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Physics Today

Evaluating musical instrumentsD. Murray Campbell

Citation: Physics Today 67 (4), 35 (2014); doi: 10.1063/PT.3.2347

View online: http://dx.doi.org/10.1063/PT.3.2347 View Table of Contents: http://scitation.aip.org/content/aip/magazine/physicstoday/67/4?ver=pdfcov Published by the AIP Publishing

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Even in the womb, unborn children hear andrespond to music; and throughout life,music accompanies and enriches many ofour most significant activities. From ancienttimes, philosophers and scientists have

studied the nature of musical sounds and the instru-ments that generate them. To a physicist, there is afascination in exploring how to apply the relativelysimple laws of classical physics to obtain a detaileddescription of the complex and rapidly varyingsound fields produced by performers and transmit-ted through a concert hall to the listeners. Many ofthe musically important aspects of sound produc-tion in instruments involve subtle features of the un-derlying physics, and those subtleties can reveal thelimitations that arise from simplifications and ap-proximations in physical models of the instruments.

Beyond the general appeal of physical under-standing, another important motivation behindcurrent research into the physics of musical instru-ments is the desire to offer helpful guidance to in-strument designers and manufacturers. Tradition-ally, those craftsmen have used techniques and rulesof thumb developed over generations by trial anderror; the aim of many scientists working to under-

stand musical instruments is to supplement suchhard-won empirical knowledge with scientific prin-ciples and useful tools that will enable excellent in-struments to be built in a more reliable and cost- effective way. Although considerable progress has

been made in recent decades, the goal remains un-expectedly elusive. One reason is that achieving ahighly accurate physical model of any musical in-strument is a significant technical challenge: Detailsare still not fully understood, for example, of thefrictional processes in the bow–string interaction ina violin or of the fluid–structure interaction drivingan oboe reed.

A more profound problem lies in the definitionof the very nature of the goal: If we are to optimizea musical instrument, who determines success? Ul-timately, of course, it must be the musician who willplay the instrument or the listener who will hearits sound. The problem is that musicians and physi-cists usually approach the evaluation of the qualityof an instrument in very different ways, and they

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Murray Campbell is Professor Emeritus and Senior HonoraryProfessorial Fellow in the school of physics and astronomy atthe University of Edinburgh, Scotland.

Evaluatingmusical instrumentsScientific measurements of sound generation and radiation by musical instrumentsare surprisingly hard to correlate with the subtle and complex judgments ofinstrumental quality made by expert musicians.

D. Murray Campbell

Evaluatingmusical instruments M K D U G O

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frequently lack a common vocabulary in which todiscuss their differences.

Early studies of musical instrument qualityconcentrated on the frequency spectra of the steady-state parts of the radiated sound and ignored thecomplicated transients at the beginning and end ofnotes. Later it became clear that such transients werenot only critical cues for the identification of partic-ular instruments but also important aspects of whatmusicians describe as “timbre” or “tone color”—thequality of sound that distinguishes a note played ona clarinet from a note of the same pitch and loudnessplayed on a trumpet. In more recent years, scientificattention has increasingly turned toward the inter-action between player and instrument and has rec-ognized that “playability”—a measure of how wellthe player can evoke the desired response from theinstrument—is a key attribute for a performer.

The task for the musical instrument acousticianis thus to establish the objectively measurableproperties of a given instrument that correspondstrongly to timbre perception and playability as

judged by musicians, and then to propose methods by which those properties can be optimized. Such an

approach has been used in studies of pianos, violins, brass, and other musical instruments.

A further problem emerged from such studies.The interaction of the player with an instrumentinvolves several senses: Sight, hearing, touch, andeven smell can contribute to the player’s experience,and those different sensations can become confusedin judgments of quality. The possibilities of such“cross- modal interference” must be understood andminimized. Only then can player judgments be con-

sidered sufficiently robust to correlate with scientif-ically measured data. Let’s look at three examples.

What makes an excellent piano?At first glance, the piano might seem to be one ofthe simplest musical instruments to model scientif-ically. When the performer presses a key, a hammerpivoted at one end is thrown toward a string, asshown in the opening figure; the escapement mech-anism ensures that the hammer head rebounds fromthe string after striking it, and a check then holds thehammer to prevent multiple strikes. The vibrationsof the string are communicated through a bridge tothe wooden soundboard, which vibrates and radi-ates sound into the room.

