INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 22 (2001) 89–101 www.iop.org/Journals/ej PII: S0143-0807(01)16272-0

The physicist’s guide to the orchestra

Jean-Marc Bonard

Departement de Physique, Ecole Polytechnique Federale de Lausanne,CH-1015 Lausanne EPFL, Switzerland

E-mail: jean-marc.bonard@epfl.ch

Received 10 August 2000, in final form 28 November 2000

AbstractAn experimental study of strings, woodwind (organ pipe, flute, clarinet,saxophone and recorder) and the voice was undertaken to illustrate the basicprinciples of sound production in musical instruments. The setup used issimple and consists of common laboratory equipment. The canonical examples(standing wave on a string, in an open and closed pipe) are easily reproduced.The characteristics of the instruments are outlined and discussed.

Introduction

Being a clarinet enthusiast, I wanted to share the richness of music (and the physicist’s way ofapproaching it) to engineering students within the frame of their general physics classes. Thebasics are reported in nearly every physics textbook and many details are provided in morespecialized contributions [1–8]. I could not, however, find a comparison of the waveforms andspectra of different instruments in any of these sources. I thus decided to directly measure andcompare these data.

The whole orchestra being too ambitious for a start, I restricted myself to two instrumentfamilies, namely strings and woodwind (the former are given a rather perfunctory treatment),and included the voice. The waveform and spectra presented in the first part illustrate thebehaviour of standing waves on strings and in closed and open pipes (variations 1–2), andhighlight the basic differences between the timbre of the instruments (variations 3–5). Inthe second part, we will note that the differences in timbre between instruments are reflectedby the measurements, but that variations between models or performers remain hard to assess(variation 6). The characteristics of some instruments will be adressed in more detail (variations6–8).

Theme

Physicists have been studying the spectrum of musical instruments at least since the 1940s.The first available methods were based on heterodyne analysis [9] or on sonographs [1, 10].The development of the fast Fourier transform (FFT) algorithm coupled with the appearanceof fast and relatively cheap microprocessors has greatly facilitated the task of musicallyinclined physicists. Quite sophisticated analysers have been realized [11] but setups basedon commercial instruments work just as well for basic analysis [12, 13].

0143-0807/01/010089+13$30.00 © 2001 IOP Publishing Ltd Printed in the UK 89

90 J-M Bonard

Figure 1. The experimental setup.

The waveforms have been acquired with the setup presented in figure 1. It consists of acondenser microphone connected directly to a 125 MHz LeCroy9400 digital oscilloscope. Themicrophone was positioned at a distance of 20 cm from the instruments, above the bridge for thestrings and in front of the bell for the woodwind. The data acquisition program LabView wasused to retrieve and store the waveforms on a computer, as well as to calculate the spectrum1

(note that this task can be directly performed on most modern oscilloscopes). The sampling rateused was typically 100× and 1000× the frequency of the played sound for the spectrum andthe waveform, respectively, giving approximatively 100 and 1000 sampled points per periodfor a total of 4096 points per acquisition (these specifications are met by every oscilloscopeand/or spectrum analyser).

The waveforms and corresponding spectra analysed here represent the continuous part ofthe sound only, played at medium loudness (mf). This permanent regime is only a small partof the character of a note. The dependence on the type of attack, duration or loudness was notconsidered, and would be a fascinating study in itself. It is also clear that a more profoundanalysis would require a careful consideration of instrument and microphone position, as wellas a calibration of the room acoustics and of the microphone response [9]. None of these pointshas been taken into account, so the experimental results have to be considered with care.

Variation 1: vibrating strings

A string instrument in its simplest form is composed of a stretched string over a resonancebody that transmits the vibration of the string to the air. Despite the simplicity of the vibratingsystem, string instruments show a phenomenal diversity of timbre2. This arises from thevariety of excitation as the string can be plucked (guitar, harp, harpsichord), bowed (violin,viola. . . ), or struck (piano). The resonance body also plays a great role as is attested by thetimbre difference between, e.g., a guitar and a banjo.

