This page intentionally left blank GREEK REFLECTIONS ON THE NATURE OF MUSIC In this book, Flora R. Levin explores how and why music was so important to the ancient Greeks. She examines the distinctions that they drew between the theory of music as an art ruled by number and the theory wherein number is held to be ruled by the art of music. These perspectives generated more expansive the- ories, particularly the idea that the cosmos is a mirror-image of music’s structural elements and, conversely, that music by virtue of its cosmic elements – time, motion, and the continuum – is itself a mirror-image of the cosmos. These opposing perspectives gave rise to two opposing schools of thought, the Pythagorean and the Aristoxenian. Levin argues that the clash between these two schools could never be reconciled because the inherent con- ict arises from two different worlds of mathematics. Her book shows how the Greeks’ appreciation of the profundity of music’s interconnections with philosophy, mathematics, and logic led to groundbreaking intellectual achievements that no civilization has ever matched. Flora R. Levin is an independent scholar of the classical world. She is the author of two monographs on Nicomachus of Gerasa and has contributed to TAPA, Hermes, and The New Grove Dictionary of Music. GREEK REFLECTIONS ON THE NATURE OF MUSIC Flora R. Levin Independent scholar CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-51890-1 ISBN-13 978-0-511-54001-1 © Flora R. Levin 2009 2009 Information on this title: www.cambrid g e.or g /9780521518901 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org eBook ( EBL ) hardback To Sam vii Figures page viii Preface ix Introduction xiii Abbreviations xix Texts xxi All Deep Things Are Song We Are All Aristoxenians The Discrete and the Continuous Magnitudes and Multitudes The Topology of Melody Aristoxenus of Tarentum and Ptolemaïs of Cyrene Aisthēsis and Logos: A Single Continent The Innite and the Innitesimal ΣФPAΓIΣ Bibliography Index Contents viii . The Immutable or Changeless (Ametabolon) System page . Names of Ratios . Circle of Fifths . Paradigmatic System with Octave Species . Greater Perfect System: Lesser Perfect System . The Greater Perfect System Projected on the Zodiac . Six Meson Tetrachords Distributed Over Thirty Equal Parts . The Family of Ptolemaïs of Cyrene (?) . The Seven Tonoi of Ptolemy . The Harmonic Series Figures [...]... number From the intrinsic properties of this – the tetraktys of the decad – the sum of whose terms equals the number 10, there were harvested in turn the theory of irrationals, the theory of means and proportions, the study of incommensurables, cosmology, astronomy, and the science of acoustics.14 Because 14 The original Pythagorean tetraktys or quaternary represented the number 10 in the shape of a perfect... obtain between the side of a square and its diagonal In the one case, that of the whole-tone in the ratio 9:8, the division for obtaining a semi-tone, or one half of a whole-tone, produces the square root of 2 In the case of the geometric square, the Pythagorean theorem demonstrates that if the length of the side is 1 inch, the number of inches in the diagonal is also the square root of 2 As George... J. Godwin, The Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition Pythagoras’ reputation is summed up accordingly by G E Owen, The Universe of the Mind, 6 Greek Reflections on the Nature of Music On the basis of the truths arrived at by mathematical means, Pythagoras and his followers could think of music s elements as concrete realities linked by number to nature s own divine proportions At the. .. from the fragments of The Pythagorean Doctrine of the Elements of Music by the littleknown Ptolemaïs of Cyrene have been cast into the form of a dialogue This results in an interesting discussion among three experts on the virtues and limitations of the various theories under examination Of the three, it is Ptolemaïs who seems to me to have grasped the uniqueness of Aristoxenus’ Aristotelian type of theoretical... the number 2, say, is not the same as or identical with anything in the world It is 2 and, as such, is not identical with a duo of musicians; rather, the duo of musicians is an instance of the number, 2; and the number, 2, is an instance of itself The note C, is in this sense an instance of itself.9 Given such an instance, the subject matter of music, like that of pure 7 The convention for indicating... came to music, the Greeks showed the same organic point of view, the same instinct for formulating laws governing reality that appears in every phase of their culture and art As we learn from the evidence presented to us, the Greeks were the first to intuit music s essence, and the first to discover the universal laws governing its structure They were the first to perceive the elements of music not... type of acute sensitivity to sound bespeaks a whole other realm of perception So deep a penetration of music into almost every aspect of life presupposes a musically gifted public and a long tradition of musical education The evidence appears in fact to depict a society concerned with music more than anything else The truth is, of course, that music was only one of the myriad products of the Greek. .. experiments, which he repeated at home, and came upon the elegantly simple truth about musical sound: the pitch of a musical sound from a plucked string depends upon the length of the string This led him to discover that the octave, the fifth and the fourth, as well as the whole-tone, are to each other as the ratios of the whole numbers These, the harmonic ratios, as they came to be called, are all comprehended... over the course of many years Professor Cook’s writings on Plato are especially compelling to me, not least for being full of dialectical arguments; but above all, for their acute appraisal of the poetic and musical aspects of Plato’s style For Professor Cook, Plato was the Beethoven of Philosophy He demonstrated this most vividly in his analysis of Plato’s use of the Greek particles – the riot of particles,”... time, they could think of music itself as the expression in sound of those same proportions by which nature asserts her divine symmetry This being the case, the universal order of things could be said to have its counterpart in the underlying structures of harmonic theory The notion that music owes its life to mathematics, and that the universe, by the same agency, owes its soul to harmonia – the attunement . to think of music in the way of nature. This is to think of music in the way of Aristoxenus of Tarentum, a student of Aristotle, and the greatest musician of antiquity. Aristotle’s famous dictum. the manifold forces of nature. The unity of reason organizes and sets limits to things musical, while the forces of human nature create things musical and set them free. These are the two principles. analysis of Plato’s use of the Greek particles – the riot of particles,” as he so aptly called them (in The Stance of Plato) – which make for the powerfully polyphonic texture of the Platonic