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 First published in print format ISBN-13 978-0-521-87951-4 ISBN-13 978-0-511-36650-5 © Andrew Barker 2007 2007 Information on this title: www.cambridge.org/9780521879514 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 p ermission of Cambrid g e University Press. ISBN-10 0-511-36650-7 ISBN-10 0-521-87951-5 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 g uarantee that any content on such websites is, or will remain, accurate or a pp ro p riate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback eBook (EBL) eBook (EBL) hardback 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, Andr ´ eBarbera, Franca Perusino, Eleonora Rocconi, Donatella Restani, Annie B ´ elis, Angelo Meriani, David Creese, Egert P ¨ ohlmann, 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. [...]... 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... point New intellectual enterprises do not spring into being in a single bound, fully armed and fully recognisable; typically they trickle together as a confluence of tributaries which are themselves side-shoots of other, pre-existing traditions of thought, none of which ‘is’ this new thing, but whose developing combination can be seen, in retrospect, gradually to have become it In any case, such information... deal remains to be done, both in interpreting the theorists’ work and (still more) in unravelling its contexts, and again in trying to communicate the significance of sometimes arcane researches to a wider readership the agenda of greek harmonics Non-specialist readers will be getting impatient with my repeated and unexplained references to harmonics It is high time I said something to explain what... differ from attunements in two principal ways A scale is a series of notes set out in order of pitch, while an instrument’s strings need not always be arranged with the highest note at one end and the lowest at the other, and the remainder set out in pitch-order between them Secondly, to think of a set of notes as a scale is to think of it as a sequence of steps, unfolding successively in time; an attunement... happened for the rest of the millennium A great deal did But the history of changes in those later centuries is to a large extent a history of shifting contexts It is a story about the ways in which inherited ideas were used, abused, recombined and inserted into new settings, new forms of discussion and new patterns of enquiry In the earlier period, while the discipline was inventing itself, there is... source is probably in the observation that an increase in tension raises the pitch of an instrument’s string At first sight it seems more promising, since degrees of tension can be measured and compared with mathematical exactitude But that is true only in principle The Greeks had no reliable way of measuring tension, and the one method to which writers on harmonics refer, that of suspending larger or smaller... points of reference are put aside, experience with my own students has convinced me that one does not need an unusually eccentric turn of mind to find harmonic theory as delightfully fascinating in its own right as any other discipline, once one has been lured into the labyrinth In other publications, and in lectures and seminars here and there around the globe, I have therefore tried to find ways of. .. objective and discernible line of demarcation, independent of human whim, decision or ingenuity, between musically well-ordered relations and transformations on the one hand, and on the other the indeterminate chaos of the non-musical The distinction is not one of convention or taste, but is somehow fixed in the order of things, awaiting discovery, and from this perspective the innovative composers discussed... consider the work of one group in more detail in Chapter 2 and that of the other in Chapter 10; for the present let the issue of measurement take centre-stage m usical intervals as linear distances Responding (inappropriately, as it turns out) to a remark of Socrates, Glaucon describes the procedures of one set of theorists as follows What they do is ridiculous, when they call certain things ‘pykn¯mata’,... crucially of all, are all the schemes of relations which harmonics identifies unified and governed by some fundamental principle or some coherent group of principles, so that all structures conforming to those principles, and no others, are thereby constituted as properly musical? If so, what kinds of principle are involved? What gives them their authority? Are they somehow rooted in human nature, or in the . 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. 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. 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