When a continuous transverse wave is generated on a stretched string of length l, a standingwave forms following the superposition of the waves after reflection at the stops. Simple

1 LabView (National Instruments, http://www.ni.com/labview/) is a powerful graphical programming environmentfor data acquisition, control, and analysis. Every oscilloscope (or other suitable apparatus) with a GPIB (IEEE-488)interface can be controlled by a LabView program. The routines for data retrieval are specifically designed for theinterfaced apparatus and can be downloaded from the website http://zone.ni.com/idnet97.nsf/browse/. The routinesused for the calculation of the power spectrum were the standard LabView routines (AutoPowerSpectrum.vi).2 The timbre is usually defined as the quality of tone distinctive of a particular singing voice or musical instrument,and is essentially determined by the relative intensity of the different harmonics.

The physicist’s guide to the orchestra 91

-100

-80

-60

-40

-20

0

20

0 5 10 15 20 25 30

Inte

nsity

[dB

]

Harmonic

plucking

bowing

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Inte

nsity

[a.u

.]

Time [s]

plucking

bowing

A2 - 110 Hz

Figure 2. Waveforms and corresponding spectra of a sonometer (A2, 110 Hz) plucked and bowedat 1/10 of the string length (open string). Inset, the first three vibration modes. The thick lineabove the waveforms indicates the period of an oscillation.

considerations show that the only allowed modes of vibration correspond to wavelengths ofλ = 2l/n where n � 1 [2–6], which form a harmonic series3 (inset of figure 2). The vibrationof a string will be a superposition of the different modes with varying amplitudes determinedby the mode of excitation, time since the excitation etc.

One of the simplest string instruments, the string sonometer, is formed by a simplestretched string (open string) over a box-shaped resonance body. Figure 2 shows the soundproduced by a plucked and bowed sonometer tuned to A2

4 and demonstrates the richness of thesound produced by a vibrating string as many intense upper harmonics are detected. Duringbowing for example, the fifth, 16th and 23rd harmonics are stronger than the fundamental.

3 The frequency ratios given by the modes of vibration on a string are pleasing to the ear and form the basis of westernmusic (e.g. between the first and second mode, 2:1 or an octave; the second and third mode, 3:2 or a fifth). The seriesof modes with wavelengths λ = 2l/n and corresponding freqencies f = v/(2l/n) (where v is the velocity of thewave) is hence called the harmonic series, and the term mode and harmonic is used interchangeably.4 The names of the notes refer to the tempered scale, formed of 12 semitones per octave with a frequency ratiobetween each semitone of 12√2 = 1.059464. The A4 is defined as the tuning A at 440 Hz, and the index of the notegives the octave. For further information on scales and intervals see, e.g., [3].

92 J-M Bonard

The point and manner of excitation along the open string (i.e. the initial conditions)decisively influence the timbre: the spectra displayed in figure 2 differ markedly, especially inthe higher modes. By plucking a string, we impose an initial condition such that the shape ofthe string is triangular with zero and maximal displacement at the stops and at the position ofthe finger, respectively. The relation between position and intensity of the excited harmonicscan be easily predicted (but is not easy to reproduce experimentally). This relation is not assimple for the bowed string, since the bow imparts both displacement (a fraction of mm) andvelocity (∼0.5 m s−1) [14].

Finally, we always consider that the properties of the vibrating string are ideal. A realstring, however, has some stiffness, which causes an increase in the frequency of the highermodes with respect to an ideal string [5]. This can be detected on figure 2, especially forthe plucked string (presumably because of the larger displacement). As a consequence, theharmonics of a string are systematically sharp, be it for plucked [13], bowed [15] or struckstrings such as in a piano [11, 16].

Variation 2: vibrating air columns

The principle of wind instruments is more complicated than that of strings. The vibratingmedium is the air inside a pipe that acts as a resonator where a standing wave can form.The ends of the pipe determine the boundary conditions. At a closed end, the amplitude ofvibration is zero and the pressure variation is maximal (the displacement of the air and theresulting variation of pressure are in antiphase). Conversely, the pressure will remain constantat the end of an open pipe and the standing wave shows a pressure node and a displacementantinode. This is schematically shown in the insets of figure 4.

