The focus of this book is the role of the monochord—lit. ‘a single string’ divisible at measured points through movable bridges—as a demonstrative tool in ancient theoretical investigation of music. During its course, the text becomes a comprehensive study of the Greek harmonic science and its place among mathematical sciences in antiquity. Through a careful and detailed analysis of all the relevant evidence about the instrument, which ancient theorists used for establishing arithmetical relationships between musical sounds and for making them audible to the ear and measurable as visual distances (i.e., as lengths of strings), David Creese writes a remarkable history of mathematical harmonics, the branch of harmonics dealing with musical intervals as ratios of numbers. In doing this, he evaluates the theoretical consequences of introducing a geometrical tool into the realm of mathematics and examines in detail how such an instrument could affect the development of scientific methods of Greek harmonicists.

The documentary evidence for the monochord (in Greek *kanōn*) examined in the book covers a chronological span which, debunking the myth of its invention by Pythagoras in the late sixth century BC, goes from its first literary reference in *The Division of the Monochord* (usually quoted by its Latin title *Sectio Canonis*) attributed to Euclid (c. 300 BC) to its fullest and most detailed description in Ptolemy’s *Harmonics* (second century AD), where the instrument is given a key role in the Ptolemaic conception of harmonic science, revealing the agreement of the two faculties he regards as fundamental for his inquiry: reason and perception. The book is divided into six chapters which, following an introduction outlining the agenda of the volume and starting with a general survey of the role of scientific instruments in ancient theoretical inquiry, deal chronologically with the sources while gradually creating an articulated picture which reveals an increasing importance of the monochord and its peculiar and unequaled (though imperfect) rigor as a demonstrative tool in mathematical investigations.

In Chapter One (‘Hearing numbers, seeing sounds: the role of instruments and diagrams in Greek harmonic science’), Creese compares the monochord to other explanatory devices (such as mathematical diagrams or the armillary spheres used in astronomy) employed by ancient scientists for their theoretical investigations, with the intention of showing the peculiarities and advantages of the *kanōn* as a diagrammatic instrument which displays geometrically the arithmetical relationships between musical sounds. It thus acts as a logical intermediary between their mathematical properties and their existence as audible entities, since it not only visually represents but also generates and ‘produces’ the intervals between them. Creese firstly sketches the different approaches to the analysis of musical intervals in antiquity: on the one hand there is the arithmetical approach here under discussion, which conceived intervals as whole-number ratios, for instance, 2:1 = octave, 3:2 = a fifth, 4:3 = a fourth, and hence regarded irrational relationships as ‘unmusical’. On the other hand, there is a perception-based approach, adopted by Aristoxenus, which considers musical intervals only according to their melodic usage and perceptible qualities, and hence enabled Aristoxenus to treat the tone as divisible into two, three or four equal parts. Based on this sketch, Creese casts off some of the misunderstandings which are still persistent within the musicological tradition, such as for example the thesis that Aristoxenus was the first proponent of modern equal temperament. The controversy about the possibility of dividing the tone (expressed by the epimoric ratio 9:8) into two equal semitones (mathematically impossible, since there is no geometric mean between the two terms of an epimoric ratio) is connected with the more important issue of the inconsistencies among the different definitions of semitones admitted by Aristoxenus (a:‘the fourth contains two and a half tones’, hence five equal semitones make up a fourth; b: the semitone is half of the tone which is ‘that by which the fifth is greater than the fourth’—but this tone (3:2 / 4:3) is bigger than the tone formed by two of the five equal semitones—; c: the semitone is ‘the interval which remains when two tones have been taken from a fourth’, i.e. it corresponds to the *leimma* of the mathematical theorists: 256:243), showing how modern tempered semitone (i.e. 100 cents: d) is actually different from the three semitones admitted by Aristoxenus: b > d > a > c (for the mathematical quantities of these four different semitones, see p. 30).

Chapter Two (‘Mathematical harmonics before the monochord’) tries to establish when the instrument first came into use in Greek antiquity, ridding us of anecdotes and their vastly over-interpreted reception. One such anecdote has Pythagoras making the chance discovery of the ratios of concords at a blacksmith’s shop and then testing his discovery through empirical experiments, including one on the monochord. Establishing that the earliest empirical demonstration of harmonic ratios by Hippasus of Metapontum (early fifth century BC)^{1} does not imply knowledge of the monochord, Creese analyzes the pioneering work in the harmonic field by Philolaus and Archytas, the two earlier Pythagoreans often credited with the use of such an instrument, showing that the harmonic ratios attributed to them were actually never matched with string lengths.

