This page intentionally left blank THE SCIENCE OF HARMONICS IN CLASSICAL GREECE The ancient science of harmonics investigates the arrangements of pitched sounds which form the basis of musical melody, and the principles which govern them. It was the most important branch of Greek musical theory, studied by philosophers, mathematicians and astronomers as well as by musical specialists. This book examines its development during the period when its central ideas and rival schools of thought were established, laying the foundations for the speculations of later antiquity, the Middle Ages and the Renaissance. It concentrates particularly on the theorists’ methods and purposes and the controversies that their various approaches to the subject pro- voked. It also seeks to locate the discipline within the broader cultural environment of the period; and it investigates, sometimes with sur- prising results, the ways in which the theorists’ work draws on and in some cases influences that of philosophers and other intellectuals. andrew barker is Professor of Classics in the Insititute of Archae- ology and Antiquity at the University of Birmingham. THE SCIENCE OF HARMONICS IN CLASSICAL GREECE ANDREW BARKER CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521879514 © Andrew Barker 2007 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. First published in print format 2007 ISBN-13 978-0-511-36650-5 eBook (EBL) ISBN-10 0-511-36650-7 eBook (EBL) ISBN-13 978-0-521-87951-4 hardback ISBN-10 0-521-87951-5 hardback 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. O dear white children, casual as birds, Playing among the ruined languages, So small beside their large confusing words, So gay against the greater silences... W. H. Auden, Hymn to Saint Cecilia Contents List of figures page ix Preface xi part ipreliminaries Introduction 3 1 Beginnings, and the problem of measurement 19 part ii empirical harmonics 2 Empirical harmonics before Aristoxenus 33 3 The early empiricists in their cultural and intellectual contexts 68 4 Interlude on Aristotle’s account of a science and its methods 105 5 Aristoxenus: the composition of the Elementa harmonica 113 6 Aristoxenus: concepts and methods in Elementa harmonica Book 1 136 7 Elementa harmonica Books ii–iii: the science reconsidered 165 8 Elementa harmonica Book iii and its missing sequel 197 9 Contexts and purposes of Aristoxenus’ harmonics 229 part iii mathematical harmonics 10 Pythagorean harmonics in the fifth century: Philolaus 263 vii viii Contents 11 Developments in Pythagorean harmonics: Archytas 287 12 Plato 308 13 Aristotle on the harmonic sciences 328 14 Systematising mathematical harmonics: the Sectio canonis 364 15 Quantification under attack: Theophrastus’ critique 411 Postscript: The later centuries 437 Bibliography 450 Index of proper names 461 General index 469 Figures 1 The central octave page 13 2 The Greater Perfect System 14 3 Disjunction and conjunction 15 4 The Lesser Perfect System 16 5 The Unchanging Perfect System 17 6 Modulation through a perfect fourth 220 7 Modulation through a semitone 221 8 The harmonia of Philolaus 266 9 The ‘third note’ in Philolaus’ harmonia 276 10 Sectio canonis proposition 19 396 11 Sectio canonis proposition 20 402 ix Preface I did most of the research for this book and wrote the first draft during my tenure of a British Academy Research Professorship in the Humanities in 2000–2003.Itwas a great privilege to be awarded this position, and I am deeply indebted to the Academy for its generous support of my work, which would otherwise have been done even more slowly or not at all. I am grateful also to the University of Birmingham for freeing me from my regular duties for an extended period. In that connection I should like to offer special thanks, coupled with sympathy, to Matthew Fox, for uncomplainingly taking over my role as Head of Department at a particularly difficult time, and to Elena Theodorakopoulos, Niall Livingstone and Diana Spencer for shouldering a sack-full of other tasks that would normally have come my way. Many others have been splendid sources of help, encouragement and advice. I cannot mention them all, but here is a Mighty Handful whose members have played essential parts, whether they know it or not: Geoffrey Lloyd, Malcolm Schofield, David Sedley, Ken Dowden, Carl Huffman, Alan Bowen, AndreB´ arbera, Franca Perusino, Eleonora Rocconi, Donatella Restani, Annie Belis,´ Angelo Meriani, David Creese, Egert Pohlmann,¨ Panos Vlagopoulos, Charis Xanthoudakis. My sincere thanks to all these excellent friends. Jim Porter and another (anonymous) reader for