The simplicity is deceptive: Many aspects of thepiano must be modeled with great care to give a re-alistic sound output. 1 Because the hammer’s head iscovered by several layers of felt with nonlinear com-pressive behavior, the spectral distribution of theenergy communicated to the string is strongly de-pendent on the hammer’s impact velocity. The finitestiffness of the steel piano string results in a signif-icant inharmonicity of the natural transverse vibra-tional modes of the string, which affects both thetimbre of the radiated sound and the overall tuningof the instrument (see box 1). The soundboard’s res-onant frequencies, radiation efficiencies, and damp-ing parameters are influenced by its size, shape, and

bracing. Upright pianos typically have smallersoundboards and shorter bass strings than grand pi-anos, which are preferred by most performers. Thefact that inharmonicity is less for a long string thanfor a shorter string at the same pitch is sometimescited as the reason for that preference, but carefulstudies by Alexander Galembo suggest that the influ-ence of the larger soundboard on the spectral enve-lope of notes in the bass register is the main reason. 2

Over most of the piano’s compass, three stringsare struck simultaneously when a key is pressed.Musicians describe them as “unison” strings. Inter-estingly, however, precise measurements of grand-piano strings after expert tuning reveal that typi-cally there are pitch differences of about 1.5 cents(1.5% of a musical semitone) between the strings.Gabriel Weinreich has shown that such mistuningscan encourage mode locking between the strings, 3

reducing the rate at which energy is transferredfrom the strings to the soundboard. Consciously orunconsciously, the master tuners are enhancing thesinging quality associated with a slow decay of the

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Listeners

Cross-modalinterference

Cross-modalinterference

Visual

V i s u a l

Auditory

A u d i t o r y

Kinesthetic

Kinesthetic

Roomconditions

Performer

Figure 1. A pianist creates and modifies a musicalperformance in response to information receivedthrough three senses: hearing (the auditory channel),sight (the visual channel), and touch (the kinestheticchannel). The player’s perception of the tonal qualityof the piano sound can be biased by the other senses,so-called cross- modal interference from visual andkinesthetic cues, depicted here as a tangled input. Alistener’s perception of the sound quality can likewisedepend on the appearance of the instrument, theplayer, and even the hall. (Adapted from ref. 4.)

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piano sound by introducing subtle deviations fromthe nominal target of unison tuning.

Some pianists believe that by merely alteringthe manner in which the key is depressed, it is pos-sible to change the timbre of a single note, withoutaltering its loudness. It is hard to see how that can

be true, since the only variable affecting the natureof the string vibration is the speed of the hammer atimpact. The player has no direct contact with thestrings and relies on the key mechanism (supple-mented by judicious use of the sostenuto and unacorda pedals) to create and control an expressivemusical performance. But haptic—which includesfeedback from both muscles and pressure receptorsin the skin—and auditory piano cues can becomeentangled: For example, the tiny sound made by afinger contacting the key surface may be moresalient to the player because it is accompanied by thesensation of touch in the fingertip.

The extent to which cross- modal interferencecan influence judgments of quality by pianists wasrevealed in a series of experiments carried out in the1970s in the Leningrad Conservatory concert hall. 4

Twelve professional pianists were asked to rankthree grand pianos, including a Steinway and aninstrument made by the Leningrad piano factory, onscales of tone quality, dynamic range, and playingcomfort. When allowed to make a judgment basedon free playing, the pianists agreed that the Stein-way was distinctly superior in tone quality to theother pianos. However, when sample single notes,scales, and arpeggios were played for them behindan acoustically transparent curtain, the same pi-anists were unable to reliably identify which instru-ment was being played.

In another experiment, the three pianos wereplaced so that the keyboards formed a triangle witha rotating piano stool at the center. Pianists wereeach blindfolded to suppress visual cues, and pre-sented with the pianos in random order; they wereable to identify the instruments with a high degreeof accuracy. The sound was then suppressed byusing white-noise headphones; surprisingly, even tothe subjects, the sight- and hearing-impaired playerswere still able to correctly identify the instruments.Apparently, the discriminating factor was containednot in the auditory signal but in the haptic sensa-tions, which contained information about the me-chanical actions of the instruments. The players per-ceived the sensations as differences in tone quality

because of cross- modal interference between audi-tory and haptic (kinesthetic) channels (see figure 1).

Have we lost Stradivari’s secret?The evaluation of musical instrument quality has anunusual financial significance in the case of the vi-olin. Instruments made around three centuries ago

by Antonio Stradivari and Giuseppe Guarneri “delGesù” now command prices two orders of magni-tude greater than those charged by the best modernviolin makers. Although various nonmusical factorsare involved in such a gross disparity in price, manyplayers believe that violins made by the old Italianmasters have special qualities that cannot be foundin modern instruments. Numerous attempts to dis-

cover a supposedly lost “secret of Stradivarius,”perhaps in the choice of wood or the chemical com-position of the varnish, have all been inconclusive.

