Sadly, Flora Levin passed away this spring; her death deprives the field of ancient music and music theory of an original and deep thinker. This volume offers provocative interpretations of Aristoxenian music theory while providing a context in modern mathematics, philosophy, and musicology for the Aristoxenian and other schools of ancient music theory. Levin held that the magic and wonder of music transcends time and unites ancient Greek writers with modern readers. As she says in her preface (p. xvi) “Greeks were the first to intuit music’s essence, and the first to discover the universal laws governing its structure.” Levin had the ability to make intellectual history seem an adventure and wrote this book to be both accessible and of interest to non-specialists. However, the large number of typographical and/or editorial errors justifies caution in recommending this edition to non-specialists.

In her introduction, Levin attributes to the ancient Greeks a uniquely acute sensitivity to pitch on the basis of their use and discussion of microtones in music. She overlooks the fact that modern violinists, for example, must develop a similar sensitivity to microtonal pitch differences and display a similar sophistication in discussing pitch differences as small as that between a major tone with the ratio 9/8 and a minor tone with the ratio 10/9. In subsequent chapters, her over reliance on the experience of modern pianists leads her to underestimate the difficulties in Aristoxenus’ position that the interval of a fourth equals 2 1/2 tones. Levin goes on to present the thesis that in his calculation of musical intervals, Aristoxenus was “practicing analytic number theory centuries before its foundations were laid by such luminaries as Peter Gustav Lejeune Dirichler, Bernhard Riemann, Georg Cantor, Leopold Kronecker, and Karl Weierstrass.” (p. xvii) While she does explain Aristoxenus’s method of calculating intervals and compares it to the methods of other ancient Greek music theorists in the following chapters, she does not address the question of what it would mean to practice number theory without an explicit logical foundation such as these modern mathematicians developed.

In the first chapter, “All Deep Things Are Song,” Levin contrasts the definition of music given by the 1st-2nd century AD music theorist Bacchius the Elder (whom she later identifies as a representative of Aristoxenian music theory) with the approach of Pythagoras and his followers. While Pythagoras had uncovered the mathematical relationships between consonant musical tones, Bacchius avoided explaining music through numbers. Instead, he defined music as the knowledge of melody, and then gave a circular definition of melody as a sequence of melodious notes. Levin (p. 3) endorses this circularity, citing Ludwig Wittgenstein’s statement that music “is a form of tautology.”^{1} She traces the influence of the Pythagorean approach, which united music theory with astronomy, but concludes that Bacchius’s approach represents a more profound philosophy. She quotes a wide range of modern writers and composers on the validity of an analogy between music and language, concluding that such an analogy ultimately breaks down; music is best considered a reality unto itself.

The second chapter, “We Are All Aristoxenians,” eloquently observes that Aristoxenus rejected attempts to extend Pythagoras’s identification of ratios with musical intervals, the approach exemplified by works such as Euclid’s *Divison of the Canon* and Ptolemy’s *Harmonics*. Levin makes the valid point that followers of Aristoxenus such as Bacchius and Cleonides lack Aristoxenus’s application of Aristotelian concepts such the as infinite divisibility of continuous magnitudes to musical intervals and the Aristoxenian theory of melodic function. Levin strongly attacks modern scholars such as Winnington-Ingram and West who, she feels, fail to appreciate Aristoxenus’s use of intuition as a source for musical theory. Levin then quotes a number of modern musicologists who, though they are not writing about Aristoxenus, express ideas similar to Aristoxenus’s about the limitations of musical notation and the role of perception and memory in music. Her justification of the chapter’s title is that Aristoxenus sought after and discovered universal principles of music, “the links that bind the music of all ages and cultures together and that make music intelligible on its own terms.” (p. 86)