As a consequence, a complete series of harmonics can form in a pipe of length l open atboth ends with wavelengths equal to 2 · l/n with n � 1. If the pipe is closed at one end, thewavelength of the fundamental corresponds to four times the length of the pipe and only theodd harmonics of wavelengths equal to 4 · l/(2n + 1) with n � 0 are allowed [2–6].

Figure 3. Excitation systems for woodwind instruments: (a) the edge of a flue organ pipe and (b)the reed and mouthpiece of a clarinet.

The vibration of the air can be excited by different means. The most simple one isto produce an edge tone by steering an airjet over an edge (like the top of a bottle or theedge of the organ flue pipe shown in figure 3(a)). The edge forms an obstacle for the jet

The physicist’s guide to the orchestra 93

and generates periodic vortices at the mouth of the instrument. The vortices produce in turnperiodic displacements of the air molecules. When the edge forms the upper portion of a pipe,the edge tone is locked by resonance to the modes of the pipe. The pitch can then only bechanged by increasing the frequency of the vortices (i.e. by blowing faster) to lock the edgetone in a higher mode (which is exactly how flautists change the register and reach higheroctaves with their instruments). Such an edge excitation acts like an open end, as the vorticesinduce air displacement but no pressure variations.

The other means of excitation in wind instruments involve a mechanical vibration, thatof the performer’s lips for brass instruments or of a reed for woodwind. Similarly to the edgetones, the vibration of the lips or of the reed is locked to the resonances of the pipe. The simplereed of the clarinet (see figure 3(b)) and of the saxophone, and the double reed of the oboeand bassoon, actually acts as a pressure regulator by periodically admitting air packets into thepipe. A reed is therefore equivalent to a closed end as it produces a pressure antinode.

-160

-140

-120

-100

-80

-60

-40

0 2 4 6 8 10 12 14

0 1 2 3 4 5 6 7

Inte

nsity

[dB

]

Harmonic

open

closed

0 0.005 0.01 0.015 0.02

0 0.0025 0.005 0.0075 0.01

Inte

nsity

[u.a

.]

Time [s]

open

closed

Figure 4. Waveform and corresponding spectra of a closed and open organ flue pipe (B3, 235 Hzand B4, 470 Hz, respectively). The timescale of the upper waveform has been divided by two withrespect to the lower waveform. The insets show the first three vibration modes for the variation ofthe pressure.

We can verify the above principles with a square wooden flue organ pipe of 0.35 m length.The excitation system is reproduced in figure 3(a) and acts as an open end, and the other endcan be either closed or open. As shown in figure 4, the fundamental of the closed pipe is foundat 235 Hz, which corresponds well to a wavelength of λ = v/f = 4 · 0.35 = 1.4 m withv = 330 m s−1. The waveform is nearly triangular, and the even harmonics are far weakerthan the odd. The same pipe with its end open sounds one full octave higher (the wavelengthof the fundamental is shorter by a factor of two) and displays a complete series of harmonics.

94 J-M Bonard

0 0.005 0.01 0.015 0.02

0 0.0025 0.005 0.0075 0.01

Time [s]

voice “aa”

saxophone

clarinet

flute

recorder

violin

Inte

nsity

[a.u

.]

-100

-50

0

50

100

150

200

250

0 4 8 12 16 20Harmonic

voice “aa”

saxophone

clarinet

flute

recorder

violin

Inte

nsity

[dB

]

Figure 5. Waveform of a violin, recorder, flute, clarinet,saxophone and of the author singing the french vowel ‘aa’(as in sat) (A4, 440 Hz for the former and A3, 220 Hzfor the latter three instruments, at mf sound level). Thetimescale of the upper waveforms has been divided bytwo with respect to the lower waveforms.

Figure 6. Spectra corresponding to the waveforms offigure 5.

