( ( ( (('("("(" (((('((

The Science of Musical Sound

Revised Edition

John R. Pierce

·=
W. H . Freeman and Company
New York
 To Max Mathews, whose Music V and whose kind and
patient counsel started many things in many places.

Library of Congress Cataloging-in-Publication Data

Pierce, John Robinson, 1910-
The science of musical sound-rev. ed.
Bibliography: p.
Includes index.
1. Music-Acoustics and physics. 2. Music- ·
Psychology. 3. Sound. 1. Title.
ML3807.P5 1983 781'.22 82-21427
ISBN 07167-6005-3

Copyright © 1983, 1992 by Scientific American Books

No part of this book may be reproduced by any mechanical, photographic,
or electronic process, or in the form of a phonographic recording, nor may
it be stored in a retrieval system, transmitted, or otherwise copied for
public or private use, without written permission from the publisher.
Printed in the United States of America
1 2 3 4 5 6 7 8 9 0 VB 9 9 8 7 6 5 4 3 2 1

) ) ) ) ) J j ) J ) )
( ( ( ( ( ( ( ( ( r r ("("((" ( cr ( ( ('( (

Co1ttents

Preface ix
I Sound, Music, and Computers 1
2 Periodicity, Pitch, and Waves 14
3 Sine Waves and Resonance 38
4 Scales and Beats 64
5 Consonance and Dissonance 76
G Consonance, Dissonance, and Harmony 87
7 Ears to Hear With 102

- i Power and Loudness 115

.9 Masking 130
I 0 Other Phenomena of Hearing 140
I I Architectural Acoustics 150
I 2 Sound Reproduction 167
I 3 Musical Instruments, Analysis, and Synthesis 180
I 4 Perception, Illusion, and Effect 208
APPENDICES
A Terminology 223
B Mathematical Notation 225

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c Physical Quantities and Units 227

D Mathematics and Waves 228
£ Reflection of Waves 234
F Digital Generation of Sound 239
c The MIDI Standard 248
H MAX 252
I Bibliography 257
Sources of Illustrations 261
Index 265
The first edition of this book, published in 1983, was written in a different
world. Then, the computer generation of musical sound was a lusty but
tiny infant. Commercial synthesizers were chiefly analog devices, costly
and tricky. Digitally generated sound, now familiar through inexpensive
digital keyboards, was limited by cost and capability.
Today analog sound is obsolete. Audio technology is digital technol-
ogy. Digital keyboards are more common than pianos and are more suited
to our limitations of space and money. Composers produce movie and
television scores by digital means, without the intervention of instru-
mentalists.
Musical sound is musical sound, whether it is generated by acoustical·
instruments or by digital hardware, and musical sound is what this book is
about. The Science of Musical Sound describes the physical and mathemat-
ical aspects of sound waves that underlie our experience of music as well as
the psychoacoustics of musical sound- the relation of the physical aspects
of sounds to their perceptual features.
The historical and scientific basis of this book isn't all that different
from the first edition, for science and sound don't change that rapidly.
Technology does; new technology and new work have added to our
knowledge, and I have sometimes amplified and sometimes corrected what
I said in the first edition. There are other changes in the book. All chapters
have been updated but the most heavily reworked are Chapter 1 on the
role of computers in music, Chapter 12 dealing with sound reproduction,
and Chapter 13 on synthesized sound and equipment for producing it. I
have added an appendix on MIDI, that surprising standard through which
ix
any commercial keyboard can operate any commercial digital synthesizer, frequently with Earl Schubert. Jay Kadis, CCRMA's audio engineer, was
and which in general allows interconnection of digital sound gear of very helpful in connection with Chapter 12, and so was James A. Moorer
various manufacturers. Andrew Schloss, who urged the republication of of Sonic Solutions. Above all, John Chowning, director, and Patte Wood,
this book, suggested this addition and drafted the appendix. It was revised administrative director, have been essential to rhis book- as to everything
with his aid and that of David Jaffe. Further, David Zicarelli has supplied else that happens at CCRMA.
an appendix on MAX, a popular programming language widely used in the I also wish to thank Linda Chaput, president of W. H. Freeman, for
comrol of MIDI-compatible synthesizers. making this new edition possible, and Christine H astings and others at
I have drastically revised, pruned, and added to a bibliographical Freeman for their patient and diligent help in the revision of the manu-
appendix. I have cited books and pertinent recordings on compact discs script and in seeing it through to publication.
both digital sound examples and of a very few examples of computer mustc
that make important points. john R. Pierce
Sound examples are essential to our understanding of sound. Today, a
fair amount of computer music has appeared on compact discs, through
Wergo and other publishers. And, there are a few recordings of sound
examples which ar:e referred to in the bibliography. Beyond this,
commercial synthesizers are available, and many of these can be used tn
producing the sine waves and combinations of sine waves described in
various chapters. Thus, diligent readers can find sound examples on var-
ious discs or, better yet, synthesize them on their own.
The preface to the first edition rold how I used the fifth Marconi
International Fellowship award in preparing the first edition and of my
gratitude to Gioia Marconi Braga, Marconi's daughter, for the opportunity
the award honoring her father gave me. It told of my indebtedness to
Jean-Claude Risser, to Elizabeth Cohen, and to many at Stanford's
CCRMA (pronounced karma, the Center for Computer Research in Music
and Acoustics) for their work in preparing sound examples. It told of
people who, in my division at Bell Laborarories, had inspired my interest in
sound-of E. E. David, Jr., and Max Mathews, whose early work, culmi-
nating in his Music V program, launched computer music to the world in
1957. Today both Max Mathews and I are professors in the music depan-
ment of Stanford University, invired to join CCRMA by its director, John
Chowning. His invention of fm synrhesis made possible the first reasonably
priced digital keyboard (Yamaha's DX7).
In the preface to the first edition I mentioned the happy month that I
spent at Pierre Boulcz's IRCAM (Institute for Research and Coord ination
of Acoustics and Music) in Paris in 1979. I acknowledged the help and
inspiration of many: including the late Gerald Strang, Manfred Schroeder,
and Earl Schubert, Jr. And of Gerard Piel, Linda Chaput, and ochers at
Scientific American Library who had made the book possible.
Several colleagues at CCRMA and elsewhere have been helpful in
preparing this revised edition . I have already mentioned the contributions
of Andrew Schloss, David Jaffe, and David Zicarelli. J have consul ted
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This book is about musical sound. Perhaps we should say acous-

