remarks have suggested, is essentially one of measurement. We can specify precisely the system of relations which gives a well designed building, for instance, its pleasing form, because these relations hold between measurable distances of height, length and breadth. (Of course this is an over-simplification, but measurement has an essential part to play.) It is much harder, and impossible without a sophisticated array of equipment and techniques, to give a comparable account of the relations between colours and tones in a skilfully balanced painting, since we have no way, outside a modern laboratory, of measuring shades and intensities of green, blue and the rest of them, and comparing them with one another on a single, objective scale.
8For references to this procedure see e.g. Theo Smyrn. 57.2–4, 60.7–9, Porph. In Ptol. Harm. 119.29–
120.7, and the famous story of Pythagoras and the ‘harmonious blacksmith’, told most elaborately (and misleadingly) in Nicomachus, Harm. ch. 6.
9Difficulties involved in the procedure are noted by Ptolemy at Harm. 17.7–17 (cf. Porph. In Ptol.
Harm. 120.33–121.10); but even he does not mention the fundamental problem, that the pitches of
Serious harmonic analysis cannot begin without a system of measurement which allows relations between pitches to be expressed in quantitative terms. (Since the science focuses on the relations between pitches, rather than on the absolute pitches of the notes themselves, the capacity to measure pitches absolutely is unnecessary.) I do not mean that everything in harmonic science reduces to quantification; Aristoxenus, as we shall see later, does his best to minimise its role. But even he cannot proceed without it.
Methods of measurement in harmonics figure prominently in a well- known passage of Plato’s Republic (530c–531c), which sketches the proce- dures of adherents to two very different schools of thought. The dramatic setting of the dialogue (itself written around 385 bc) would place their activ- ities in the later decades of the fifth century. We shall consider the work of one group in more detail in Chapter2and that of the other in Chapter10; for the present let the issue of measurement take centre-stage.
m u s i c a l i n t e rva l s a s l i n e a r d i s ta n c e s
Responding (inappropriately, as it turns out) to a remark of Socrates, Glaucon describes the procedures of one set of theorists as follows. What they do is ridiculous, when they call certain things ‘pykn¯omata’, and bend their ears to the task as if trying to catch a sound from next door, some of them declaring that they can still just hear a sound in between, and that this is the smallest interval, by which measurement should be made, while others disagree, claiming that the notes sounded are already the same. (Rep. 531a4–8)
The task these people have set themselves, then, is to identify a unit ‘by which measurement should be made’. It is to be the smallest gap between pitches that the human ear can pick out; larger ones will be ‘measured’ as multiples of that unit. Their procedure, as Socrates’ sarcastic metaphors in his next speech make clear (531b2–6), involves adjusting the pitches of strings on an instrument by twisting the tuning-pegs, until two strings give notes so nearly identical with one another that they can approach no closer without reaching an apparent unison. When that situation is achieved, the unit of measurement has been found.
The crucial tools of the trade here are the ears. Under this procedure the unit involved is accessible to the hearing and to nothing else; there is no question, for example, of measuring the tensions exerted by the tuning- pegs. Since acuity of hearing varies from one individual to another, it is hardly surprising that this approach generated disagreements of the sort that Glaucon describes; and no system of measurement based directly upon it
can be fully objective, or remain demonstrably constant when it is deployed by different researchers. There will be substantial difficulties, too, even for a single scientist who is undisturbed by others’ doubts and has established such a unit of measurement to his own satisfaction, in using it for the purposes for which it was intended. In determining how many times the unit fits into the ‘gap’ between two notes of a scale, for example, he will apparently have laboriously to repeat the procedure by which the unit was established as many times as it takes to fill up the vacant space.
Another crucial implication of the passage is that these people were working with – or perhaps fumbling towards – a quasi-linear conception of pitch, in which the relations between pitches are thought of as gaps or spaces, some larger and some smaller, and hence measurable. I have already suggested that this mode of representation does not arise naturally from imagery inherent in the Greek language; it demands a degree of conceptual detachment from inherited cultural norms. Here we must consider the noun pykn¯oma, which Glaucon evidently regards as a piece of pretentious jargon. The adjective pyknos means ‘compressed’ or ‘dense’, and is applied to things whose constituents are packed closely together. Its usual converse is araios (sometimes manos), ‘loosely packed’, ‘diffuse’. It plays significant roles in mainstream harmonic science, as we shall see later in this book; and in Chapter2we shall encounter another way in which cognate expressions are linked specifically to the people we are considering here (see p. 42 below).
Pykn¯omata, then, are ‘densifications’, complexes of items stacked tightly up
against one another. The word entered the language of harmonics, if we may judge by Glaucon’s reaction to it, as an abstruse technicality, not as part of the common coinage of every-day musical talk. Even the adjective pyknos, which is common in literature of every kind from Homer onwards, makes its first surviving appearance in connection with music in the technical writings of the fourth century. Despite their many descriptions of music, the poets of the preceding period never exploit, in this context, a contrast between ‘dense’ and ‘diffuse’.10But the contrast does play important roles
in the cosmological and scientific speculations of Presocratic philosophers
10Although the first occurrence of pyknos in harmonic works that have come down to us is in
Aristoxenus’ Elementa harmonica (late fourth century), where it is common, the present passage of the Republic allows us to infer its use a century earlier; it seems unlikely that theorists who talked of pykn¯omata failed to describe these same complexes of notes, adjectivally, as pykna. The noun
pykn¯oma itself occurs here for the first time in Greek of any sort (with the doubtful exception of
a disputed reading at Aeschylus, Suppl. 235). Subsequently, like other nouns from the same stem,
pyknot¯es and pykn¯osis, and the related verb pyknoein, it is used almost exclusively in philosophical,
scientific or technical contexts, though the adjective was still used frequently in other settings. (Pyknot¯es is contrasted with manot¯es in a musical context at Plato, Laws 812d6–7, but in this instance is more likely to refer to the ‘close packing’ of notes in time than in pitch.)
and in the medical tradition;11 and I suspect, though of course I cannot prove, that it was from them that it passed into the language of harmonics. If so, the contrast and the conceptual associations that it brings with it mark a link between the musical theorists and those who pursued rarefied intellectual research in other fields. We shall find in due course that it would not be the only stowaway from Presocratic philosophy to have embarked on a new career in harmonics.
There is no great distance between a representation of pitches as densely or loosely packed together, and a more explicitly linear conception of the ‘dimension’ of pitch. But I shall postpone any further examination of this issue, and of other evidence about this group of theorists, until Chapter2. Let us now take a preliminary look at the other group mentioned in this passage of the Republic.
m u s i c a l i n t e rva l s a s r at i o s
The direct information Plato provides here about their methods of mea- surement is slight and enigmatic. We are told that they ‘measure audible concords against one another’ (531a1–2), and that they ‘search for numbers in those audible concords’ (531c1–2). If we had to interpret those remarks in isolation we would be hard pressed to do it. We could tell that these people were engaged in attempts at measurement; that the objects measured were audible ‘concords’ (symph¯oniai) and notes; that the measurements were relative, not absolute, (they are measured ‘against one another’); and that these relative measurements were somehow expressed in terms of numbers. From these modest certainties some inferences might hesitantly be drawn. Fortunately, however, we have other information that helps us to construe Plato’s allusive remarks. We can identify it because the Republic identifies this group of harmonic specialists by name. They are the Pythagoreans