Chapter Three (‘The monochord in context’) is mostly devoted to the *Sectio canonis*, the earliest mathematical treatise where this instrument is shown in its demonstrative role. Creese starts by reconstructing the fourth-century demonstrative and argumentative background of this text, showing its debts both to harmonic *epideixeis* (lit. performances or displays where lecturers in music —called *harmonikoi*, lit. ‘experts in *harmonia* ’— made theoretical demonstrations on stringed instruments) and to the Aristotelian formulation of apodeictic demonstration (that is, a scientific procedure whose method follows the formal demonstrative logic of Aristotle’s *Analytics*) and its application to mathematics in Euclid’s *Elements*. Then he examines the influence exerted on this treatise by the fourth-century harmonic and acoustic theoretical developments, evident, for instance, in the choice of using the ‘ditonic’ scale of Plato’s *Timaeus*, 9:8 × 9:8 × 256:243, instead of other interval models, to construct the division of the *kanōn*; or in the replacement of Archytean acoustics with a theory that describes pitches as variable rates of vibration, instead of as variable speeds of travel, echoing Peripatetic writings like the *De audibilibus* and the *Problems*.

Chapter Four (‘Eratosthenes’), the briefest of the book, deals with the only datable author among mathematical harmonicists who worked during the Hellenistic time. A careful and detailed analysis of the surviving fragments^{2} of his work reveals the lack of firm evidence that Eratosthenes really used the monochord, since his fragments only vaguely refer to a canonic division and provide a set of tetrachord ratios that show many affinities with those assigned to Aristoxenus and seem to have had just the intention to convert the Aristoxenian data (based on a ‘spatial’ conception of musical intervals) into the rational language of mathematical harmonics.

In Chapter Five (‘Canonic Theory’), which I consider the most valuable section of the book, Creese makes a significant survey of all the evidence we have about the monochord from the time between Eratosthenes and Ptolemy, since it is precisely between the late third century BC and the second century AD that the harmonic science came to be called ‘canonics’ ( *kanonikē*, lit. ‘monochord science’). The term itself reveals the instrument’s unquestionable centrality at the time in shaping the science of mathematical harmonics. This chapter combines a study of the work of little-known theoreticians like Panaetius ‘the younger’, Ptolemaïs of Cyrene, Didymus ‘the musician’ and Adrastus ‘the Peripatetic’,^{3} in order to present a broader picture of the animated debate between the supporters of the roles granted to reason and to perception in different approaches to harmonic science. The author rightly emphasizes here the difficulties resulting fromthe empirical employment of the *kanōn* in a theoretical inquiry. After all, this was an instrument that presented sensory evidence to reason through a ‘physicality’ which could never completely rule out the imperfections of material objects.^{4} This tension is by the author used firstly to show to what extent ancient theorists were aware of the dangers inherent in premising the correctness of a scientific procedure on the defective accuracy of an empirical device, and then to better introduce the last chapter of the book, which examines the peculiar attention paid by Ptolemy to these problems.

Chapter Six (‘Ptolemy’s Canonics’) points out how Ptolemy’s remarks on canonic science, though greatly relying on the previous tradition (especially on Didymus’ work),^{5} was definitely instrumental in establishing the reliability and accuracy of the monochord as scientific instrument: he associated it equally with the realms of reason and perception, and resolved the discrepancies between the results obtained through its theoretical and its empirical usage by reducing these discrepancies to an imperceptible degree. To this purpose, Ptolemy proposed to build more complex and larger instruments, with eight or even fifteen strings (on which each pitch could have its own string), and to employ them empirically, judging the intervallic ratios through the wider melodic context of the most common scale structures. Moreover, he gave the *kanōn* a place within his broader approach to harmonic science (conceived as a reflection of the mathematical structure of the universe), presenting ‘the universe itself as a kind of perfectly tuned *kanōn* ’ (p. 355).

The results achieved by this book go well beyond those suggested by the title (a subtitle would have perhaps done more justice to its contents). Despite the complexity of the topic, the text is well written. It makes an important contribution not only to current scholarship on Greek musical theory (nowadays no longer a peripheral discipline within musicology and classical studies), but also to the study of ancient Greek science in general.

**Notes**

1. Aristox. fr. 90 = Schol. Pl. *Phd.* 108d4 = DK 18.12. Hippasus is said by Aristoxenus to have struck four bronze discs to make them resound; they had equal diameters but varied thickness according to the ratios corresponding to musical concords (12:9:8:6).

2. Nicom. *Harm.* 260.12-17; Ptol. *Harm.* II.14.

3. Although the author admits his debts to other scholars who examined specific questions related to this topic, the wider picture emerging from this chapter constitutes an important advance in scholarly work on harmonic science.

4. These difficulties are made explicit, even if not fully developed, by Adrastus (ap. Porph. *in Harm.* 71.4-72.3), who points out how the bridge of the monochord always has some width and hence makes the accuracy of demonstration slightly imperfect (p. 254 ff.).

5. On this, see also S. Hagel, ‘The Context of Tuning: Thirds and Septimal Intervals in Ancient Greek Music’, in E. Hickmann, A.A. Both and R. Eichmann, *Studien zur Musikarchäologie V. Music Archaeology in Context. Archaeological Semantics, Historical Implications, Socio-Cultural Connotations* (Rahden/West. 2006), pp. 281-304.