tics, a word that entered the English language in the seventeenth century,
the century in which Galileo wrote about musical sound.
Today papers in the] ournal of the Acoustical Society of America are
divided into many categories, some of which have little relation to music,
such as "Underwater Sound," or "Ultrasonics, Quantum Acoustics, and
Physical Effects of Sound." "Music and Musical Instruments" is a cate-
gory, but much that is essential to an understanding of musical phenomena
appears under "Psychological Acoustics" and "Physiological Acoustics."
In today's world, knowledge has become so divided and extended that it
sometimes seems hard to make sense of it.
In the Greek and Roman worlds music, including whatever was
known about musical acoustics, held a high place in science and philoso-
phy. In the liberal arts of the Middle Ages, music was a part of the higher
quadrivium, along with arithmetic, geometry, and astronomy. The place of
music in the liberal arts was above that of grammar, rhetoric, and logic,
which constituted the lower trivium that dealt with words rather than
numbers.
As time passed, music's status became more complicated. The roman-
tics tended to associate music with the grandiose sentiments rather than
the grand scientific insights of their age. Still, through the nineteenth
century, scientists studied music and musical sound with insight as well as
aesthetic appreciation. In 1862 Hermann von Helmholtz, physiologist,
physicist, great scientist on a grand scale, published On the Sensations of
Tone as a Physiological Basis for the Theory of Music. Not a word in the
1
 Sound, Music, and Computers 3
2 THE SCIENCE OF MUSICAL SOUND