The acoustical quality of a violin is closelylinked to its structural resonances. 5 The violin bodyis in essence a hollow box whose top plate is pierced

by two f-shaped holes. When a string is bowed, fric-tional stick–slip interactions generate an oscillatingforce with a sawtooth waveform that is transmittedto a thin wooden bridge sitting on the top plate be-tween the f-holes; the bridge’s motion communi-cates the vibration to the body, which radiates thesound. Various researchers, notably Erik Jansson atKTH in Stockholm 6 and Jim Woodhouse at the Uni-versity of Cambridge, 7 have studied violin body res-onances by measuring the bridge admittance, de-fined as the ratio of the bridge velocity to the forceapplied to the bridge. In Woodhouse’s technique, il-lustrated in figure 2a, an impulsive force is applied

by a small pendulum hammer carrying an ac-celerometer, and the resulting bridge velocity ismeasured with a laser vibrometer.

A typical bridge-admittance curve for a good-quality modern violin is shown in figure 2b. At firstsight it may seem surprising that the response is souneven—the drop of 30 dB between 550 Hz and700 Hz is particularly striking. Yet that feature ischaracteristic of high-quality instruments, old andnew, as are the group of discrete “signature modes”

between 200 Hz and 600 Hz and the “bridge hill” ofclosely spaced and overlapping modes between1500 Hz and 3000 Hz.

A consequence of that nonuniformity of response

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An idealized flexible stretched string, fixed rigidly at both ends, providesa classic textbook case of a vibrating system with a set of harmonicallyrelated modes. 14 The second mode has twice the frequency of the first,and the pitch interval between the two modes is an octave. The twostatements in the previous sentence are generally considered to beequivalent, the term “octave” defining a frequency ratio of exactly 2. Yetthat is not quite right. The finite stiffness of the steel wire used in pianostrings introduces a small but significant inharmonicity in the stringmodes: In the middle of the piano keyboard, the frequency ratio of thefirst two modes is typically around 2.001, and the degree of inharmonicityincreases with mode number. Since the modes decay freely after thehammer head has rebounded from the string, the frequency componentsof the radiated sound are also inharmonic.

In the traditional method of piano tuning, octaves are regulated byminimizing beats between the first frequency component of the uppernote and the second frequency component of the lower note. The inhar-monicity of those components leads to “octave stretching”: The pitchinterval between the fundamental frequencies of the notes A0 and A7on a well-tuned piano is typically around half a semitone greater than it

would be if each of the seven octaves involved had a frequency ratio of exactly 2. While a high degree of inharmonicity in piano strings is unde-sirable, experiments in synthesizing piano sounds 15 have revealed thatthe level of inharmonicity found in good-quality grand pianos and theassociated degree of octave stretching are considered by musicians tobe essential features of the sound of the instrument. (For a tale of RichardFeynman’s encounter with piano tuning, see P HYSICS TODAY, December2009, page 46; relevant letters followed in June 2010, page 8.)

Box 1. Inharmonicity and octave stretching

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The work by Fritz and her colleagues suggestsstrongly that when distracting visual cues and priorexpectations are suppressed, expert players judgethe best modern instruments to have a level of qualityat least as great as the classic instruments made bythe old Italian masters. The remaining scientific chal-lenge is to identify which aspects of the physics ofthe violin are responsible for the performance of aninstrument, whether old or new, that is judged to beexcellent by an internationally renowned performer.

Must brass instruments be made of brass?Trumpets, trombones, and tubas are known as brassinstruments and are commonly made from themetal alloy known as brass. The dominant acousti-cal characteristic of the musical brass family is notthe material of construction, however, but themethod by which the sound is generated: the vibra-tion of the player’s lips, usually pressed against acup- or funnel-shaped mouthpiece. 10 (Saxophonesare also generally made from brass, but because themouthpiece holds a reed, they are considered wood-wind instruments.)

The relationship between the player and theinstrument is illustrated in figure 3. Air from theplayer’s lungs flows past the lips and into the instru-ment. The air column bounded by the wall of the in-strument can be treated to a first approximation asa passive linear system, whose resonant propertiesare described by the input impedance (the ratio, atthe mouthpiece input plane, of acoustic pressure tothe oscillating volume velocity of the air flow thatproduced it). When a note is sounded, the player’slips open and close at the frequency of the playednote. An oscillation regime builds up in which themodulated airflow supplies energy to the acousticresonances of the air column; the resulting pressurevariation in the mouthpiece provides feedback thatstabilizes the lip vibration.