the Press read two versions of the entire typescript in draft; without their comments, to which I have done my best to respond, the book would have been a good deal less satisfactory than it is. I appreciate the magnitude of the task they generously undertook, and though they added substantially to my labours Iamexceedingly grateful for theirs. This is the fourth book of mine to which the staff of Cambridge University Press have served as midwives, and they have amply lived up to the standards of efficiency, courtesy and patience which I have come to expect and appreciate. My thanks to all concerned on this occasion, and especially to my admirable copy-editor, Linda Woodward, both for her careful work on the lengthy typescript and for the gratifying interest she took in its contents. Thanks, too, to my xi xii Preface oldest son, Jonathan Barker, who showed me how to solve certain vexing mathematical conundrums; and as always, to the rest of my family and especially my wife, Jill, for their continuing patience and encouragement. I can only regret that David Fowler is no longer here to be thanked. His untimely death has deprived me and many others of a friend and colleague whose enthusiasm and insatiable curiosity were infectious and inspiring, and whose lively and sympathetic humanity put some warmth and light into this cynical world. He was one of the most charming people who ever trod the earth, and he will be sadly missed. part i Preliminaries Introduction Few books have more splendidly informative titles than Theon of Smyrna’s Mathematics useful for reading Plato.Atitle modelled on his, perhaps Har- monic theory useful for reading classical Greek philosophy and other things would have given a fair impression of my agenda here. But that’s a little cumbersome; and for accuracy’s sake, I would have had to tack the phrase ‘and indications of the converse’ onto the Theonian title, since I shall be trying to show not only how harmonics can be ‘useful’ to students of other fields, but also how the preoccupations of Greek writers who tilled those fields can shed light on the development of harmonics itself, and can help us to understand its methods and priorities. More importantly, this hypothetical title would have been dangerously hubristic; it has the air of presupposing a positive answer to one of the book’s most serious questions. Leaving one or two exceptional passages aside (the construction of the World-Soul in Plato’s Timaeus, for example), does a knowledge of the specialised science of harmonics, and of its historical development, really give much help in the interpretation of texts more central to the scientific and philosophi- cal tradition, or in understanding the colourful environment inhabited by real Greek musicians and their audiences, or indeed in connection with anything else at all? Can such knowledge be ‘useful’, and if so, in which contexts, and how? I intend to argue that it can, though not always in the places where one would most naturally expect it. There is a point I should like to clarify before we begin, to avoid misun- derstandings and to help explain some of this book’s unavoidable limita- tions. Specialists in the ancient musical sciences may be few (though there are many more swimmers in these tricky waters now than there were when I took my first plunge over twenty-five years ago); but they are nevertheless various. By and large, they fall into two main groups. Some are profes- sional musicologists, who may have worked their way upstream into these reaches from a starting point in the Middle Ages or the Renaissance. Others set out from a training in Classics, within which broad church I include 3 4 Introduction devotees of ancient philosophy and science. Musicologists, of course, are sometimes proficient in Greek and Latin, and some classicists are excellent musicians; but when tackling their professional work, each group brings to it the equipment, the presuppositions and the puzzlements of their own academic tribe. I am no exception, and I make no bones about the fact. I am a classicist and a student of Greek science and philosophy. As it hap- pens, I have made a good deal of music in my time, but I am not a trained musicologist. American colleagues have sometimes chided me, no doubt rightly, for my lack of a properly musicological perspective. So be it; each of us does what he or she can. Most work published nowadays in this field is written by specialists for specialists. From time to time, over the years, I have contributed my own penny-worth to these esoteric conversations; but I have always had another objective in mind.