title about acoustics, though we would say that the book is about musical
acoustics and psychoacoustics and, in addition, about music in general.
In this century some musicians have looked to science and technology
for new directions in music without concentrating on the word acoustics.
These have included Edgard Varese and Hermann Scherchen. This has not
been the chief current of musical thought, nor has music been a part of the
mainstream of science. The greatest influence of science on music has
come through the development of means for recording and reproducing
the sounds of music played on conventional musical instruments. The
phonograph, with its later electronic advances, and radio revolutionized
the role of music in our lives as radically as photography, motion pictures,
and television have changed our world of visual experience. Today the
computer and digital technology in general are working fantastic changes
in the recording and transmission of sound, and in the generation of
musical sounds.
This book is indeed about acoustics- both the physical acoustics
pertinent to the understanding of conventional musical instruments and
the sounds they produce, and the psychoacoustics that helps us to under-
stand the perception of musical sounds. But it is about acoustics in relation
to music and musical ideas.
The changes that computers and their descendants will continue to
work in music will come partly through fresh insights of musicians who
work with new sounds. But computers have opened up new ways of
analyzing and experimenting with sounds, and new ways of investigation
the response of human beings to sound, including musical sounds. Today
we know far more about sounds and their perception than we did in the
pre-electronic era. And we will know more in the future.
Figure 1-1 K; ummhoms, a Renaissance
We can be sure from past experience that new sounds and new
group of double-reed instruments.
understanding of sound will affect the course of music profoundly. Better
sounds have always produced different music.
Musical instruments have improved greatly in range and quality in the
past few centuries, and certainly up to the beginning of this century. In
part, this improvement resulted from (1) the development of better, more nization and sound itself, the music that they produced was different in
easily playable instruments, especially brass instruments with valves and style and sound. They did not confront the past on its own ground.
woodwinds with better key mechanisms; (2) the increasing skill of instru- In our century, electronic sounds in general have had a profound
mentalists; and (3) an expansion of the range of musical sound, as com- effect on some nonelectronic music. When Edgard Varese wrote Deserts in
posers and performers developed and exploited new effects and new 1954 for taped sound and orchestra, he was proud that he had provided
idioms. such continuity of sound quality that it is hard to detect transitions from
Whether or not we wish to call such change progress, it brought an tape to orchestra. Some of Krzysztof Penderecki's music of the 1960s for
expansion in the variety of orchestral sound. Think for a moment of the conventional orchestra deliberately imitates "electronic" sound quality, as
sounds of Bach, Mozart, Wagner, Debussy, and Stravinsky. As successive does some orchestral writing of Yannis Xenakis a little earlier. In such
generations of composers expanded into new territories of concept, orga- works, written at a time before synthesized and computer-produced
4 THE SCIENCE OF MUSICAL SOUND Sound, Music, and Computers 5

Figure 1-3 Edgard Varese {left).

Computers and Music

When, in 1957, Max Mathews first used a digital computer to produce
complex musical sounds, it seemed that this could have a liberating effect
on composers. In principle, a compu ter can produce any sound. Its poten-
tial is limited only by the composer's imagination. But of this Milton
Babbitt said, " It's like a grand piano in the hands of a group of savages.
You know that wonderful sounds can come fro m it, but will they?" The
computer will produce novel and fine music, but only through human skill
and effort.
Figure 1-2 A modern B-Aat bass clarinet. Today the ubiquitous personal computer can be adapted to generate
musical sounds. Inexpensive digital keyboards are more widespread than
pianos, and a seemingly endless variety of commercial digital hardware is
available for producing, modifying, and analyzing musical sounds. Com-
sounds escaped from their early "electronic" timbre, we hear the orchestra
posers of many different styles of music are making themselves heard in
refining sounds whose somewhat harsh qualities many listeners found
homes, in concerts, and through TV and film scores.
objectionable. Does the future of musical sou nd lie in digital synthesis? Will the
T he influence of electronic music on some composers has been more computer have any influence on other aspects of music, for example form
subtle. Gyorgy Ligeti, w ho had worked with Karlheinz Stockhausen at the or organization? Some have thought it might.
West German radio electronic studios in · Cologne from 1958 to 1960, In 1957 Lejarin A. Hiller, Jr., and his collaborator L. M. Isaacson took
abandoned the limited and difficult electronic means of tone production
their inspiration from Johann J oseph Fux (1660- 1741), w ho codified rules
available at that time. Nevertheless his music for voices and conventional
to describe the stylistic practices of earlier contrapuntalists, most notably
instruments shows that he is acutely aware of the subtle qualities of
Palestrina. Hiller and Isaacson programmed a computer to make random
electronic sounds and of the musical value of the sophisticated control of
choices constrained by some of Fux's rules for first-species counterpoint,
sound quality.
6 THE SCIENCE OF MUSICAL SOUND Sound, Music, and Computers 7

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Rules in music are not canned algorithms that we can use in making a
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Figure 1-4 A page from the score of Krzysztof Penderecki's Polymorphia.

Figure 1-6 Lejarin A. HiUer.