Variation 3: tutti

We are now ready to study the behaviour of most strings and woodwind. Figures 5 and 6show the waveforms and spectra of six different instruments. Table 1 also summarizes thecharacteristics of the woodwind instruments studied here. A quick glance shows numerousdisparities between the instruments, and we will try now to understand these timbre variationsand their origin in more detail. Note, however, that the character of an instrument cannotbe reduced to one note in the permanent regime. As we will see below, the timbre of theinstrument can change drastically with the pitch and depends furthermore on other factorssuch as the sound level [10], the acoustics of the room or the placement of the microphone [9].

Table 1. Characteristics of the woodwind instruments studied in this work.

Instrument Bore Excitation

flute cylindrical edgeclarinet cylindrical single reedsaxophone conical single reedrecorder cylindrical edge

The physicist’s guide to the orchestra 95

Variation 4: the violin

The violin produces a very rich sound with at least 20 strong harmonics and complexwaveforms, as was the case for the string sonometer in figure 2. The strongest mode isnot the fundamental, but the seventh harmonic in the case of figure 6.

Variation 5: the flute

As can be seen in figure 5, woodwind shows simple waveforms and spectra when comparedto string instruments. The flute (flauto traverso) is a textbook example of an wind instrumentwith open ends as the pipe is (nearly) cylindrical over the whole length. The most salientfeature of the flute is the limited number of harmonics (∼7) with an intensity that decreasesmonotonously [17, 18]. The timbre is also very similar for the first two registers (not shownhere).

Variation 6 (menuetto): the clarinet

We have seen that a pipe closed at one end shows only odd harmonics. The clarinet, with itssimple reed and (nearly) cylindric bore, should be a prototype of such a pipe.

At first sight, this is indeed the case. In the low register (figure 5 for an A35), the odd

harmonics are clearly the strongest modes. The even harmonics, although present, are stronglyattenuated (at least up to the sixth harmonic). There are other marked differences with the flute.First, the sound is far richer in higher harmonics. Second, the waveform varies considerablywith the pitch as displayed in figure 7. The contents of higher harmonics strongly decreasesfrom 20 for the A3, to 9 and 5 for the A4 and A5. The contribution of the even harmonicsbecomes also increasingly important. The third mode remains more intense than the secondfor the A4, but this is no longer the case for A5. The three notes considered here are producedwith different fingerings, but the same tendency is observed with notes obtained with identicalfingerings in different registers (e.g. B3, F5, D6). Note also that variations in the sound levelinduce considerable changes in timbre [10].

The clarinet thus shows a fascinating behaviour: it responds like a pipe closed at oneend in the lower register but gives a sound with strong even harmonics in the higher registers.The timbre, therefore, varies as the pitch is increased, with a very distinctive sound for eachregister. This is due to several facts. First, the bore of the clarinet is not perfectly cylindricalbut has tapered and slightly conical sections [2]. Second, the flared bell, the constrictingmouthpiece (figure 3(b)) and the toneholes (even if they are closed) perturb significantly thestanding waves. Third, the transfer function of the pipe of the clarinet shows maxima for evenharmonics, and weak even harmonics produced by the reed can become significant. Finally,for wavelengths comparable to the diameter of the toneholes, the sound wave is no longerreflected at the open tonehole but continues to propagate down the pipe. This correspondsto a frequency of ∼1500 Hz in typical clarinets [2, 19], and the sound will show increasingnumbers of even harmonics with increasing pitch, as found in figure 7.

Figure 7 leaves out one important feature. The clarinet does not change from the first tothe second register by an octave (i.e. by doubling the frequency), but by a duodecime (triplingthe frequency). This feature is due to the excitation system alone (the reed acts as a closedend), as can be easily demonstrated by replacing the mouthpiece of a flute (or of a recorder)with a clarinet mouthpiece mounted on a section of pipe such that the overall length of theinstrument remains identical. The instrument sounds a full octave lower and changes registers

5 The clarinet is a transposing instrument, meaning that a written A3 does not sound at the pitch of an A3. Mostof today’s clarinets are either B or A instruments, implying that a written C sounds like a B or an A, respectively.When not specified, the pitch given in the text corresponds to the actual sounding pitch.

