The Science of String Instruments Thomas D. Rossing Editor The Science of String Instruments Editor Thomas D. Rossing Stanford University Center for Computer Research in Music and Acoustics (CCRMA) Stanford, CA 94302-8180, USA [email protected] ISBN 978-1-4419-7109-8 e-ISBN 978-1-4419-7110-4 DOI 10.1007/978-1-4419-7110-4 Springer New York Dordrecht Heidelberg London # Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection 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 in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer ScienceþBusiness Media (www.springer.com) Contents 1 Introduction............................................................... 1 Thomas D. Rossing 2 Plucked Strings ........................................................... 11 Thomas D. Rossing 3 Guitars and Lutes ........................................................ 19 Thomas D. Rossing and Graham Caldersmith 4 Portuguese Guitar ........................................................ 47 Octavio Inacio 5 Banjo ...................................................................... 59 James Rae 6 Mandolin Family Instruments........................................... 77 David J. Cohen and Thomas D. Rossing 7 Psalteries and Zithers .................................................... 99 Andres Peekna and Thomas D. Rossing 8 Harpsichord and Clavichord ............................................ 123 Neville H. Fletcher and Carey Beebe 9 Harp ....................................................................... 145 Chris Waltham 10 Burmese Arched Harp ................................................... 167 Robert M. Williamson v vi Contents 11 Plucked String Instruments in Asia..................................... 173 Shigeru Yoshikawa 12 Bowed Strings............................................................. 197 Thomas D. Rossing and Roger J. Hanson 13 Violin ...................................................................... 209 Joseph Curtin and Thomas D. Rossing 14 Cello ....................................................................... 245 Eric Bynum and Thomas D. Rossing 15 Double Bass ............................................................... 259 Anders Askenfelt 16 Bows, Strings, and Bowing............................................... 279 Knut Guettler 17 Viols and Other Historic Bowed String Instruments .................. 301 Murray Campbell and Patsy Campbell 18 The Hutchins–Schelleng Violin Octet After 50 Years ................. 317 George Bissinger 19 Hammered Strings ....................................................... 347 Thomas D. Rossing 20 Some Remarks on the Acoustics of the Piano .......................... 353 Nicholas Giordano 21 Hammered Dulcimer..................................................... 371 David Peterson 22 Electric Guitar and Violin ............................................... 393 Colin Gough 23 Virtual String Synthesis.................................................. 417 Nelson Lee and Julius O. Smith III Index............................................................................ 457 Contributors Anders Askenfelt Royal Institute of Technology (KTH), Department of Speech, Music, and Hearing, Stockholm, Sweden Carey Beebe Carey Beebe Harpsichords Australia, Factory 35/17 Lorraine Street, Peakhurst, NSW 2210, Australia George Bissinger Acoustics Laboratory, East Carolina University, Howell Science Complex, Room E208, Greenville, NC 27858, USA Eric Bynum 214 Macoy Ave, Culpeper, VA 22701, USA Graham Caldersmith 12 Main Street, Comboyne, NSW 2429, Australia Murray Campbell School of Arts, Culture and Environment, University of Edinburgh, Edinburgh EH1 1JZ, UK Patsy Campbell School of Arts, Culture and Environment, University of Edinburgh, Edinburgh EH1 1JZ, UK David J. Cohen 9402 Belfort Rd, Henrico, VA 23229, USA Joseph Curtin Joseph Curtin Studios, 3493 W. Delhi Road, Ann Arbor, MI 48103, USA Neville H. Fletcher Research School of Physics and Engineering, Australian and National University, Canberra, ACT 0200, Australia vii viii Contributors Nicholas Giordano Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036, USA Colin Gough 44 School Road, Birmingham B13 9SN, UK Knut Guettler The Norwegian Academy of Music, Eilins vei 20, Jar 1358, Norway Roger J. Hanson 2806 Edgewood Drive, Cedar Falls, IA 50613-5658, USA Octavio Inacio ESMAE, Rua da Alegria, 503, Porto 4000-045, Portugal Nelson Lee Stanford