QUANTIFYING MUSIC THE UNIVERSITY OF WESTERN ONTARIO SERIES IN OF SCIENCE

ASERIES OF BOOKS

IN PHILOSOPHY OF SCIENCE, METHODOLOGY,

EPISTEMOLOGY, LOGIC, HISTORY OF SCIENCE,

AND RELATED FIELDS

Managing Editor

ROBER T E. BUTTS Dept. ofPhilosophy, University of Western Ontario, Canada

Editorial Board

JEFFREY BUB, University ofWestern Ontario

L. JONATHAN COHEN, Queen's College, Oxford

WILLIAM DEMOPOULOS, University ofWestern Ontario

WILLIAM HARPER, UniversityofWestern Ontario

JAAKKO HINTIKKA

CLiFFORD A. HOOKER, University ofNewcastle

HENRY E. KYBURG, JR., UniversityofRochester

AUSONIO MARRAS, University ofWestern Ontario

JÜRGEN MITTELSTRASS, University of Konstanz

JOHN M. NICHOLAS, University ofWestern Ontario

GLENN A. PEARCE, University ofWestern Ontario

BAS C. VAN FRAASSEN, University ofToronto & Princeton University

VOLUME 23 H. F. COHEN Technical University Twente, Department o[ Social History o[ Seien ce and Technology, Enschede The QUANTIFYING MUSIC The Science oi Music at the First Stage oi the Scientific Revolution, 1580-1650

Springer-Science+Business Media, B.Y. Library of Congress Cataloging in Publication Data

Cohen, H. F. Quantifying music.

(The University of Western Ontario series in philosophy of science ; v.23) Bibliography: p. Includes indexes. I. Music- and -16th century. 2. Music­ Acoustics and physics-17th century. I. Title. 11. Se ries. ML3807.C63 1984 781'.1 84-3270 ISBN 978-90-481-8388-3 ISBN 978-94-015-7686-4 (eBook) DOI 10.1007/978-94-015-7686-4

All Rights Reserved © 1984 by Springer Science+Business Media Dordrecht Originally published by D. Reidel Publishing Company 1984. Softcover reprint of the hardcover 1st edition 1984 and other copyright owners as specified on appropriate pages within No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording or by any information storage and retrieval system, without written perrnission from the copyright owner The ineffably heartfelt quality of music, owing to which it flows along as an intimately known yet perpetually remote Paradise, so fully intel­ ligible yet so inexplicable, comes from its reflecting all stirrings of our innermost Being, though quite devoid of reality and far removed from its pain.

Arthur Schopenhauer (1819)

Sounds can shed more light on Philosophy than any other quality, which is why the science of Music should not be neglected, even if all singing and playing were completely abolished and forbidden.

Marin Mersenne (1636) TABLE OF CONTENTS

PREFACE xi

ACKNOWLEDGEMENTS xvii

CHAPTER 1 / DEFINING THE PROBLEM SITUATION 1.1. The Problem of Consonance 1 1.1.1. Zarlino's Redefinition of the Problem 3 1.1.2. Objections to the Senario 6 1.2. The Nature of the Scientific Revolution 7 1.2.1. The Science ofMusic Around 1600 10 1.3. Outline of Chapters 2 through 7 11

CHAPTER 2 / THE MATHEMA TICAL APPROACH 13 2.1. 13 2.1.1. The Empirical Foundation 15 2.1.2. Distinguishing Consonance from Dissonance 16 2.1.3. The Genesis of Harmony 23 2.1.4. Passing by Acoustics 29 2.1.5. Conclusions 32 2.2. The Division of the 34 2.2.1. The Incompatibility of the Pure Consonances 37 2.2.2. Summary 43 2.3. Simon Stevin 45 2.3.1. 'On the Theory ofMusic' 48 2.3.2. Preliminary Defmitions 48 2.3.3. The Octave Comprises 6 Equal Tones 51 2.3.4. The Octave Comprises 12 Equal 53 2.3.5. Sustaining Arguments 57 2.3.6. A Musician's Critique 61 2.3.7. Contemporary Music 63 2.3.8. Conclusions 67 CHAPTER 3 / THE EXPERIMENT AL APPROACH 75 3.1. Giovanni Battista Benedetti 75 3.2. 78

