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The Physics of Musical Instruments

Neville H. Fletcher Thomas D. Rossing

The Physics of Musical Instruments

With 408 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Neville H. Fletcher Thomas D. Rossing CSIRO Australia Department of Physics Research School of Physical Sciences Northern Illinois University Australian National University De Kalb, IL 60115 Canberra, A.c.T. 2601 USA Australia

Library of Congress Cataloging-in-Publication Data Fletcher, Neville H. (Neville Horner) The physics of musical instruments/Neville H. Fletcher, Thomas D. Rossing. p. cm. Includes bibliographical references and indexes. ISBN-13:978-0-387 -94151-6 I. Music-Acoustics and physics. 2. Musical instruments- Construction. l. Rossing, Thomas D., 1929- II. Title. ML3805.F58 1993 784.19'OI'53-dc20 93-27442

Printed on acid-free paper.

© 1991 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in con• nection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

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ISBN-13:978-0-387-94151-6 e-ISBN-13:978-1- 4612-2980-3 DOl: 10.1007/978-1-4612-2980-3 Preface

The history of musical instruments is nearly as old as the history of civilization itself, and the aesthetic principles upon which judgments of musical quality are based are intimately connected with the whole culture within which the instruments have evolved. An educated modem Western player or listener can make critical judgments about particular instruments or particular per• formances but, to be valid, those judgments must be made within the appro• priate cultural context. The compass of our book is much less sweeping than the first paragraph might imply, and indeed our discussion is primarily confined to Western musical instruments in current use, but even here we must take account of centuries of tradition. A is designed and built for the playing of music of a particular type and, conversely, music is written to be performed on particular instruments. There is no such thing as an "ideal" instrument, even in concept, and indeed the unbounded possibilities of modem digital sound-synthesis really require the composer or performer to define a whole set of instruments if the result is to have any musical coherence. Thus, for example, the sound and response of a violin are judged against a mental image of a perfect violin built up from experience of violins playing music written for them over the centuries. A new instrument may be richer in sound quality and superior in responsiveness, but if it does not fit that image then it is not a better violin. This set of mental criteria has developed, through the interaction of musical instruments makers, performers, composers, and listeners, over several cen• turies for most musical instruments now in use. The very features of particular instruments that might be considered as acoustic defects have become their subtle distinguishing characteristics, and technical "improvements" that have not preserved those features have not survived. There are, of course, cases in which revolutionary new features have prevailed over tradition, but these have resulted in almost new instrument types-the violin and in place of the viols, the Boehm flute in place of its baroque ancestor, and the saxophone in place of the taragato. Fortunately, perhaps, such profound changes are rare, vi Preface and most instruments of today have evolved quite slowly, with minor tonal or technical improvements reflecting the gradually changing mental image of the ideal instrument of that type. The role of acoustical science in this context is an interesting one. Centuries of tradition have developed great skill and understanding among the makers of musical instruments, and they are often aware of subtleties which are undetected by modern acoustical instrumentation for lack of precise technical criteria for their recognition. It is difficult, therefore, for a scientist to point the way forward unless the problem or the opportunity, has been adequately identified by the performer or the maker. Only rarely do all these skills come together in a single person. The first and major role of acoustics is therefore to try to understand all the details of sound production by traditional instruments. This is a really major program, and indeed it is only within the past few decades that we have achieved even a reasonable understanding of the basic mechanisms determining tone quality in most instruments. In some cases even major features of the sounding mechanism itself have only recently been unravelled. This is an intellectual exercise of great fascination, and most of our book is devoted to it. Our understanding of a particular area will be reasonably complete only when we know the physical causes of the differences between a fine instrument and one judged to be of mediocre quality. Only then may we hope that science can come to the help of music in moving the design or performance of contemporary instruments closer to the present ideal. This book is a record of the work of very many people who have studied the physics of musical instruments. Most of them, following a long tradition, have done so as a labor oflove, in time snatched from scientific or technical work in a field of more immediate practical importance. The community of those involved is a world-wide and friendly one in which ideas are freely exchanged, so that, while we have tried to give credit to the originators wherever possible, there will undoubtedly be errors of oversight. For these we apologize. We have also had to be selective, and many interesting topics have perforce been omitted. Again the choice is ours, and has been influenced by our own particular interests, though we have tried to give a reasonably balanced treatment of the whole field. The reader we had in mind in compiling this volume is one with a reason• able grasp of physics and who is not frightened by a little mathematics. There are fine books in plenty about the history of particular musical in• struments, lavishly illustrated with photographs and drawings, but there is virtually nothing outside the scientific journal literature which attempts to come to grips with the subject on a quantitative basis. We hope that we have remedied that lack. We have not avoided mathematics where pre• cision is necessary or where hand-waving arguments are inadequate, but at the same time we have not pursued formalism for its own sake. Detailed Preface vii physical explanation has always been our major objective. We hope that the like-minded reader will enjoy coming to grips with this fascinating subject. The authors owe a debt of gratitude to many colleagues who have contri• buted to this book. Special thanks are due to Joanna Daly and Barbara Sullivan, who typed much of the manuscript and especially to Virginia Plemons, who typed most of the final draft and prepared a substantial part of the artwork. Several colleagues assisted in the proofreading, including Rod Korte, Krista McDonald, David Brown, George Jelatis, and Brian Finn. We are grateful to David Peterson, Ted Mansell, and other careful readers who alerted us to errors in the first printing. Thanks are due to our many colleagues for allowing us to reprint figures and data from their publica• tions, and to the musical instrument manufacturers that supplied us with photographs. Most of all, we thank our colleagues in the musical acoustics community for many valuable discussions through the years that led to our writing this book.