96 J-M Bonard

-100

-80

-60

-40

-20

0

20

40

60

0 4 8 12 16 20

Inte

nsity

[dB

]

Harmonic

A3 - 220Hz

A4 - 440Hz

A5 - 880Hz

0 0.005 0.01 0.015 0.02

Inte

nsity

[a.u

.]

Time [s]

Figure 7. Waveform and corresponding spectra of a clarinet (A3, A4, A5 at 220, 440 and 880 Hz,respectively, played mf). The timescales of the second and third waveforms have been divided bytwo and four with respect to the lower waveform.

in duodecimes, not in octaves. The reverse effect can be demonstrated by mounting a flutemouthpiece on a clarinet.

Trio I: timbre quality

It appears from figure 5 that one can recognize an instrument family by its waveform or spectraat a given pitch in controlled experimental conditions. It would be tantalizing if one could alsorecognize one clarinet from another, for example to choose and buy a good instrument.

Figure 8 shows the spectra of my three clarinets playing the same written note (i.e. thesame fingering), with the same mouthpiece, reed, embouchure and loudness. The uppercurve corresponds to a wooden B student model. The two lower curves were obtained withprofessional (pro) grade B and A clarinets. I can identify each instrument by playing afew notes from the produced sound and from muscular sensations in the embouchure andrespiratory apparatus.

At first glance, the spectra of the three clarinets are readily comparable. Closer inspectionshows that the spectra begin to differ from the 10th harmonic on! There are actually fewervariations in relative intensity between the two B instruments than between the two proclarinets. The pro B seems to be slightly richer in harmonics than the student model, butthat difference was not systematically observed. In contrast, the A has no strong harmonics

The physicist’s guide to the orchestra 97

beyond the 11th in figure 8, and this lack of high harmonics was found for different pitches.This leads to the conclusion that the B and A clarinets are (slightly) different instruments(many clarinettists will agree with that point). The measured differences between two Bclarinets remain quite subtle despite the huge and easily audible difference in timbre.

Figure 8. Spectra of the written C4 of a student B and a professional grade B and A clarinetplayed with the same fingering, mouthpiece and reed (sounding B3, 235 Hz, and A3, 220 Hz,respectively).

Trio II: tone quality

Is it possible to tell apart a good from a bad tone? This question is of utmost importance forevery musician to obtain the desired tone quality. Figure 9 shows two clarinet tones obtainedon the same instrument, with the same fingering, mouthpiece and reed. The first is a goodtone: one could describe it as fullbodied, agreeable to the ear. The second is a beginner’stone: emitted with a closed throat and weak. The difference is instantly audible but difficultto quantify from figure 9. The variations appear again in the higher harmonics: the bad toneshows no harmonics beyond the 12th, which is at least five modes fewer than the good tone.

Figure 9. Spectra of a good and a bad sound on the clarinet (B3, 235 Hz).

98 J-M Bonard

It is quite astonishing that the quality of the sound is determined by the presence (orabsence) of high harmonics with amplitudes that are at least 40 dB (a factor 104) weaker thanthe fundamental! Musicians are sensitive to very subtle effects which are difficult to (a) linkto a physically measurable value and (b) to quantify precisely. Conventional statistics haveproven ineffective for classifying the sound quality. Interestingly, efficient solutions based onneural networks have recently been demonstrated [20].

Variation 7: the saxophone

The saxophone is a single-reed instrument with a truncated conical bore, and a short glance atfigures 5 and 6 shows that the timbre of the saxophone is very different from that of the flute orthe clarinet. The even harmonics are as strong as the odd [21]. The sound remains very rich inharmonics even in the higher registers, far more than for the clarinet, and the timbre changesonly slightly between the first and the second registers. The saxophone does not behave at alllike a clarinet!

The main reason is the form of the bore. In a cone, the standing waves are not plane butspherical [2, 21–23]. This has profound implications for the standing wave pattern [22]. Inshort, the intensity of a wave travelling down or up the pipe is, in first approximation, constantalong the pipe, which implies that the amplitude scales with the inverse of the distance to thecone apex. The waves interfere to form a spherical standing wave with pressure nodes separatedby the usual half-wavelength spacing, but with an amplitude that varies as the inverse of thedistance to the cone apex. This is true for a closed as well as an open end [22]. A conicalbore shows, therefore, a complete harmonic series, be it excited with a reed, the lips or anedge [23]! It would seem also that the conical pipe of the saxophone favours the higherharmonics as compared with the cylindric bore of the clarinet.

