# Aristoxenus

# Aristoxenus

(*b.* Tarentum, *ca*. 375-360 B.C.; *d*. Athens [?]), harmonic theory.

Aristoxenus was a native of Tarentum, a Greek city in southern Italy. He flourished in the time of Alexander the Great (reigned 336-323), and can hardly have been born later than 360. His father’s name was either Mnaseas or Spintharus; the latter was certainly his teacher, a musician whose wide acquaintance included Socrates, Epaminondas, and Archytas. Aristoxenus studied at Mantinea, an Arcadian city that had a strong conservative musical tradition, and later became a pupil of Aristotle in Athens. His position in the Lyceum was such that he hoped to become head of the school upon the death of Aristotle (322 B.C.); he is said to have vented his disappointment in malicious stories about his master. The date of his death is unknown, but the attribution to him of 453 published works (even if many were spurious)suggests a long life.

Of this vast production little has survived except for three books that have come down under the title *Harmonic Elements*. Modern scholars are agreed, however, that these represent two or more separate treatises. There is a substantial fragment of the second book of *Rhythmical Elements*. Aristoxenus’ numerous other writings on music are all lost, except for quotations, but much of our scattered information on early Greek musical history must derive from him. He also wrote biographies (he was one of those who established this kind of writing as a tradition of the Peripatetic school); treatises on educational and political theory and on Pythagorean doctrine; miscellanies; and memoranda of various kinds.

It is proper, if paradoxical, that Aristoxenus should be included in a dictionary of scientific biography. It is proper because music under the form of “harmonics” or the theory of scales was an important branch of ancient science from the time of Plato onward, and because Aristoxenus was the most famous and influential musical theorist of antiquity. It is paradoxical because he turned his back upon the mathematical knowledge of his time to adopt and propagate a radically “unscientific” approach to the measurement of musical intervals.

When the Pythagorean oligarchs were expelled from Tarentum, Archytas, the celebrated mathematician and friend of Plato, remained in control of the new democracy and may still have been alive when Aristoxenus was born. From Archytas’ pupils in Tarentum-or from the exiled Xenophilus in Athens Aristoxenus must have become familiar with Pythagorean doctrine. The Pythagoreans recognized that musical intervals could be properly measured and expressed only as ratios (of string lengths or pipe lengths). Pythagoras himself is said to have discovered the ratios of the octave, fifth, and fourth; and the determination of the tone as difference between fifth and fourth must soon have followed. In each case the ratio is superparticular and incapable, without the aid of logarithms, of exact division in mathematical terms. Thus, “semitone” (Greek, *hemitonion*)is a misnomer, when two tones were subtracted from the fourth, the Pythagoreans preferred to call the resulting interval “remainder” (leimma). The Pythagorean diatonic scale, consisting of tones (9:8)and *leimmata* (256:243), was known to Plato. Archytas worked out mathematical formulations for the diatonic, chromatic, and enharmonic scales upon a different basis.

Aristoxenus, however, turned his back upon the mathematical approach and stated that the ear was the sole criterion of musical phenomena. To the ear, he held, the tone was divisible into halves (and other fractions); the octave consisted of six tones, the fifth of three tones and a half, the fourth of two tones and a half, and so on. His conception of pitch was essentially linear; the gamut was a continuous line that could be divided into any required fractions, and these could be combined by simple arithmetic. It was for the cultivated ear to decide which intervals were “melodic” i.e., capable of taking their places in the system of scales.

The division of the octave into six equal tones and of the tone into two equal semitones recalls the modern system of “equal temperament,” and it is held by some authorities that Aristoxenus envisaged such a system or sought to impose it upon the practice of music. If that were so, he might well have rejected a mathematics that was still incapable of expressing it. Equal temperament, however, was devised in modern times to solve a specific problem: how to tune keyboard instruments in such a way as to facilitate modulation between keys. No comparable problem presented itself to Greek musicians: although modulation was exploited to some extent by virtuosi of the late fifth century B.C. and after, there is no reason to suppose that it created a need for a radical reorganization of the system of intervals or that such could have been imposed upon the lyre players and pipe players of the time. Furthermore, such a “temperament” would distort all the intervals of the scale (except the octave) and, significantly, the fifths and fourths; but Aristoxenus always speaks as though his fifths and fourths were the true intervals naturally grasped by the ear and his tone the true difference between them. It seems more likely, then, that he took up a dogmatic position and turned a blind eye to facts that were inconsistent with it; this would be in keeping with the rather truculent tone he sometimes adopted.

Writers on harmonics, from Aristoxenus on, fall into two schools: his followers, who reproduced and simplified his doctrines in a number of extant handbooks, and the “Pythagoreans” such as Eratosthenes, Didymus the musician, and-notably-Ptolemy, who elaborated ratios for the intervals of the scale. It is perhaps doubtful whether any writer of the mathematical school prior to Ptolemy produced a comprehensive theory of scales in relation to practical music, and it may well have been the inadequacy and limited interests of the Pythagoreans that set Aristoxenus against this approach. Nor would it be fair to deny Aristoxenus’ scientific merits because of his disregard of mathematics. He was not in vain a pupil of Aristotle, from whom he had learned inductive logic and the importance of clear definition: and what he attempted was, in the words of M.I. Henderson, “a descriptive anatomy of music.” His arguments are closely reasoned, but, lacking his master’s breadth and receptivity, he can be suspected of sacrificing musical realities to logical clarity.

The details of his system, which can be found in standard textbooks and musical encyclopedias, are not in themselves of primary interest to the historian of science. The quality of his thinking at its best can, however, be illustrated from his work on rhythm. Earlier writers had tended to discuss rhythm in terms of poetic meters, but, since rhythm also manifests itself in melody and in the dance, there was some confusion of thought and terminology. Aristoxenus drew a clear distinction between rhythm, which was an organized system of time units expressible in ratios, and the words, melodies, and bodily movements in which it was incorporated (*ta rhythmizomena*) and from which it could be abstracted. This was a much-needed piece of clarification worthy of Aristotle.

## BIBLIOGRAPHY

I. Original Works. Modern editions of Aristoxenus’ works are *The Harmonics of Aristoxenus*, edited, with translation, notes, introduction, and index of words, by H. S. Macran (Oxford, 1902); and *Aristoxeni elementa harmonica*, Rosetta da Rios, ed. (Rome, 1954).

II. Secondary Literature. Works dealing with Aristoxenus include Ingemar Düring, *Ptolemaios und Porphyrios über die Musik* (Göteborg, 1934), which contains, in German translation, Ptolemy’s criticisms of Aristoxenus; C.von Jan, “Aristoxenos,” in Pauly-Wissowa, *Real Encyclopädie*, II (Stuttgart, 1895), 1057 ff.; L. Laloy, *Aristoxène de Tarente* (Paris, 1904), an outstanding work; F. Wehrli, Die *Schule des Aristoteles, Texte und Kommentar*, Vol. II, *Aristoxenos* (Basel, 1945), for the shorter fragments; and R. Westphal, *Aristoxenos von Tarent* (Leipzig, 1883-1893), which includes the fragment on rhythm.

R. P. Winnington-Ingram

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**Aristoxenus**