December 1988 Neville H. Fletcher Thomas D. Rossing Contents

Preface v

PART I Vibrating Systems 1

CHAPTER 1 Free and Forced Vibrations of Simple Systems 3 1.1. Simple Motion in One Dimension 4 1.2. Complex Amplitudes 6 1.3. Superposition of Two Harmonic Motions in One Dimension 7 1.4. Energy 10 1.5. Damped Oscillations 10 1.6. Other Simple Vibrating Systems 12 1.7. Forced Oscillations 16 1.8. Transient Response of an Oscillator 20 1.9. Two-Dimensional Harmonic Oscillator 22 1.10. Graphical Representations of Vibrations: Lissajous Figures 23 1.11. Normal Modes of Two-Mass Systems 25 1.12. Nonlinear Vibrations of a Simple System 26

APPENDIX A.1. Alternative Ways of Expressing Harmonic Motion 29 A.2. Equivalent Electrical Circuit for a Simple Oscillator 30 References 32

CHAPTER 2 Continuous Systems in One Dimension: Strings and Bars 33 2.1. Linear Array of Oscillators 33 2.2. Transverse Wave Equation for a String 35 2.3. General Solution ofthe Wave Equation: Traveling Waves 36 2.4. Reflection at Fixed and Free Ends 36 2.5. Simple Harmonic Solutions to the Wave Equation 37 x Contents

2.6. Standing Waves 38 2.7. Energy of a Vibrating String 39 2.8. Plucked String: Time and Frequency Analyses 39 2.9. Struck String 42 2.10. Bowed String 45 2.11. Driven String: Impedance 47 2.12. Motion of the End Supports 48 2.13. Damping 50 2.14. Longitudinal Vibrations of a String or Thin Bar 53 2.15. Bending Waves in a Bar 54 2.16. Bars with Fixed and Free Ends 57 2.17. Vibrations of Thick Bars: Rotary Inertia and Shear Deformation 60 2.18. Vibrations of a Stiff String 61 2.19. Dispersion in Stiff and Loaded Strings: Cutoff Frequency 61 2.20. Torsional Vibrations of a Bar 63 References 64

CHAPTER 3 Two-Dimensional Systems: Membranes and Plates 65 3.1. Wave Equation for a Rectangular Membrane 65 3.2. Square Membranes: Degeneracy 68 3.3. Circular Membranes 69 3.4. Real Membranes: Stiffness and Air Loading 70 3.5. Waves in a Thin Plate 71 3.6. Circular Plates 72 3.7. Elliptical Plates 74 3.8. Rectangular Plates 75 3.9. Square Plates 78 3.10. Square and Rectangular Plates with Clamped Edges 81 3.11. Rectangular Wood Plates 82 3.12. Bending Stiffness in a Membrane 85 3.13. Shallow Spherical Shells 86 3.14. Nonlinear Vibrations in Plates and Shallow Shells 88 3.15. Driving Point Impedance 89 References 92

CHAPTER 4 Coupled Vibrating Systems 95 4.1. Coupling Between Two Identical Vibrators 95 4.2. Normal Modes 96 4.3. Weak and Strong Coupling 98 4.4. Forced Vibrations 100 4.5. Coupled Electrical Circuits 103 4.6. Forced Vibration of a Two-Mass System 107 4.7. Systems with Many Masses 109 Contents xi

4.8. Graphical Representation of Frequency Response Functions 110 4.9. Vibrating String Coupled to a Soundboard 112 4.10. Two Strings Coupled by a Bridge 114

