Andrew Barker,

*The Science of Harmonics in Classical Greece.*Cambridge: Cambridge University Press, 2007. Pp. xii, 481; figs. 11. ISBN 9780521879514. $115.00.

Reviewed by Nathan Sidoli, Waseda University

Barker has written an important book for anyone interested in ancient Greek music theory and its relationship with other intellectual activities of the time, such as philosophy and the empirical or mathematical sciences. *Harmonics* was the branch of ancient music theory that sought to give a rational account of the relationships between pitched sounds and to describe the systems of pitched sounds that were, or according to the theorists should have been, used in musical compositions. Although harmonics was not the only field of ancient music theory, it was in a sense the central field and hence attracted considerable attention from those ancient theorists who wished to found music theory on a philosophically acceptable rational basis. In this book, as the title suggests, Barker focuses on the classical period of the discipline, beginning with the *harmonikoi* and the Pythagoreans and ending with a discussion of Aristotle's successor, Theophrastus.

The book is divided into three parts. Part 1, "Preliminaries," introduces the reader to the basic terminology and systems of ancient Greek music theory and to the fundamental challenge of ancient harmonics--the problem of measuring musical intervals. The rest of the work is divided into two parts according to the two fundamental ways of conceiving of musical intervals, as linear distances or as ratios. Part 2, "Empirical Harmonics," covers the theorists who sought to measure musical intervals by some unit measure or by conceiving of the interval as a linear distance, starting with the early *harmonikoi* and focusing primarily on the work of Aristoxenus. Barker gives a thorough discussion of the *harmonikoi*, discussing them first through the testimony of Plato and then at greater length through that of Aristoxenus. Finally, he situates their work in the context of the sophists and more importantly in the domain of working musicians. This is followed by Barker's treatment of Aristoxenus, which is centered on a sustained and close reading of *Elementa harmonica*. Part 3, "Mathematical Harmonics," treats the theorists who sought to account for musical intervals with whole number ratios and to give some sort of mathematical or rational account of the different types of intervals based on these ratios. This section treats the harmonic work of the Pythagoreans, Philolaus and Archytas, the harmonic passages of Plato and Aristotle, a full analysis of the Euclidean *Sectio canonis*, and concludes with a treatment of Theophrastus' critique of the general project of quantification. Since this book covers a large number of texts and offers detailed, and in many cases new, discussions of all of them, I will focus on just a few of the texts and point out how Barker takes a fresh approach.

In Barker's reading, Aristoxenus developed his theory of harmonics, not only as purely theoretical discipline, but also so that it might be of use to those who are appraising the value of musical compositions. For Aristoxenus, the fundamental objects of harmonics are not notes and intervals but "melodic structures, and the principles that govern their construction" (138). As argued in Chap. 9, Aristoxenus did not believe that knowledge of harmonics alone would be sufficient to make one widely knowledgeable of music (231), hence his harmonics should be read as part of a broader approach of practice and aesthetic evaluation, of which technical expertise forms only one component (Chap. 9).

After a detailed discussion of the arrangement of the *Elementa harmonica*, in which Barker argues, as have others before him (Gibson 2005, 40), that Books II and III are a revision of the project set out in Book I with the new addition of, among other things, the concept of *dynamis* (115-122). In particular, Barker argues that the text as we have it derived from two lecture series, the later of which revisited the material in the first from a different perspective, but also presupposed an understanding of the material covered in the first (123-134).

Barker provides a useful discussion of Aristoxenus's elusive concept of *dynamis* by enumerating all of the passages in which the term appears and discussing them in turn (183-192). Aristoxenus uses the term *dynamis* with a range of meanings, which center around the relationship of notes to one another, or the function of a note within some larger system of notes. It is the *dynamis* of a note that allows us to give it a proper name (such as *lichanos* or *mese*). For Aristoxenus, the *dynamis* of notes, or a tetrachord, can be described numerically, in the sense that a range for the possible sizes of the intervals can be set out, but the *dynamis* itself is more fundamental and is based, in some sense, on the nature of melody itself (189-191).

