Greek Reflections on the Nature of Music

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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.

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This is imprecise; in Elements of Harmony , Aristoxenus uses tasis simply of a rise in pitch and anesis of a drop in pitch. 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 a However, there is no problem with a continuous magnitude being divided at any point, though it is not composed of points.

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Aristotle himself makes this clear at On Generation and Corruption a 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. 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.

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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.

The first half of Chapter 6, "Aristoxenus of Tarentum and Ptolemais of Cyrene," continues the examination of mathematical implications of Aristoxenus's method.

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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. This hypothesis is intriguing, elegantly expressed, and couched with appropriate acknowledgement that it cannot be proven; alternative viewpoints are cited p.

Greek reflections on the nature of music

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.

Ptolemais attributes to Aristoxenus a statement on the importance of rational thought as applied to objects of perception that Levin p. Due to the brevity of the quotes from Ptolemais, Levin supplements them with the most extended citations in the book from Aristoxenus's 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.