Antiquity knew Aristoxenus [fl. 4th c. BCE] as
I must clarify. Aristoxenus’s surviving work, and this is especially true of his Harmonics, does not deal with actual music, but with the material of music — not with the nature of sound but with the nature of musical intervals and scales. In our days, the investigation of such things comes under the rubric of musicology. Sophie Gibson in her excellent study makes a convincing case that Aristoxenus was the inventor of this discipline.
For sheer abstruseness, Aristoxenus’s writings must lie near the top in ancient Greek. Aristoxenus’s subjects, the nature of the musical material and of rhythm, are of themselves difficult. Scientific investigation of them at the time he wrote was at its very beginning, speculative, and confused.1 Though Gibson demonstrates that Aristoxenus followed Aristotle in method, she makes it clear that in the musical sphere he was determined not to owe anything to anybody. He lashed his predecessors, and elsewhere even Aristotle, says the Suda, when the former passed him over as successor for leadership of the Lyceum. Intellectual perplexity and odium scholasticum give his books a double edge; irritation is added to difficulty, rather like the Clarke-Leibniz correspondence.
Furthermore, the most substantial piece of Aristoxenus’s work we have, the Harmonics, survives in three incomplete books, of which Book I appears written at hazard, and Book II is a recasting of the same material, though exhibiting contradictions with Book I. Book III breaks off just when we might have been getting to something to do with real melody. From Books I and II of his Rhythmics there survive chunks, and only much smaller fragments from the rest of Aristoxenus’s work.
Notwithstanding these problems of fragmentation and opaqueness, Aristoxenus is an extremely important source for the study of ancient Greek music, where the evidence is so scarce that we need to squeeze out all we can anywhere we find it. I was delighted to read Sophie Gibson’s excellent discussion of Aristoxenus’s work, and I am most grateful for her heroic labor in elucidating it. Lucid indeed her exposition is, though necessarily requiring constant and diligent attention. A secondary work such as Gibson’s is vital to the study of Aristoxenus, for in many places in his text it is not in the least obvious what he is trying to do, and in such a technical subject we need plenty of context.
Gibson examines Aristoxenus’s work minutely, point by point in a chain of many complex small arguments, and her book is best reviewed chapter by chapter.
In her Introduction Gibson states Aristoxenus’s principal influences — his musical training (the Suda says he was the son of a musician), Pythagoreanism, Aristotle — and his ambition, not simply to describe music empirically, but to “establish an autonomous science” [p. 4]. Others had looked at music before Aristoxenus, notably the Pythagoreans, who were less concerned with music itself than “about its position in a mathematically coherent universe.” [p. 6] The same might be said of Plato, and both Plato and Aristotle are mainly interested in music’s effect on character, again not in music itself. Aristoxenus in his Harmonics ignores both cosmology and ethics, and examines music — at least its pitch structure — as a system in itself. Here, says Gibson, musicology was born.
Gibson gives Chapter One to a survey of harmonic theory before Aristoxenus. Music theory began when Pythagoras discovered that the fundamental consonances of ancient Greek music, the octave, the fifth, and the fourth, can be expressed in numerical ratios, 2:1, 3:2, and 4:3 respectively. Later the tone was found to be 9:8. (The elegance of this series quickly got spoiled when the semitone and diesis, a fraction of the semitone kin to the modern quartertone, were found not to be so tidily definable. Theorists up to Claudius Ptolemy kept fiddling with the numbers.)
And yet, though basic consonances could be represented in number, that they were consonances was judged by the hearing. Music theory quickly began to busy itself with the antithesis of reason versus perception. Aristoxenus rejected the mathematical approach as the absolute criterion of defining sound as musical, Gibson explains, because “we do not perceive musical sound as ratios or relative speed.” [p. 16] Aristoxenus prefers to study the relationship between notes “as multiples of a particular unit of measurement” [p. 18]:
“… for example, the interval of a fifth as three and a half tones without reference to the ratio of 3:2 and in this way it is more compatible with the way that the human mind perceives music.” [p. 18]
Earlier empiricists had accepted a similar logic, and tried to identify the “atom” (so to speak) of musical space from which they thought to build larger intervals. Plato ridicules these harmonikoi listening intently over their monochords in Republic 531a-c. Aristotle takes the diesis as his “least element.” This Aristoxenus rejects; how, he says, can one comprehend 28 dieses in a row when one cannot sing even three? [ Harm. I.28.6ff.] Gibson notes that Aristoxenus tends to use the tone as his unit of measure. By this he avoids the impossibility of mathematically dividing a superparticular2 ratio such as that of the tone, 9:8 — “… a semitone could be perceived as an interval half the size of a tone” [p. 18].
In Chapter Two Gibson examines Aristotle’s influence on Aristoxenus. Aristotle said almost nothing about music itself — he mentions music as it relates to other subjects and never devoted a work to it. It was Aristotle’s method that Aristoxenus took over, especially from the Posterior Analytics, and “applied [it] with a diligence unseen in Aristotle” to music [p. 31]. Aristoxenus worked by the book: he limits his subject, he considers what is observed by the hearing and what can be established by reason, he defines parts, he searches for first principles, he constructs axioms and draws demonstrations from them. He even follows Aristotle’s injunction to pay attention to things of value in predecessors’ work, though he did that in order to trash it.