Variation 8: the recorder

To predict the spectra of the recorder should be easy. Since the bore is nearly cylindrical, it isexcited by an edge and should therefore behave in a way similar to the open organ flue pipe. Iexpected a limited number of harmonics and a full harmonic series. Figure 5 shows that I waswrong. The alto recorder indeed has a limited number of harmonics, and a similar timbre inthe two registers. But it shows the spectrum of a closed pipe—the even harmonics are moresuppressed than for the clarinet—and despite that it changes registers in octaves!

What is the explanation for the odd behaviour of the recorder? The player generates anairjet by blowing into a rectangular windcanal, which is then cut by the edge (see figure 3). Itappears from calculations that the position of the edge relative to the airjet influences criticallythe intensity of the different harmonics [24]. When the edge cuts the side of the jet, the fullharmonic series is observed. The even harmonics are, however, completely absent when theedge is positioned in the centre of the jet, as is the case for most modern recorders (includingthe one I used). This of course does not affect the modes of resonance of the instrument:the second harmonic can be excited easily by increasing the speed of the airjet, which raisesthe pitch by an octave. It also follows that I have been very lucky with the open organ fluepipe—which follows the expected behaviour shown in figure 4 thanks to a favourable positionof the edge with respect to the airjet [24]!

Variation 9: a capella

We perform frequently with an incredibly versatile musical instrument, namely our voice.Few instruments have such varied expressive possibilities and ability to change the timbreand loudness. A sound wave is generated in the larynx by the mechanical vibrations of thevocal folds (commonly called vocal cords) that modulate the flow of air from the lungs to the

The physicist’s guide to the orchestra 99

-50

0

50

100

0 4 8 12 16 20

Inte

nsity

[dB

]

Harmonic

“ii”

“ou”

“aa”

0 0.005 0.01 0.015 0.02

Inte

nsity

[a.u

.]

Time [s]

A3 - 220 Hz

“ii”

“ou”

“aa”

Figure 10. Waveform and corresponding spectra of the author singing an A3 (220 Hz) on threedifferent vowels: the french ‘ii’ (as in this), ‘ou’ (as in shoe) and ‘aa’ (as in sat). The formants areindicated for each spectrum by a dotted line in linear scale for qualitative comparison [3].

vocal tract [4]. The motion of the vocal folds is similar to that of the lips of brass instrumentplayers, with the difference that the resonances of the vocal tract do not influence the vibrationof the folds. A complete harmonic series is generated, with a mode intensity that decreasesmonotonically. However, the timbre is essentially determined by the shape of the vocal tractthat acts as a resonator. Depending on the position of the tongue and on the mouth opening,the position and width of the formants of the vocal tract (the broad resonances, indicated infigure 10) can be varied and some harmonics produced by the vocal folds are favoured withrespect to others [3, 25]. Since vocal tract and vocal folds are independent of each other, thetimbre of the voice will change with the pitch for a given tract shape as the harmonics areshifted towards higher frequencies while the position of the formants remains constant.

The effect of the vocal tract shape is displayed in figure 10 for three vowels sung at thesame pitch (the formants are also indicated). The tongue is placed closed to the palate toproduce the ‘ii’: it is nearly sinusoidal with weak upper harmonics. The first formant peaksaround 200 Hz and decreases rapidly. The second and third formants around 2000 and 3000 Hz

100 J-M Bonard

are also visible. The ‘ou’ results from a single formant with a maximum around 300 Hz and aslowly decreasing tail: the waveform is more complex and richer in higher harmonics, givinga flute-like sound. The ‘aa’ is obtained with an open tract and is far more complex. The mostintense harmonic is the third because of the relatively high position (∼800 Hz) and large widthof the first formant. The second and the third formants are as intense as the first and give asignificant amount of higher harmonics to the sound.