Although the variation of the open area betweenthe lips is approximately sinusoidal, the nonlinearnature of the pressure-flow relationship results inpressure variations within the mouthpiece at har-monic multiples of the basic lip oscillation fre-quency. When the acoustic resonances of the air col-umn also occur at harmonic multiples of thatfrequency, a strong oscillation regime is established.The player’s perception is of a “well-centered” note:The playing frequency is clearly defined and stable.Harmonic alignment of the acoustic resonances istherefore usually considered to be a measure of highquality in brass instruments.

Subtle modification of intonation is, however,an important feature of expert musical perfor -mance, and brass players frequently “bend” or “lip”a note by small adjustments of the facial musclesthat control the mechanical resonance frequency ofthe lips. Judgments of the playability of brass instru-ments involve a balance between strongly centeredpitches and flexibility of intonation; the relationship

between those properties and measured input-impedance curves requires further investigation.

One of the most striking features of the soundof a trumpet or trombone is the way in which thetimbre changes during a crescendo. When a trum-

pet is played quietly, the frequency spectrum of thesound is dominated by the first four or five harmon-ics; as the loudness increases, the spectrum becomesincreasingly rich in upper harmonics. At the fortis-simo level, a particularly hard and brilliant timbreis generated; such sounds are described by musi-cians as “brassy,” reflecting a common assumptionthat the effect arises from structural resonances ofthe metal bell of the instrument (also see box 2). Al-though the spectral enrichment in very loud playingis now known to arise primarily from nonlinearsound propagation in the instrument’s air column, 11

many makers and players continue to insist that thewall material of a “brass” instrument is importantin determining the musical quality of the instru-ment’s sound.

Some 30 years ago, acoustician and brass in-strument maker Richard Smith investigated the re-liability of players’ judgments of brass instrumentquality. 12 Ten professional trombonists were askedto play and evaluate a number of tenor tromboneswith brass bells of different wall thickness. All the

bells were made on the same mandrel to ensure thatthey had the same internal dimensions, and the massdistributions of the instruments were equalized. Vi-sual cues were eliminated by blindfolding the play-ers. Under those circumstances the players couldnot distinguish between the instruments. A purecopper bell included in the test set was not identi-fied as having a noticeably different timbre, yetSmith reported that “when subsequently played innon-blind tests it gained magical properties!” Thatclassic experiment provides yet another example ofthe way in which quality judgments by players can

be biased by cross- modal interference.The question of the extent to which wall vibra-

tions contribute to the sound radiated by brass

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When a trombone is played loudly, the amplitude of the internal pressurewave can exceed 10 kPa. At that level, linear acoustics is inadequate todescribe the wave propagation, since the local speed of sound becomesdependent on pressure. The crest of the pressure wave travels faster thanthe trough, which results in a progressive steepening of the wavefront.At a critical distance along the tube—a distance that depends on therate of change of the pressure signal in the mouthpiece and the boreprofile of the instrument—the almost instantaneous pressure jumpcharacteristic of a shock wave appears.

The sound radiated from the trombone’s bell when an internal shock wave is created has a very wide spectrum of harmonics that can extendwell into the ultrasonic range. Even at moderate playing levels wellbelow the threshold for shock formation, distortion arising from non -linear sound propagation can make a significant contribution to thetimbral palette of the brass player. Instruments with an approximatelyconical bore, like the cornet and the euphonium, are less susceptible tononlinear distortion than instruments with long sections of cylindricaltubing, such as the trumpet and trombone. That is because the progres-sive increase in the cross- sectional area of the wavefront propagating in a

conical tube corresponds to a reduction in the pressure amplitude. Studiesof nonlinear propagation effects have led to increased awareness of thescientific reasons for some of the subtle differences in timbre and playa-bility among instruments of the brass family. 16

Box 2. Shock waves in trombones

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instruments remains controversial. In experimentswith a mechanical sound source, Smith measuredthe sound output from the trombones used in his

blind tests. He found that differences of severaldecibels in specific frequency components, meas-ured at a position corresponding to the left ear ofa human player, were correlated with changes inthe wall thickness of the bell. The differences werenot, apparently, recognized by the players in the

blind tests.Recently Wilfried Kausel, Daniel Zietlow, and

Thomas Moore have undertaken systematic studiesof wall-vibration effects in brass instruments, 13

using an artificial lip excitation source and an exper-imental arrangement that allows the wall vibrationsto be damped by sand (figure 4). They found thatdamping the wall vibrations resulted in several-decibel increases in the first few harmonics of theradiated sound, along with reductions in the ampli-tudes of higher harmonics. Further blind tests usingprofessional performers will be necessary to clarifythe importance of wall vibrations in musical practice.