8 THE SCIENCE OF MUSICAL SOUND Sound, Music, and Computers 9

computer solve cut-and-dried problems over and over again. What, indeed,
is the place of computers in music?
Hiller and others have pursued the idea of the computer as a com-
poser, or at least, as a tool in manipulating musical material. I will say little
about the computer as a composer's aid, for that is a subject unrelated to
understanding and generating musical sounds.
We may note, however, that some help that the composer needs can
be and is supplied by a computer. One of the most useful tasks that a
computer can do is to produce musical scores of high quality in a day when
even clear hand copies of music have become excessively expensive. Leland
Smith's pioneer SCORE program is now available for IBM (or IBM-com-
patible) personal computers. It produces scores of excellent, publishable
quality. It is easy to make changes and to extract instrumental parts from a
full orchestral score. T he input to SCORE is the computer keyboard.
Many later programs have been developed for producing musical scores on
both Macintosh and IBM computers, including Finale, Professional Com- Figure 1-7 Herbert A. Deutch, with the earliest prototype
poser, and NoteWriter. Some allow playing on a pianolike keyboard of the Moog synthesizer, which he helped invent.
input.
T he role of the computer in composition goes beyond the production
of a neat final score. Computers are used to store and manipulate musical menr. Therefore, analog synthesis did not have a musical impact compara-
materials, including lists of notes or their equivalent. ble to that which digital synthesis has had.
I think that the chief challenge of the computer lies in another
direction, that of new sounds and their use. In the past, many composers
have responded to the challenge of creating new tone colors. Harry Partch
New Technology and Computer Music
invented both a new scale and an entire orchestra of new instruments to The computer offers a wide range of sounds, along with the means for
play his music, but Partch's instruments were difficult to build and are not controlling them very accurately. The challenge is how to master a con-
commercially available. In their search for new timbres, some composers stantly changing medium of unlimited acoustic potential, and how to find
have evoked strange sounds from conventional instruments. Perhaps the aesthetic reasons for realizing these new capabilities.
strangest was the sound of a violin burning on a New York stage, an event In the early days of computer music, composers encountered a num-
staged by Lamont Young and Charlotte Moorman. ber of problems. There were no instruments available. Composers had to
Two early-twentieth-century analog instruments, the T heremin and create their own instruments as computer software. And they had to play
the Ondes Martenot, were recognized as unique musical resources by a them; no performers were available either. The composer had to supply all
number of composers who wrote idiomatically for them. We have noted input through a typewriterlike keyboard. All these factors proved awkward
that electronic analog (as opposed to digital) synthesizers appeared in the in performances. Once a composition was completed, who wanted to sir in
mid-twentieth century. They played an appreciable role in music through an auditorium and listen to music coming from loudspeakers? An audience
the 1960s. Robert Moog's analog synthesizers had a distinct musical im- could not even be sure when to clap unless the composition gave a clue, or
pact, for example, through Walter Carlos' Switched On Bach. unless the house lights came up. Was there an alternative to the concert?
Electronics was an essential part of musique concrete of Pierre Only a few commercial recordings were made, and none had a wide
Schaeffer and others at the Studio d'Essai of the French radio system. distribution.
Analog synthesis was pursued in the West German radio studios at Co- Some of the troubles with concerts were overcome in various ways.
logne by Karlheinz Stockhausen and others. One solution was to couple recorded sounds with projected images, as in
Analog electronics tended to be expensive and not to stay in adjust- Andy Moorer's Lions Are Growing, a setting of a poem by Richard Brauti-
10 THE SCIENCE OF MUSICAL SOUND Sound, Music, and Computers 11

gan. In this effective piece composed in 1978, a computer-processed voice

speaks, sings one line of music and chords, and roars, accompanied by
appropriate slides. This made computer music into a component of a
multimedia event, like later digitally synthesized film and TV scores.
Today, computer film and TV scores are no more and no less recorded and
replayed than are scores played by musicians on conventional instruments.
Another concert alternative has been to couple recorded synthesized
sound with a singer or instrumentalist. This can be very successful, but it
has not been the only resource. In this day of digital keyboards (Figure
1-10), parts or all of a score of digital sound can be evoked using a
conventional keyboard. Through the world-standard MIDI interface, a
standard commercial keyboard can control commercial digital synthesis
hardware. Or a keyboard player can supplement or interact with or control
and modify the process of digital sound synthesis.
Max Mathews's Radio Baton, a sort of computerized musical drum,
offers another solution. In all modes of operation the player strikes or

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Figure 1-8 A working computer score for Andy Moorer's Lions are Growing.