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3.2.1. The Singer's Dilemma 79 3.2.2. Smashing the Senario 82 3.2.3. Summary and Conc1usions 83 3.3. 85 3.3.1. and Resonance 87 3.3.2. The Coincidence Theory of Consonance 90 3.3.3. Conc1usions 92 3.4. The Nature of the Coincidence Theory 94 3.5. Marin Mersenne 97 3.5.1. The 'Abstract ofMusical Theory' 100 3.5.2. Some Properties of 101 3.5.3. The Coincidence Theory Put to the Test 103 3.5.4. The Division of the Octave 111 3.5.5. Quantifying All Possible Music 112 3.5.6. Conc1usions 114

CHAPTER 4 / THE MECHANISTIC APPROACH 115 4.1. 116 4.1.1. The Corpuscular Theory of Sound 120 4.1.2. The Nature ofConsonance 127 4.1.3. Musical Instruments 147 4.1.4. Intermezzo: Consolations for the Physicist 149 4.1.5. The Division of the Octave 151 4.1.6. Conc1usions 157 4.2. Rene Descartes 161 4.2.1. The 'Compendium of Music' 161 4.2.2. The Scientific Analysis ofMusical Beauty 166 4.2.3. The Perception of Consonance 172 4.2.4. Conc1usions 175

CHAPTER 5 / CONT ACTS AND CRITICISMS 180 5.1. The Theorists 181 5.2. The Early Physicists 182 5.2.1. Benedetti 183 5.2.2. Vincenzo Galilei 183 5.3. The Mathematicians 184 5.3.1. Stevin 184 5.3.2. Kepler 185 5.4. The Mersenne Circ1e 187 T ABLE OF CONTENTS ix

5.4.1. Prologue: Beeckman Meets Young Descartes 188 5.4.2. Beeckman, Descartes, and Mersenne 190 5.5. Galileo Galilei 201 5.6. Conc1usion 202

CHAPTER 6 / AN EXAMPLE FROM THE SECOND GENERATION 205 6.1. The Prevalence of the Coincidence Theory 206 6.2. 209 6.2.1. The Theory ofConsonance 210 6.2.2. The Division of the Octave 214 6.2.3. The Consonance of the Intervals with 7 225 6.2.4. Conc1usion 228

CHAPTER 7 / CONCLUSIONS 231 7.1. 2500 Years since 231 7.1.1. What Had Been Accomplished and What Not 231 7.1.2. What Was To Be Accomplished 234 7.2. Music as a Science: Implications and Perspectives 243 7.2.1. The Scientific Revolution 243 7.2.2. Music as an Art and Music as a Science 250 7.2.3. The Victory of the Coincidence Theory: An Example of Theory Replacement 254 7.2.4. Quest Without End 258

NOTES 260

BIBLIOGRAPHY 296

NAME INDEX 303

SUBJECT INDEX 305 PREFACE

The soul rejoices in perceiving harmonious sound; when the sound is not harmonious it is grieved. From these affects of the soul are derived the name of consonances for the proportions, and the name of dissonances for the unharmonic proportions. When to this is added the other harmonie proportion whieh consists of the longer or shorter duration of musical sound, then the soul stirs the body to jumping dance, the tongue to inspired speech, according to the same laws. The artisans accommodate to these harmonies the blows of their hammers, the soldiers their pace. As long as the harmonies endure, everything is alive; everything stiffens, when they are disturbed.!

