QUANTIFYING MUSIC THE UNIVERSITY OF WESTERN ONTARIO SERIES IN PHILOSOPHY 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 Netherlands 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-Acoustics and physics-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. Johannes Kepler 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 Octave 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 Semitones 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. Vincenzo Galilei 78 vii viii T ABLE OF CONTENTS 3.2.1. The Singer's Dilemma 79 3.2.2. Smashing the Senario 82 3.2.3. Summary and Conc1usions 83 3.3. Galileo Galilei 85 3.3.1. Pendulums 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 Sound 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. Isaac Beeckman 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 Renaissance 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. Christiaan Huygens 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 Pythagoras 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 harmonic 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 mathematics, 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).