Figure 1-9 Max Mathews and his Radio Baton. Among other things, the
Radio Baton enables the performer to beat out the rhythm, loudness, and
sound balance of a piece whose notation is stored in computer memory.
12 THE SCIENCE OF MUSICAL SOUND Sound, Music, and Computers 13

strokes a surface with one or more drumsticks. The velocity of striking can Many talented young composers use computers and digital synthesis
control loudness; the position can control timbre in one direction and hardware as experimental tools to study the intricacies of musical sounds.
pitch in another. The successive pitches and durations can also be stored in Some will produce digitally synthesized music. Others will compose music
the computer. Rate of striking or position of stroking can control tempo, for conventional instruments. All will be influenced, all will learn many
and position of striking or stroking can change sound quality and instru- new and useful things. And so, I hope, will the reader of this book.
mental balance. In this way the Radio Baton brings the performer into the
realm of computer-generated sound, but the skills required are perhaps Although this book discusses computers and the digital analysis and syn-
nearer to those of a conductor than of a traditional percussionist. A thesis of musical sounds, it is really about the well-known aspects of all
number of similar devices have appeared. aspects of musical sounds, about pitch, scales, consonance, harmony, and
Today, digital synthesis of sound is used by composers all over the timbre, and about some less-known aspects of perception. We can't have a
world, in universities, in conservatories, in computer-music institutions, useful understanding of musical sounds without considering these aspects.
and in commercial music. The MIT press publishes a quarterly Computer We will starr with periodicity, pitch, and waves.
Music journal. The worldwide Computer Music Association holds an
annual International Computer Music Conference, as well as other meet-
ings, and issues a publication called ARRAY. There are commercial publi-
cations, including Keyboard and Electronic Musician. When I look back
over more than thirty years to the time when Max Mathews generated the
first computer music piece in 1957 I am struck by obstacles overcome and
progress made.

Figure 1-10 The Yamaha SY99, a late model digital keyboard. The earlier
Yamaha DX7, which reached the market in 1983, was the first completely
digital keyboard synthesizer, and the first such synthesizer that sold at an
affordable price (around $2,000). Early "digital" keyboard devices sold for
about ten times this price. The success of the DX7 was due partly to japanese
persistence and ingenuity, partly to the use of special integrated circuit chips,
and partly to the usc of fm (frequency modulation) synthesis, an invention of
John Chowning. Now discontinued, the DX7 was a landmark that ushered in
a new era. (Photo courtesy of Yamaha Corp. of America)
 Periodicity, Pitch, and Waves 15

"concert pitch" is that the A above middle C sounds at 440 vibrations each
second. More on pitch can be found on pages 36-37.
Sounds that have a definite, unambiguous pitch are called periodic,
because something happens over and over again at a constant rate. Galileo
found by accident that he could produce a sound having definite pitch by
Periodicity, Pitch, attd Waves scraping a brass plate with a sharp iron chisel. The tiny parallel and
equidistant ridges left on the brass were a permanent witness to the
vibrations of the screeching chisel that had engraved them. An old book on
musical acoustics relates that Galileo also produced a pitched sound by
rubbing a knife rapidly around the edge of a milled coin. You can try this
by scratching the edge of a quarter with your fingernail. The sound
produced as your nail encounters the ridges around the edge of the coin
does have some pitch. The faster you scratch the coin, the higher the pitch.
The siren provides a clear illustration of the periodicity related to
A !though wind instruments have been known for nearly five
musical pitch. The very siren that Varese used in Ionisation stood in his
New York studio. When I turned the crank faster, the pitch of the sound
thousand years, and harps for almost as long, sounds of a definite pitch are that the siren produced rose. Why was this? In order to understand, we
not necessary to music. The earliest musical instruments that archeologists must examine the mechanism of the siren, which was invested by Charles
have found in Egypt are clappers. Perhaps song accompanied their rhyth-
mic bear, but rhe music may have been largely rhythmical. A knowledge- p
able friend of mine tells me that, in very primitive music, the chief interval
used is the fifth (seven semitones), though sometimes an indefinite musical
third (four semitones) is also used.
Rhythm by itself can make music. In our time, Carlos Chavez com-
posed a toccata for percussion alone. A siren is heard in Edgard Van!se's
Ionisation, but that fine work achieves its effect chiefly by rhythm and
timbre (sound quality).

Pitch and Periodicity

What is pitch? Psychologists insist that pitch is a name for our subjective
experience of periodic waveforms, rather than a physical property of the
sound wave that reaches our ears. Loosely, we can use the word pitch to
denote the shrillness of a sound. In this sense, the hissing sound s has a
higher pitch than the shushing sound sh. In this chapter, pitch is consid-
ered to be a definite quality related to musical tones, such as those
produced by the violin, the clarinet, the tuba, the piano, or the human
voice. We hear such sounds as having definite pitches that correspond to
particular notes of the musical scale. T he present musical standard for
Figure 2-1 De Ia Tour's siren.