APPENDIX A.l. Structural Dynamics and Frequency Response Functions 117 A.2. Modal Analysis 121 A.3. Finite Element Analysis 122 References 123

CHAPTERS Nonlinear Systems 125 5.1. A General Method of Solution 126 5.2. Illustrative Examples 128 5.3. The Self-Excited Oscillator 130 5.4. Multimode Systems 131 5.5. Mode Locking in Self-Excited Systems 133 References 135

PART II Sound Waves 137

CHAPTER 6 Sound Waves in Air 139 6.1. Plane Waves 139 6.2. Spherical Waves 143 6.3. Sound Pressure Level and Intensity 145 6.4. Reflection, Diffraction, and Absorption 147 6.5. Acoustic Components at Low Frequencies 150 References 155

CHAPTER 7 Sound Radiation 156 7.1. Simple Multipole Sources 156 7.2. Pairs of Point Sources 159 7.3. Arrays of Point Sources 161 7.4. Radiation from a Spherical Source 163 7.5. Line Sources 165 7.6. Radiation from a Plane Source in a Baffle 166 7.7. Unbaffled Radiators 169 References 171 xii Contents

CHAPTERS Pipes and Horns 172 8.1. Infinite Cylindrical Pipes 172 8.2. Wall Losses 175 8.3. Finite Cylindrical Pipes 178 8.4. Radiation from a Pipe 183 8.5. Impedance Curves 183 8.6. Horns 186 8.7. Finite Conical and Exponential Horns 191 8.8. Bessel Horns 194 8.9. Compound Horns 196 8.10. Perturbations 198 8.11. Numerical Calculations 200 8.12. The Time Domain 200 References 203

PART III String Instruments 205

CHAPTER 9 Guitars and Lutes 207 9.1. Design and Construction of Guitars 207 9.2. The Guitar as a System of Coupled Vibrators 208 9.3. Force Exerted by the String 209 9.4. Modes of Vibration of Component Parts 212 9.5. Coupling of the Top Plate to the Air Cavity: Two-Oscillator Model 216 9.6. Coupling to the Back Plate: Three-Oscillator Model 217 9.7. Resonances ofa Guitar Body 218 9.8. Response to String Forces 221 9.9. Sound Radiation 223 9.10. Resonances, Radiated Sound, and Quality 224 9.11. Electric Guitars 226 9.12. Frets and Compensation 228 9.13. Lutes 228 9.14. Other Plucked String Instruments 230 References 232

CHAPTER 10 Bowed String Instruments 235 10.1. A Brief History 235 10.2. Research on Violin Acoustics 236 10.3. Construction of the Violin 237 10.4. Motion of Bowed Strings 238 10.5. Violin Body Vibrations 247 10.6. The Bridge 255 Contents xiii

10.7. Sound Radiation 258 10.8. The Bow 270 10.9. The Wolf Tone 271 10.10. Tonal Quality of Violins 272 10.11. Viola, Cello, and 275 10.12. Viols 278 10.13. A New Violin Family 279 References 283

CHAPTER 11 Harps, Harpsichords, and Clavichords 287 11.1. The Koto 287 11.2. The Harp 290 11.3. The Harpsichord 293 11.4. Harpsichord Design Considerations 296 11.5. Harpsichord Characteristics 299 11.6. The Clavichord 300 References 304

CHAPTER 12 The 305 12.1. General Design of 306 12.2. Piano Action 308 12.3. Piano Strings 315 12.4. String Excitation by the Hammer 319 12.5. The Soundboard 322 12.6. Sound Decay: Interaction of Strings, Bridge, and Soundboard 331 12.7. Tuning and Inharmonicity 334 12.8. 337 12.9. Electric Pianos 342 References 343

PART IV Wind Instruments 345

CHAPTER 13 Sound Generation by Reed and Lip Vibrations 347 13.1. Basic Reed Generators 347 13.2. Detailed Discussion of a Reed Generator 350 13.3. Effect of Reservoir Impedance 355 13.4. Reed Generators Coupled to Horns 357 13.5. Nonlinearity 360 13.6. Time-Domain Approach 362 References 363 xiv Contents

CHAPTER 14 Lip-Driven Brass Instruments 365 14.1. Historical Development of Brass Instruments 365 14.2. Horn Profiles 367 14.3. Mouthpieces 369 14.4. Radiation 373 14.5. Slides and Valves 375 14.6. Small-Amplitude Nonlinearity 378 14.7. Large-Amplitude Nonlinearity 381 14.8. Input Impedance Curves 384 14.9. Transients 385 14.10. Acoustic Spectra 387 14.11. Mutes 388 14.12. Performance Technique 390 References 392