Considered as an intellectual structure, Aristoxenus’s exposition looks neat. However, Gibson concludes, the attempt to apply the Aristotelian axiomatic method to harmonic science is not “entirely appropriate” [p. 38], and in her following chapter she demonstrates how this more geometrico reasoning breaks down when applied closer and closer to real music as Aristoxenus tries to bull his way through contradictions. Even though flawed, such a method was (she says) a program that later theorists could and did extend and refine.
In Chapter Three, the focus of her book, Gibson surveys the Harmonics, explicates the work’s contents and develops her discussion of Aristoxenus’s Aristotelian method. The state of the text is problematic, and so Gibson attends usefully to the relationship of Books I and II, and the organization of the whole. To broadly characterize Gibson’s approach to the Harmonics, she is what Aristoxenus so badly needed, a good, “blue-pencil” editor.
The heart of this heart-chapter, in my opinion, lies in Gibson’s explanation of Aristoxenus’s concept of dynamis, that is, how any note, bounded by intervals, is what it is by how it functions within the whole scale. This way of looking at the note liberates it from the ungiving rigor of mathematical definition: “the perception of a structure is more important than its measurement” [p. 46]. Gibson is right to consider this Aristoxenus’s most important conceptual contribution; it is as though in treating the note he had discovered the existence of the “phonemic” as something apart from the “phonetic.”
In Chapter Four Gibson turns to the fragments of Aristoxenus’s Rhythmics. Here too Aristoxenus attempted to organize his theory axiomatically. He makes a separation between rhythmics and metrics, between rhythm in the abstract and as it is embodied in the rhythmizomena. He again rejects earlier views which based rhythm on the syllable or analyzed it by feet, and posits instead a
Aristoxenus’s other works occupy Chapter Five. Only fragments survive, and a few titles of the 453 books the Suda says Aristoxenus wrote. Later authors cite Aristoxenus mainly for his pronouncements on music, dance, instruments, tragedy. What remains of his work on music and education shows his conservatism and his attachment to Pythagorean ethics. Aristoxenus wrote the life of Pythagoras and his disciples, and on Pythagorean doctrine. He was also known in antiquity as a skillful, albeit often harsh and sour-tempered, biographer of other philosophers and poets.
None of Aristoxenus’s other works ever approached the influence and important of the Harmonics. Gibson in her final chapter surveys the transmission and development of Aristoxenus’s ideas throughout antiquity, and their reception by Boethius, through whose De Institutione Musica they passed to the Middle Ages and in some sense to the present.
Here, and in her general conclusion, Gibson sums up Aristoxenus’s contributions to musicology. First, he invented musicology as a discipline in itself, not subservient to cosmology or ethics. He brought method to the investigation of harmonics, and, even though the method he used was flawed, his example suggested how it could be used better. Aristoxenus devised essential conceptions: that musical sound is that which proceeds by stepped pitches; that note is the first element of melody and that how it functions in a scale is what makes it musical; that larger intervals can be put together from the tone and smaller intervals defined as parts of a tone.
Aristoxenus did not abjure mathematics altogether. Later ancient theorists blended his synthetic approach to intervals with the Pythagorean method, and this is very much what every musician accepts today, a Both/And way of thinking: a fifth is both pitches in a ratio of 3:2 as to their frequencies, and also three tones and a semitone. The concept of dynamis seems intriguingly close to the way the modern theory of harmony defines notes of the scale as “tonic,” “dominant,” and so forth, and how they change functions in modulation, say in a classical sonata.
Aristoxenus is responsible for the limitation of scales, which at times threatened to proliferate uselessly, to the seven possible species of the octave.
Most of all, says Gibson, Aristoxenus created a language in which the empirical discussion of music could take place. Surely, if we had a good sample of the music itself whose harmonic material Aristoxenus attempted to described, we would understand him much better, and perhaps even admire how well he managed in his pioneer attempt. Unfortunately, when Aristoxenus speaks, it as if we are hearing a lecture on biology without a fish on the plate; we hear much about what Aristoxenus thinks about his personal approach to harmonics, but have no music to relate his opinions to. Yet it is important to understand these
Notes
1. P. 20. As I read Gibson here, I had a small insight on a conceptual problem which hindered ancient harmonicists from a clear understanding of the nature of pitch. This will I hope confirm Gibson’s characterization of early harmonic theory as “confused,” and I would be grateful for the liberty to air it here. Gibson [p. 16] quotes Aristoxenus’s dismissal of those who define notes and sound as “movements,” and she refers us [note 45] to Archytas’s [Fr. 1 (KD 47B1)] theory that difference in pitch was due to difference in force of breath. Other theorists speculated on the “speed” of high and low pitches. Much confusion, I conjecture, came into acoustics because the Greeks lacked, though perhaps were groping for, the concept of “frequency.” The English word itself is late, 1838 [ OED ]. It may be telling that the Greek words for “low” and “high” pitch,
2. Or “epimoric”: a “ratio … such that the greater term is equal to the smaller term plus an integral ‘part’ or ‘factor’ of the smaller term. All such ratios have the form n+1:n.” (Chapter One, note 15).