Finale

We have seen that the physicist’s approach to musical instruments opens fascinating andcomplex possibilities. The classical examples (closed and open pipe, for example) are easyto reproduce, but one steps quickly into territory uncharted by the classical physics textbooks,which makes the exploration all the more exciting. We have also seen that instruments canbe identified by their timbre, but that it is quite difficult to tell two different models from oneanother and to classify the quality of the produced sound. It may be even more difficult (not tosay impossible) to examine the quality of an interpretation and to understand why well-playedmusic touches us so deeply.

Musical acoustics is a beautiful subject to teach at every level. Music appeals to everybodyand a lot of students play or have played at some stage an instrument: this often makes for livelydemonstrations in front of the class. It involves both wave mechanics and fluid mechanics inquite complex ways, and a simple experimental setup can offer direct and compelling insightsin the physics of sound production. I hope that this excursion in the basic physics of musicalinstruments will motivate some readers to include the subject in their curriculum and that itmay provide helpful material for those who already do.

Acknowledgments

I thank heartily the different people who either lent me their instrument or who took sometime to come and play in the lab: Ariane Michellod (flute), Severine Michellod (recorders),Stephan Fedrigo (violin—handcrafted by the performer!) and Lukas Burgi (saxophone). I amalso greatly indebted to Paul Braissant, Bernard Egger and Yvette Fazan, who maintain andexpand an impressive collection of physics demonstration experiments at EPFL, and who arenever put off by the sometimes strange requests of physics teachers. I wish also to acknowledgethe referee for very useful comments and suggestions.

References

[1] Leipp E 1975 Acoustique et musique (Paris: Masson)[2] Benade A H 1976 Fundamentals of Musical Acoustics (New York: Dover Publications)[3] White H E and White D H 1980 Physics and Music (Philadelphia: Sauders College)[4] Campbell M and Greated C 1987 The Musician’s Guide to Acoustics (London: J M Dent and Sons Ltd)[5] Rossing T D 1990 The Science of Sound 2nd edn (Reading: Addison-Wesley)[6] Fletcher N H and Rossing T D 1991 The Physics of Musical Instruments (Berlin: Springer)[7] Taylor C 1992 Exploring music (Bristol: Institute of Physics Publishing)[8] Hirschberg A, Kergomard J and Weinreich G 1996 Mechanics of Musical Instruments (Wien: Springer)[9] Benade A H and Larson C O 1985 J. Acoust. Soc. Am. 78 1475

[10] Laloe S and Laloe F 1985 Pour la science Mai p 73[11] Brown J C 1996 J. Acoust. Soc. Am. 99 1210[12] Matsres M F and Miers R E 1997 Am. J. Phys. 65 240[13] Smedley J E 1998 Am. J. Phys. 66 144[14] Broomfield J E and Leask M J M 1999 Eur. J. Phys. 20 L3[15] Halter C and Zufferey J-C 2000 Regard scientifique sur le monde artistique du son (Ecole Polytechnique Federale

de Lausanne)[16] Fletcher H 1964 J. Acoust. Soc. Am. 36 203[17] Klein W L and Gerritsen H J 1975 Am. J. Phys. 43 736

The physicist’s guide to the orchestra 101

[18] Smith J R, Henrich N and Wolfe J 1997 Proc. Inst. Acoustics 19 315[19] Benade A H and Kouzoupis S N 1988 J. Acoust. Soc. Am. 83 292[20] Fasel I R, Bollacker K D and Ghosh J 1999 Proc. Int. Joint Conf. on Neural Networks vol 3 (Piscataway: IEEE)

p 1924[21] Benade A H and Lutgen S J 1988 J. Acoust. Soc. Am. 83 1900[22] Ayers R D, Eliason L J and Mahgerefteh D 1985 Am. J. Phys. 53 528[23] Boutillon X and Valette C 1992 J. Phys. IV C1-2 C1-105[24] Fletcher N H and Douglas L M 1980 J. Acoust. Soc. Am. 68 767[25] Sundberg J 1982 Scientific American March p 77