Although Aristoxenus takes a strong position against the attempts of other theorists to quantify harmonics, in Book III of *Elementa harmonica*, he develops an approach to the science that is clearly based in the deductive texts of the mathematicians. The bulk of Book III is made up of four preliminary arguments, establishing first principles, and twenty-three theorems that are demonstrated on the basis of these. Barker treats this material by going through two example theorems (200-204) and then focusing on three problems that arise from reading Aristoxenus's text: (1) the elucidation of certain logical difficulties in the theorems, (2) a reconciliation of Aristoxenus's stance against quantification with the toil he exerts on proving the quantitative statements of the theorems, and (3) the general purpose of the theorems in Aristoxenus's science (208-215). Barker's analysis goes some way to answering many of the difficult issues raised by this series of propositions, which is the only example we have of deductive style Greek science not found in the applied mathematical texts.

The section on mathematical harmonics begins, not with a discussion of "Pythagorean harmonics" but rather with chapters on harmonic theory as found separately in the writings of Philolaus and Archytus, taking as point of departure the recent studies of these thinkers by Huffman (1993, 2005). This has the effect of further undermining the traditional view of the Pythagoreans engaging, as a school, in a single project on the mathematization of music theory. Indeed, Barker argues that Philolaus used both ratio conceptions of intervals along with the linear conceptions of empirical theorists and that he focused on the numbers extracted from the ratios, as opposed to the relations expressed by them in order to carry out a largely cosmological project (278-286). In contrast, it is claimed that Archytas introduced the strict correlation between ratios and musical intervals, found in the later mathematical tradition, and did so in order to advance a mathematical theory that addressed both the physical processes of pitch production and the practical realm of musical experience (287-307).

Barker argues that the Euclidean *Sectio canonis* is an integrated work composed around the turn of the 3rd century BCE and meant to secure a deductive foundation for mathematical harmonics. He begins by arguing against claims that passages in Ptolemy and Porphyry give evidence for a version of the treatise shorter than what we find in the modern editions (366-370). Barker divides his discussion of the treatise into the following sections: introduction, props. 1-9, props. 10-13, props. 14-18, and props. 19-20. Whether or not we agree with the reasoning, the brief introductory passage attempts to give a rational argument for why it is appropriate to associate ratios with pitch intervals and to map certain classes of ratios to certain classes of intervals (370-378). The next section, props. 1-9, lays out a series of mathematical theorems whose character is purely number theoretic (378-384). This is followed by four propositions, props. 10-13, that transition into harmonics by associating certain notes with the previously discussed intervals, and which presume that the reader is familiar with some basic music theory. Next follows a series of loosely related propositions, such as that the fourth is less than two and a half tones, and the so-called "enharmonic passage" which gives a method for setting out an enharmonic tuning and then applies some of the arguments developed in the foregoing to claim that certain intervals are not divided equally. Barker sees no reason to believe that the enharmonic passage was not included in the original composition (391-394). Finally, the work transitions to a geometrical model and sets out a diatonic division of the monochord. Barker's position is that the treatise as a whole, although apparently disorganized in places, is an attempt to systematize various aspects of harmonics by means of a theory of the physics of pitched sound and some contemporary developments in mathematics.

Barker concludes his treatment of mathematical harmonics with a study of the criticisms of the foundations of the mathematical approach advanced by Theophrastus in the long fragment of his *On Music* found in Porphyry's *Commentary on Ptolemy's* Harmonics. According to Barker's reading, Theophrastus objects that any kind of quantitative abstraction from the physics of the acoustics of pitch can never satisfactorily account for the quantitative difference of pitches, which should be the main concern of music theorists (414-421). Furthermore, Barker argues that Theophrastus does not put forward a coherent object to Aristoxenus's approach, and hence is probably arguing against some other theorists in the empirical tradition (421-428).

This is a densely argued work with many detailed discussions of technical sources. It is unlikely that any scholar will agree with all of Barker's readings, but we should be grateful to him for laying out his own readings with such care, and in the process, shedding light on many difficult passages.

References

Gibson, S., 2005, *Aristoxenus of Tarentum and the Birth of Musicology* (Routledge, New York).

Huffman, C. A., 1993, *Philolaus of Croton, Pythagorean and Presocratic* (Cambridge University Press, Cambridge).

Huffman, C. A., 2005, *Archytus of Tarentum, Pythagorean, Philosopher and Mathematical King* (Cambridge University Press, Cambridge).

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