Science and music: A dialogThe scientific understanding of the functioning ofmusical instruments has progressed notably in thepast few decades, aided by the growing sophistica-tion of both acoustical measuring instrumentationand computational simulation techniques. It is in-creasingly evident, however, that the transfer of sci-entific knowledge from the research community tomusical instrument makers and performers relieson relationships of mutual respect between creativemusicians and those who seek to study them scien-tifically. Musicians sometimes look with scorn onsimplified models that physicists use to explore thefundamental principles of an instrument’s behavior.Scientists need to explain that the models are stepsalong a route that could ultimately lead to insightsof great value to a maker or player. On the other side,scientists can be dismissive of performers whoseevaluations of instruments are biased and inconsis-tent, but it is essential to remember that it is the per-former who brings the instrument to musical lifeand who must be the ultimate arbiter of its quality.

Fortunately, there are now several internationalforums in which scientific researchers, performers,

and makers come together regularly to discuss suchproblems in an open and constructive way. One im-portant and very productive forum at whichstringed instrument makers and researchers get to-gether is the annual Acoustics Workshop organized

jointly by the Violin Society of America and OberlinCollege (http://www.vsa.to/oberlin-workshops).The International Symposium on Musical Acousticstakes place every two years (this summer in LeMans, France: http://isma.univ-lemans.fr), usuallyinvolving both practical workshops and academicsessions. The Institute of Musical Acoustics at theUniversity of Music and Performing Arts Vienna pe-riodically organizes meetings designed to bridge the

gap between scientists and makers (http://viennatalk.mdw.ac.at/); the next meeting will be in 2015. Suchdialogs should lead to a deeper understanding ofthe mysterious but fascinating relationships be-tween scientific and musical evaluations of the qual-ity of musical instruments.

References1. A. Askenfelt, Five Lectures on the Acoustics of the Piano ,

Royal Swedish Academy of Music, Stockholm (1990),http://www.speech.kth.se/music/5_lectures/contents.html.

2. A. Galembo et al., Acta Acust. United Ac. 90 , 528 (2004).3. G. Weinreich, J. Acoust. Soc. Am. 62 , 1474 (1977).4. A. Galembo, Piano Tech. J. 55 , 14 (2012).5. J. Woodhouse, P. M. Galluzzo, Acta Acust. United Ac.

90 , 579 (2004).6. E. V. Jansson, Acta Acust. United Ac. 83 , 337 (1997).7. J. Woodhouse, Acta Acust. United Ac. 91 , 155 (2005).8. G. Bissinger, J. Acoust. Soc. Am. 124 , 1764 (2008).9. C. Fritz et al., Proc. Natl. Acad. Sci. USA 109 , 760 (2012).

10. M. Campbell, Acta Acust. United Ac. 90 , 600 (2004).11. A. Hirschberg et al., J. Acoust. Soc. Am. 99 , 1754 (1996).12. R. Smith, Proc. Inst. Acoust. 8 (1), 91 (1986).13. W. Kausel, D. W. Zietlow, T. R. Moore, J. Acoust. Soc.

Am. 128 , 3161 (2010).14. N. Fletcher, T. D. Rossing, The Physics of Musical In-

struments , 2nd ed., Springer, New York (1998).15. H. Fletcher, E. D. Blackham, R. Stratton, J. Acoust. Soc.

Am. 34 , 749 (1962).16. A. Myers et al., J. Acoust. Soc. Am. 131 , 678 (2012). ■

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Figure 4. The experimental arrangement used atthe Institute of Musical Acoustics (IWK) in Vienna toinvestigate the influence of wall vibrations on thesound radiated by a French horn. The instrumentwas mounted in a wooden box, which could befilled with sand to damp the wall vibrations. Thehorn was driven by an artificial mouth with water-filled rubber lips, and the sound from the bell wasradiated into an anechoic chamber through a circularaperture in the front wall of the box. Significantchanges in the spectral content of the radiatedsound were measured when the walls weredamped; that confirmed similar findings in studiesof a trumpet by Thomas Moore and colleagues atRollins College. (Image courtesy of Wilfried Kausel.)

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