Thus the German astronomer, Johannes Kepler, evokes the power of music. Where does this power come from? What properties of music enable it to stir up emotions which may go far beyond just feeling generally pleased, and which may express themselves, for instance, in weeping; in laughing; in trembling over the whole body; in a marked acceleration of breathing and heartbeat; in participating in the rhythm with the head, the hands, the arms, and the feet? From the beginning of musical theory the answer to this question has been sought in two different directions. In the aesthetic approach to music, the explanation of the power of music has been sought in the analysis of compositional techniques. These may be of a nearly infinite variety. Specific musical effects can be shown to result from changes of tempo; from the choice of a certain key; from modulation; oma­ mentation; instrumentation; from the wider or narrower range of the melody; from dynamical contrasts; and so on. Surely analysis of music along such lines can reveal essential truths, and it may considerably enhance and deepen our enjoyment when we listen to music. . However, there is another, scienti[ic approach to music, which takes as its starting point, not compositional techniques, but musical sound. In this approach, music is not looked at in terms of aesthetics, but in terms of , physics and physiology. Since compositional techniques, in the last analysis, come down to - highly differentiated - applications of musical sound, it may legitimately be asked whether ultimately a complete re duc­ ti on of the musical experience to physical and physiological mechanisms might be achieved. This is a highly intriguing question, to which as yet no