In the third chapter, “The Discrete and the Continuous,” Levin compares Aristoxenus’s treatment of the interval of the semitone, which in terms of ratio theory would be the square root of 9/8 (or 3/2 the square root of 2), but which Aristoxenus says is simply half the difference between the intervals the fourth and the fifth, to Archimedes’s approximation of the irrational number *pi*. She gives an overview of Aristoxenus’s life and development as a musical theorist, and favorably contrasts his approach to the continuity of musical intervals to Theophrastus’s. Later in this chapter, Levin identifies the Aristoxenian term *tasis* with the modern concept of melodic tension and the Aristoxenian term *anesis* with the modern concept of melodic resolution (p. 110-111). This is imprecise; in *Elements of Harmony*, Aristoxenus uses *tasis* simply of a rise in pitch and *anesis* of a drop in pitch. Aristoxenus’s theory of melodic function does seem to indicate an interest in the general kinds of melodic/perceptual effects modern theory describes as tension and resolution, but he does not use the terms *tasis* and *anesis* in these senses. As she proceeds in developing a philosophical context for Aristoxenus’s position that the interval of a fourth equals 2 1/2 tones, Levin makes problematic certain issues that need not be problematic. Aristoxenus conceives of intervals as being continuous and infinitely divisible, while individual pitches are discrete and indivisible. Aristotle had established that continuous, infinitely divisible magnitudes are not composed of discrete points; see, for example, *Physics* 231a24-25. Levin (p. 113) finds this to be a logical difficulty for Aristoxenus. However, there is no problem with a continuous magnitude being divided at any point, though it is not composed of points. Aristotle himself makes this clear at *On Generation and Corruption* 317a7-8. Instead, Levin proposes that Aristoxenus used his theory of melodic function and consideration of the limits of human perception to resolve this alleged paradox. The chapter closes with a mathematical lapse on Prof. Levin’s part. In arguing that the ratio theory of concordant intervals is in a way as problematic as Aristoxenus’s measurement of the fourth as 2 1/2 tones, she says “Infinity thus pours fourth from the discontinuity of the octave with its own parts, so that the octave is to its internal semi-tones as .666666666 to infinity.” (p. 119-120)

Chapter 4, “Magnitudes and Multitudes,” begins with the claim that Aristoxenus used the twelfth-tone interval as an infinitesimal measure in a process analogous to integration in calculus. Most of the chapter is devoted to an exposition of Euclid’s *Division of the Canon* as an example of the Pythagorean approach to harmonic theory that Aristoxenus rejected.

In Chapter 5, “The Topology of Melody,” Levin discusses Johannes Kepler’s interest in the relationship between harmonic theory and astronomy Ptolemy presented in Book 3 of his *Harmonics*. This leads to a brief mention of the possibility that string theory in contemporary physics may open the door for a new appreciation of links between harmonics and cosmology. Citations from contemporary music theorists on the concept of musical space bring Levin back to aspects of Aristoxenus’s method first brought up in chapter 3, considered in more detail by incorporating specific terminology of the Greek melodic system presented in chapter 4 and consideration of other music theorists prior to Aristoxenus: Damon, Lasus, and Epigonus. Levin emphasizes that Aristoxenus privileged the mental experience of musical attunement over the musical instruments that produce musical notes, and that this priority of the mind justifies his seeking a new approach to musical theory. She presents the thesis that Aristoxenus’s central contribution was to use the twelfth-tone interval to transform “the geometrical idea of a magnitude, *megethos*, to the arithmetic idea of a collection, *plêthos*, of discrete units.” (p. 199)

The first half of Chapter 6, “Aristoxenus of Tarentum and Ptolemais of Cyrene,” continues the examination of mathematical implications of Aristoxenus’s method. Levin makes an undeniable contribution by drawing attention to Aristoxenus’s incorporation of irrational numbers along with rational numbers in a one-dimensional continuum of pitch. She expands upon Aristoxenus’s use of the twelfth-tone interval as a unit of measurement, replacing the multiplication or division of ratios with the addition or subtraction of twelfth-tone units, by drawing upon testimony from the Aristoxenian theorists Cleonides and Aristides Quintilianus as well as Ptolemy’s account of Aristoxenus’s method. Levin uses citations from the theorist Didymus to make a transition from Ptolemy to Ptolemais of Cyrene. Levin argues that she was the granddaughter of Ptolemais of Egypt, who was the daughter of Ptolemy I Soter, wife of Demetrius Poliorketes and mother of Demetrius the Fair, who ruled Cyrene during the mid-third century B.C. This hypothesis is intriguing, elegantly expressed, and couched with appropriate acknowledgement that it cannot be proven; alternative viewpoints are cited (p. 242n4).