University, Center for Computer Research in Music and Acoustics (CCRMA), Stanford, CA 94302-8180, USA Andres Peekna 5908 North River Bay Road, Waterford, WI 53185-3035, USA David Peterson University of Central Arkansas, 56 Ridge Drive, Greenbrier, AR, USA James Rae 827 Valkyrie Lane NW, Rochester, MN 55901, USA Thomas D. Rossing Stanford University, Center for Computer Research in Music and Acoustics (CCRMA), Stanford, CA 94302-8180, USA Julius O. Smith III Stanford University, Center for Computer Research in Music and Acoustics (CCRMA), Stanford, CA 94302-8180, USA Chris Waltham Department of Physics & Astronomy, University of British Columbia, 6242 Agricultural Road, Vancouver, BC V6T 1Z1, Canada Robert M. Williamson Department of Physics, Oakland University, Rochester, MI 48309-4401, USA Shigeru Yoshikawa Graduate School of Kyushu University, 4-9-1 Shiobaru, Minami-ku, Fukuoka 815-8540, Japan Chapter 1 Introduction Thomas D. Rossing String instruments are found in almost all musical cultures. Bowed string instruments form the backbone of symphony orchestras, and they are used widely as solo instru- ments and in chamber music as well. Guitars are used universally in pop music as well as in classical music. The piano is probably the most versatile of all musical instru- ments, used widely not only in ensemble with other musical instruments but also as a solo instrument and to accompany solo instruments and the human voice. In this book, various authors will discuss the science of plucked, bowed, and hammered string instruments as well as their electronic counterparts. We have tried to tell the fascinating story of scientific research with a minimum of mathematics to maximize the usefulness of the book to performers and instrument builders as well as to students and researchers in musical acoustics. Sometimes, however, it is difficult to “translate” ideas from the exact mathematical language of science into words alone, so we include some basic mathematical equations to express these ideas. It is impossible to discuss all families of string instruments. Some instruments have been researched much more than others. Hopefully, the discussions in this book will help to encourage further scientific research by both musicians and scientists alike. 1.1 A Brief History of the Science of String Instruments Quite a number of good histories of acoustics have been written (Lindsay 1966, 1973; Hunt 1992; Beyer 1999), and these histories include musical acoustics. Carleen Hutchins has written about the history of violin research (Hutchins 2000). Relatively less has been written about scientific research on other string instruments. Pythagoras, who established mathematics in Greek culture during the sixth century BC, studied vibrating strings and musical sounds. He reportedly discovered T.D. Rossing (*) Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, Stanford, CA 94302-8180, USA e-mail: [email protected] T.D. Rossing (ed.), The Science of String Instruments, 1 DOI 10.1007/978-1-4419-7110-4_1, # Springer Science+Business Media, LLC 2010 2 T.D. Rossing that dividing the length of a vibrating string into simple ratios produced consonant musical intervals. According to legend, he also observed how the pitch of the string changed with tension and the tones generated by striking musical glasses, but these are probably just legends (Hunt 1992). Most early acoustical investigations were closely tied to musical acoustics. Galileo reviewed the relationship of the pitch of a string to its vibrating length, and he related the number of vibrations per unit time to pitch. The English mathematician Brook Taylor provided a dynamical solution for the frequency of a vibrating string based on the assumed curve for the shape of the string when vibrating in its fundamental mode. Daniel Bernoulli set up a partial differential equation for the vibrating string and obtained solutions which d’Alembert inter- preted as waves traveling in both directions along the string (Beyer 1999). 1.1.1 Bowed String Instruments During the sixteenth century, two families of viols developed: the viola da gamba, or “leg viol,” and the viola da braccio, or “arm viol.” These instruments, which normally had six strings tuned in fourths (except for a major third separating the third and fourth strings), developed in different sizes from treble to bass. They have remained popular to this day, especially for