xi xii PREFACE

definitive answer can be given, since even if it would tentatively be answered in the affirmative, it is certain that such a final state has not yet been reached. In the course of this historical study we shall come across optimists and sceptics in this respect. Descartes, for instance, believed that people's varying tastes apriori preclude any such reduction to scientific analysis. As against this there were optimists like Heimholtz, who tried, for instance, to explain at least part of Palestrina's greatness by demonstrating that the spacing of chords in this composer's Masses and motets concurred with predictions that followed from Heimholtz' own physical/physiological theory of consonance. Whatever stand one may take on this issue,2 it is certain that the scientific analysis of musical sound has indeed revealed quite a lot of interesting musical truths. The present state of our knowledge in this domain may be gathered from such books as Wood's The Physics of Music (1975 7 ), or Roederer's Introduction to the Physics and Psychophysics of Music (19752). The history of the science of music, however, has not yet been written. Now it cannot possibly be the purpose of this book to fill the entire historiographical gap at one presumptuous stroke. On the other hand, so much preparatory work is already available (critical editions of relevant texts; historical studies on individual cases, etc.) that the time has become ripe for an attempt at defining the basic historical issues to be made. The best way to achieve this, or so it has seemed to me, is to concentrate on one particular episode of central importance to the history of the science of music, namely, the transition from the scientific analysis of music in terms of number to an approach on an essentially physical basis. This transition took place in the first half of the 17th century, as part of the Scientific Revolution. 'The Scientific Revolution' has become the current label for designating the sum total of the events that together, in the course of the 17th century, effected the transition from Aristotelian science (and from Renaissance ) to a science of which the basic features may broadly be termed 'modern'. Among those who played a major part in the first stage of the transition may be mentioned Kepler, Galileo, Stevin, Bacon, Descartes, Gassendi, Mersenne, and Beeckman. However different their approaches to science; however varied the domains of science on which they concentrated: the science ofmusic was part of the research es of all of them. In later stages, too, many of the most important scientists devoted part of their efforts to musical theory, for instance, Huygens, Hooke, Wallis, Newton, and Leibniz. Now the fact that the science of music belonged to the research interests of so many major scientists of the 17th century is not due to chance. In fact music is one of the oldest sciences in history , stretching backwards as far as PREFACE xiii the 6th century RC., when Pythagoras for the first time connected certain musical intervals with certain definite numerical ratios. Ever since has music been part of the sciences. In the Western world, throughout the Middle Ages music belonged, together with arithmetic, geometry, and astronomy, to the quadrivium, that is, the 'exact' portion of the seven liberal arts which con­ stituted the propaedeutic of the medieval university curriculum. This tradition was still very much alive during the 17th century. Not until the 18th century was well under way did music gradually drop out of the mainstream of what by then went under the name of 'science'. Why this happened I do not know. It may be conjectured that, whatever the original reasons of this moving out, it was greatly enhanced by the 19th century Romantic movement, with its one-sided emphasis on the autonomy of the artistic inspiration. Since then first-rank scientists have only rarely cultivated the science of music (Heimholtz is by far the most notable example), and the domain has become ever more marginal to the various standard academic disciplines, increasingly constitut­ ing a specialism in its own right. The tradition is now kept alive by the relatively small number of scientists who are pleased to work in the fascinating border area where musical theory, acoustics, physiology, and psychology come together. As a result of the science of music gradually having dropped out of the mainstream of science, historians of science have tended to overlook the prominent place it was assigned in earlier times by the scientists whose works they made the subject of their historical research. It is quite possible for monographs purporting to give an overview of the scientific achievement of men like Galileo or Descartes to be altogether silent on what these men had to say concerning . Insofar as it was noticed at all that a gap was waiting to be filled here ,3 the job has usually been done by professional students of musical his tory , who devoted much effort to analyzing the contributions made to musical theory by various 17th century scientists such as Kepler, Mersenne, and Descartes. But quite naturally most of these his­ torians failed to link their findings with what at the time was going on in the other sciences. Only a few of them did try, notably C. V. Palisca and D. P. Walker. However much these two differ from each other, and however much I have eventually found some of my own results to differ from theirs, it is to their essays that I owe the original inspiration for the attempt made in this book to show to both the historian of science and to the music historian the importance of treating the science of music as one of the sciences, on a par with more obvious domains of science like mechanics and optics. The validity of this perspective is here demonstrated for the first half ofthe 17th century. It xiv PREFACE is investigated how the transformation of science that took place in that period affected the science of music. The 'science of music' in the sense in which the term is employed he re does not encompass the whole of music theory. It covers that part of it that aims at reducing the musical experience to scientific as distinguished from aesthetic principles, which in effect me ans that it deals with the two interrelated problems of consonance and of the division of the octave. This, then, is the main subject of the book. Underneath it three counter­ subjects can be discerned, that run through the entire historical account, and are brought to the surface in the final chapter. The first is the rise of an autonomous science of acoustics out of the musical concepts developed in the course of our period (a point that has been made before by Dostrovsky (1974/5) from the acoustical point of view). Throughout our inquiry the genesis of particular acoustical concepts and theories will be dealt with insofar as these are needed to understand what went on in the science of music. The second countersubject concerns the intriguing question whether the science of music and the art of music influenced each other over our period. In particular it will be noted that the transformation of the science of music very nearly coincided with a major change in musical style, namely, the transition from Renaissance to Baroque music, and it will be inquired to what extent these developments were interrelated. Thirdly, we shall find that the theory of consonance that, within the relatively brief period covered by this book, gained general acceptance, suffered from some glaring deficiencies as compared to both its predecessor and its contemporary rivals. Therefore we shall explicitly inquire what it was that, nevertheless, enabled the theory in question to all its competitors. This inquiry will be restricted to the historical, despite the obvious implications it has in store for the philosophy of science. The implications our inquiry has for the highly debated question of how scientific theories are replaced, or what, if any, rules are followed by the scientist in choosing between a few riYal theories competing for his allegiance I intend to explore in aseparate article, provisionally entitled 'Music as a test-case'. The historical account to be given in the main part of the book is entirely based on primary source material. The extent to which, in organizing my account of the thinking of a particular scientist, I could draw on the historical literature varied a great deal as to the scientist in question. It ranges from Dickreiter's brilliant monograph on Kepler's musical theories to the near­ complete historiographical void that faces the historian interested in the musical theories of the three Dutchmen who figure prominently in my PREFACE xv account: Stevin, Beeckman, and Huygens. Therefore I have discussed the contributions these three made to the science of music in somewhat greater detail than in the other cases where at least some literature is already available. My overall aim has been, not only to contribute to our knowledge of the thinking of eight major scientists who were active around 1600, but also to bring to light the numerous interrelations between their vastly different musical theories, which become clear as a result of the particular point of view chosen in this book. This point of view, though derived from necessities imposed by the subject matter in question, would nevertheless have been out of my reach but for so me key ideas of K. R. Popper, which ever since I became acquainted with his works have exerted a decisive influence on my way of doing history . The reader will find that his notions on the importance of reconstructing the problem situation in the science of a given period have informed the entire structure and content of the book. The notes consist mainly of references; they do not contain material of interest to the non-specialist. All quotations in the main text are rendered in English; they have carefully been checked against the originals as to both literalness and context. In order to enable the reader to judge for hirnself, the original language has been printed in the notes. Anachronistic terminology has been adopted freely whenever there is no appreciable difference of mean­ ing to be detected between a modern word (for instance 'pitch'), and its cumbersome early counterpart ('difference according to the acute and the low' and such-like circurnlocutions). In all other cases I have attempted to be consistent in sticking to contemporary usage. Since the science of music itself is already at the crossroads of several distinct disciplines, its history is even more so. Having been educated as a historian, I benefited a great deal from many helpful comments made by friends-and-relations from various other academic fields. My fellow toiler in the vineyard of the history of the science of music, Penelope Gouk, certainly deserves to be mentioned in the first place, and to be thanked for her invalu­ able help in clarifying my own thought by tirelessly pointing out where and how hers differs from mine. I would also like to express my gratitude to Melissa Bowerman, Christopher Burch, Steven Engelsman, Jos Henselmans, Liesbeth de Klerk, Mark Lindley, Nancy Nersessian, Rudolf Rasch, Nora Schadee, and Gerrit Veeneman. I appreciate the assistance on various matters of language I received from Kees Bolle, Rein Ferwerda, and the late Jop Spiekerman. Much of the research for, and some of the writing of, this book was done du ring the academic year 1979/1980, when I was a Fellow of the Netherlands xvi PREFACE