In Chapter 7, “Aisthêsis and *Logos* : A Single Continent,” presents and examines the passage from Porphyry’s commentary on Ptolemy’s *Harmonics* that contains our only extant citations from Ptolemais of Cyrene, along with the passages from Ptolemy on which Porphyry comments. Levin casts these excerpts in the form of a dialogue between Ptolemy, Porphyry, and Ptolemais, indicating in footnotes where her transitions require departures from the Greek texts. This choice enhances the readability of the passage while maintaining fidelity to the musical ideas presented. The passage concerns the relative role of perception and reason in Pythagorean and Aristoxenian theory; Ptolemais describes Aristoxenus as accepting reason and perception as equal in power, though perception takes the lead. An example of Levin’s willingness to elaborate a phrase for the sake of comprehensibility is her translation of the phrase *Commentary on Ptolemy’s Harmonics* 25.23 During as “…on the basis that perception comes first in the order of events, but not on the basis of its power.” (p. 261) Levin interprets Ptolemais as sympathetic to Aristoxenus’s approach. Having argued in chapter 6 that Ptolemais probably wrote during the 3rd century BC, Levin traces the phrase “rational postulates of the canon”, *Elements of Harmony* itself. Levin shows that Aristoxenus’s position on the relationship between of reason and perception, which Ptolemais amplified, is a prerequisite for a harmonic theory that can address the theoretical ramifications of melodic modulation.

In chapter 8, “The Infinite and the Infinitesimal,” Levin briefly treats Aristoxenus’s musical conservatism, and presents the Aristoxenian *tonoi* or transposition scales attested in Cleonides. She concludes by making clear that while Cleonides and Ptolemy represent Aristoxenus’s twelfth-tone interval as a simple unit of measurement, it was in Aristoxenus’s conception based on a doctrine of melodic and mathematical limit.

ERRATA

p. 67, Figure 2 row A.4: “1 + 1/3” should read “1 + 1/5”; row B.3: “1 + 4/3” should read “1 + 4/5;” row B.4: “supertriquitus” should read “supertriquintus;” “1 + 2/3” should read “1 + 3/5.”

p. 119n58: lines of text and lists of musical notes have been jumbled out of order; there appear to be discrepancies in the lists of notes.

p. 142n24: “comprises three conjoined tetrachords” should read “comprises three conjoined tetrachords and the one-tone interval between hypatê hypatôn and proslambanomenos.”

p. 144n28: “paranêtê (next to the top)” should read “(next to the bottom)”; nêtê (the topmost string, which emits the lowest pitch)” should read “(the lowest string, which emits the highest pitch).”

p. 150n37, 150n38, 151n39: note 37 contains the citations for both references 37 and 38; the citation given in note 38 corresponds to p. 151.2, but no reference appears there; the citation in footnote 39 does not correspond to the reference.

p. 152 diagram of Chromatic tetrachord: the ratios indicated, “256:243, 256:243, 64:54” do not multiply to 4:3.

p. 153n41: the citation given corresponds with reference 43; 153n43 repeats the citation given in 153n42.

p. 196n97 diagram: “E F A C” should read “E F A B”; lower bracket of diagram should connect the lowest and highest notes, not the second-to-lowest and the highest.

p. 199.11: “6” should read “7.”

p. 201.2: statement attributed to Hermann Helmholtz is a comment by the translator of his work, in an appendix.^{2}

p. 201.12: “1.05947631” should read “1.0594631.”

p. 201.15: “Had Aristoxenus gone one to multiply 1 by 2 twelve times, he would have arrived at (the twelfth root of 2)” is not a valid statement; “1:1.0594631” should read “1.0594631”; 201.16: “84/89” should read “89/84.”

p. 202 above figure 7, the Hemiolic Chromatic: “3/4 tone” should read “3/8 tone” (bis).

p. 216.9: “1/3” should read “2/3”; 216.10: “21/12” should read “22/12”; 216.11: “1 and 3/4” should read “1 and 5/6.”

p. 217.3: “3 3/12” should read “3/12.”

p. 246 figure 8: diagram of Ptolemais’s family tree should be presented in chapter 6, not chapter 7.

p. 270.1-6 extended quote from Jon Solomon *Ptolemy* not indented.^{3}

p. 286.11: “and” should read “or;” cf. Aristoxenus *Elements of Harmony* 2.54 = 67.4-5 Da Rios.

p. 299.25: “1/2” should read “1/12.”

p. 305.23: Andrew Barker’s name missing from bibliography entries.

**Notes**

1. Ludwig Wittgenstein, *Notebooks 1914-1916*, ed. G.H. von Wright and G.E.M. Anscombe, Oxford: Blackwell 1961, p. 40, 4.3.15.

2. Hermann Helmholtz, *On the Sensations of Tone as a Physiological Basis for the Theory of Music*, Trans. A.J. Ellis, New York: Dover Publications, p. 548.

3. Jon Solomon, *Ptolemy “Harmonics”: Translation and Commentary*, Leiden: Brill 2000, p. xxxv.