CHAPTER 15 Woodwind Reed Instruments 394 15.1. Woodwind Bore Shapes 394 15.2. Finger Holes 397 15.3. Impedance Curves 401 15.4. Reed and Air Column Interaction 407 15.5. Directionality 411 15.6. Performance Technique 412 15.7. The 414 15.8. The 418 15.9. The 420 15.10. The Saxophone 420 15.11. Construction Materials 421 References 423

CHAPTER 16 Flutes and Flue Organ Pipes 426 16.1. Dynamics of an Air Jet 426 16.2. Disturbance of an Air Jet 431 16.3. Jet-Resonator Interaction 433 16.4. The Regenerative Excitation Mechanism 436 16.5. Jet Drive Nonlinearity 442 16.6. Transients and Mode Transitions 444 16.7. Simple Flute-Type Instruments 447 16.8. The Recorder 449 16.9. The Flute 454 16.10. Materials 460 16.11. Directional Characteristics 461 16.12. Performance Technique 461 References 464 Contents xv

CHAPTER 17 Pipe Organs 467 17.1. General Design Principles 468 17.2. Organ Pipe Ranks 472 17.3. Flue Pipe Ranks 474 17.4. Characteristic Flue Pipes 477 17.5. Mixtures and Mutations 479 17.6. Tuning and Temperament 481 17.7. Sound Radiation from Flue Pipes 482 17.8. Transients in Flue Pipes 483 17.9. Flue Pipe Voicing 484 17.10. Effect of Pipe Material 485 17.11. Reed Pipe Ranks 487 17.12. Analysis of Timbre 489 17.13. Tonal Architecture 490 References 491

PART V Percussion Instruments 495

CHAPTER 18 497 18.1. Kettledrums 498 18.2. Bending Stiffness 499 18.3. Piston Approximation for Air Loading 500 18.4. Green Function Method for Calculating Air Loading 502 18.5. Comparison of Calculated and Measured Modes for a Membrane 504 18.6. Timpani Sound 507 18.7. Radiation and Sound Decay 509 18.8. The Kettle 510 18.9. Bass Drums 512 18.10. Side Drums 514 18.11. Tom-Toms 515 18.12. Onset and Decay of Sound 517 18.13. Snare Action 518 18.14. Indian Drums 520 18.15. Japanese Drums 525 18.16. Indonesian Drums 527 18.17. Latin American Drums 528 18.18. 530 18.19. Friction Drums 530 References 530 xvi Contents

CHAPTER 19 Mallet Percussion Instruments 533 19.1. 533 19.2. The 535 19.3. Tuning the Bars 539 19.4. Resonators 542 19.5. The 544 19.6. Vibes 546 19.7. Mallets 547 19.8. Chimes 549 19.9. Triangles 551 19.10. Gamelan Instruments 552 19.11. Tubaphones and Gamelan Chimes 553 References 554

CHAPTER 20 , , Plates, and Steel Drums 555 20.1. Cymbals 555 20.2. Modes of Vibration of Cymbals 555 20.3. Transient Behavior 558 20.4. Tam-Tams 560 20.5. Modes of Vibration of a Tam-Tam 561 20.6. Nonlinear Mode Coupling in Tam-Tams 561 20.7. Sound Buildup and Decay in Tam-Tams 562 20.8. Gongs 564 20.9. Nonlinear Effects in Gongs 566 20.10. 567 20.11. Rectangular Plates: Bell Plates 568 20.12. Stressed Plates: and 568 20.13. Other Plate Percussion Instruments 569 20.14. Steel Drums 569 20.15. Sound Spectra and Modes of Vibration 571 20.16. Mechanical Coupling Between Note Areas 573 20.17. Tuning Steel Drums 575 References 576

CHAPTER 21 Bells 577 21.1. Modes of Vibration of Church Bells 578 21.2. Tuning and Temperament 583 21.3. The Strike Note 584 21.4. Major-Third Bells 586 21.5. Sound Decay and Warble 588 21.6. Scaling of Bells 591 21.7. Modes of Vibration of 594 21.8. Timbre and Tuning of Handbells 596 Contents xvii

21.9. Sound Decay and Warble in Handbells 596 21.10. Scaling of Handbells 597 21.11. Sound Radiation 598 21.12. Clappers 599 21.13. Ancient Chinese Two-Tone Bells 600 21.14. Japanese Temple Bells 603 References 604

Name Index 607 Subject Index 613