Institute for Advanced Study in the Humanities and Social Sciences (NIAS) at Wassenaar. I extend my heart-felt thanks to Board and Directorate of this wonderful institution for the unexpected invitation; to Marina Ben Hamida­ Voerman, Dinny Young, Harriet van Heusden and Dick van der Kooij for their secretarial and library help; to my fellow Fellows for the inspiration many of them so abundantly provided me with; to the Directorate of the Museum Boerhaave and the Netherlands' Ministry of Culture for granting me a one year leave; and to my then colleagues at the Museum Boerhaave for keeping all ongoing business away from me, thus enabling me to concentrate entirely on my research project. I dedicate this book to my daughter Esther.

The Hague, March 1983 ACKNOWLEDGEMENTS

Figures - Permission to reproduce Figure 28 was gran ted by the Koninklijke Bibliotheek, Den Haag; Figures 40 and 67 by M. Nijhoff Publishers, Den Haag; Figure 51 (which is taken from: Pierce and David (1958), p. 133) by Double­ day & Company, Garden City (NY); and Figures 65 and 75 by the Univer­ siteitsbibliotheek, Leiden.

Pictures - Permission to reproduce the portraits of Kepler (p. 14), Stevin (p. 47), Galileo (p. 86), Descartes (p. 162), and Heimholtz (p. 239), was granted by the Museum Boerhaave, Leiden. Stevin's and Descartes' portraits are undated engravings in the possession of the Museum Boerhaave, Galileo's comes from his Macchie sofari (1613), and Helmholtz's is a daguerrotypie dated 23rd of March, 1848. Permission to reproduce the portrait of Mersenne (p. 98) was granted by the Haags Gemeentemuseum, Den Haag; it is an engraving by Cl. Duflos, done at the end of the 17th century. Huygens' portrait (p. 208) is a pastel done by B. Vaillant in 1685/6; reproduced with permission of the Huygens Museum 'Hofwyck', Voorburg. Permission to reproduce one page from Stevin's manuscript (p. 49) was granted by the Koninklijke Bibliotheek, Den Haag; Beeckman's signature (in the absence of any known portrait; p. 116) by M. Nijhoff Publishers, Den Haag (it is taken from p. 193 ofVol. 1 of the Journal). Finally, the map of Breda (p. 189) is reproduced from Bfaeu's Stedenatfas (p. 164), with permission granted by the Atlas van Stolk, Rotterdam.

Quotations - Permission to quote passages from The Principal Works 01 Simon Stevin, and to use again parts of Cohen (1980), was gran ted by Swets & Zeitlinger, lisse. Permission to quote passages from the Journal tenu par Isaac Beeckman was granted by M. Nijhoff Publishers, Den Haag; from D. P. Walker (1978) by Brill, Leiden; from E. J. Dijksterhuis (1950 and 1961) by Meulenhoff, Amsterdam; and from Wood (1975 7 ) by Chapman & Hall, London.

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