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The physics of music

Article  in  Physics Education · September 1981

DOI: 10.1088/0031-9120/16/5/314

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Phys Educ , Vol 16. 1981 Prlnted In Great Brltaln

The physics of muslc

S R Hoon and B K Tanner

It has become an established tradition within the Physics Department at Durham University that a Christmas lecture is presented annually for the sixth forms in the area. These have proved so popular that despite three performances, several hundred school students have to be turned away. Last year our topic was 'The physics of music' and this proved immensely popular both with sixth formers and staff alike. In view of the very small number of articles on sound submitted to Physics Education and the lack of interest on the part of many teachers in the presen- tation of the physics of sound, we felt it worthwhile to make the text available to a wider audience. This article should be read in association with that of Professor Charles Taylor in which he sets out a model syllabus for the teaching of sound (Taylor 1979a). In his article Taylor was unable, for reasons of space, to demonstrate how many of the ideas presented related to real musical instruments although this material is excellently treated in his other writings (Taylor 1976, 1979b).

Stephen Robert Hoon is a senior demonstrator in solid state physics at the University of Durham. After graduating he spent three years teaching in Nigeria before taking up a post of research assistant at the University College of North Wales, where he subsequently gained a PhD. His research interests include magnetic materials, especially magnetic j u ids , and his other interests include guitar playing. Brian Keith Tanner is a lecturer in physics at Durham University. H e graduated from Oxford University and subsequently obtained a DPhil there. His research interests include magnetic domains and the use of x-ray topographic techniques to study interactions of domains and crystal defects. H e has written over 50 articles. He has played the organ since 1962, studying under John Webster and Nicholas Danby, and is currently assistant organist at Elvet Methodist Church, Durham.

In our lecture we concentrated on effects demon- strable in real musical instruments and attempted to relate the sound heard by the audience to the under- lying physics. A similar basis is reflected in this article and we hope it will provide a stimulus to teachers to use real musical instruments as teaching aids and also further the building of bridges between the arts and sciences.

It is a salutary exercise to display on an oscil- loscope the waveforms generated by a microphone. For half an hour prior to our lecture, as the audience assembled, we played a sequence of recorded music ranging from Praetorius to Oscar Peterson and asked the audience to attempt to correlate what they heard with what they saw. It was clear that the electrical signal amplitude increased with loudness and scratches on a record could be identified by their associated spikes in the waveform. However the detailed shape of the trace was extremely complex. Indeed, there appears little difference between that associated with a classical symphony and an orchestra tuning up. Yet the microphone responds to just the same fluctuations in air pressure that our ears detect. By some amazing process, our brain-ear system is capable of identifying from these pressure changes those associated with individual instruments playing together, audience coughs, extraneous traffic noises, etc. Although the details of the process are far from being understood, it is clear that the ear-brain system can detect changes in acoustic energy, i.e. loudness, and changes in pitch.

Pitch and frequency If we are to comprehend even the simplest properties of musical instruments we must know something about the propagation of acoustic waves through a medium. The medium of interest is of course air, capable of transmitting longitudinal vibrations or waves. Wave propagation is easily demonstrated using the time honoured slinky, a long steel spring of about 8 cm diameter. If isolated (transverse) pulses are sent down the slinky (figure la) it is apparent to all observers that the pulse velocity is independent of pulse amplitude and is a property of the medium. This result may be confirmed for an arbitrarily shaped wave packet (figure lb), that is the medium is nondispersive. The concept of wavelength i. may be introduced by the same demonstration, and reinforced by sending packets of waves of short wavelength down the slinky (figure IC). The slinky also allows the notion of frequency v to be introduced. It is not difficult to demonstrate that the relationship

U = V/" (1)

holds for the slinky if a camera is used to photograph the waveform and a stopwatch to time the interval

0031 -91 20 81 050300+ 12501 50 c 1981 The lnstltute of Physics

between reflections at each end. It must be remem- bered that acoustic waves in air are however longitudinal, and the demonstration may accordingly be repeated for this case, albeit rather less unequivocally to a large audience. These simple ideas of frequency, wavelength and velocity must now be carefully translated to explain the great richness and variety of sounds that are found in all musical instruments.

For a pure sinusoidal oscillation, pitch and frequency are closely related. The inverse proportionality of frequency to wavelength is simply shown by connecting a loudspeaker and oscilloscope to a high output audio sine wave generator. As the frequency is increased so the pitch can be heard to rise. Pitch and frequency are related logarithmically, i.e.

Pitch interval = K lg, (v2 /v l ) ( 2 )

where v1 and 11, are the frequencies of the two notes. K is usually taken as 1200, resulting in 1200 pitch units per octave or 100 per semitone. This unit is called the cent. This apparatus also lends itself to investigating the hearing range of members of the audience, nominally from 15 Hz to 25 kHz. In each of the lectures striking confirmation was made that age and gender determine the upper frequency threshold, young girls definitely hearing the highest frequencies.

The sine wave generator is of course a purely elec- tronic device. To demonstrate that the mechanically induced vibrations of a tuning fork are fundamentally no different from those produced by a vibrating loudspeaker a microphone connected to an oscil- loscope may be used. When the tuning fork is struck with a soft faced mallet, a pure 'thin' sounding note is heard and a highly regular sinusoidal waveform displayed on the oscilloscope. It is this purity of oscillation, almost totally devoid of harmonic content, that gives the tuning fork its thin, unmusical note. Its resonant frequency is determined by the shape, mass and elasticity of the fork material. It is interesting to observe that two frequencies are induced when the tuning fork is struck sharply with the hard face of the mallet. The high frequency corresponds to ringing in the fork rather than the mechanical vibration of its arms. These two frequencies may be seen to modulate one another and further decay with different characteristic time constants. The mechanical vibration is sustained the longer.

Vibration of strings Having introduced U, v and 1 for the travelling wave, the standing wave may now be discussed. The standing wave is basic to the understanding of all stringed instruments. It must be stressed that the

- 1 s + fl +2

* t

C

C

d

Figure 1 Wave motion on a string. a, Pulses. Wave velocity U = distance the pulse (or wave) moves in 1 S; b, Isolated wave of a single wavelength. Wavelength i. = I , the distance between equivalent points; c, Group of waves. Frequency v = number of wavelengths passing a given point in 1 S;

d, Standing waves. Equal number of waves travelling in both directions

standing wave is a special case where the number of waves per unit time propagating down the wire or slinky (figure Id) is equal to that reflected back and an integral number of the wavelengths is equal to the length of the string. Using the slinky once more the fundamental, second, third and fourth harmonics are easily and clearly excited. Although a standing wave transmits no energy, the air surrounding the string absorbs energy from it. In addition, as a consequence of the direct physical coupling to the bridge of the instrument, the string's vibration becomes damped, eventually, to a standstill. The forces on the bridge can be quite large and this may be readily seen from the slinky demonstration. Not only does the assistant at the stationary bridge-end have to counteract the tension in the slinky wire, he also has difficulty in keeping his node truly nodal!

The parameters which determine the pitch of a vibrating string are easily demonstrated using an acoustic or Spanish guitar. The use of the frets to shorten the vibrating length of the string and hence to increase the characteristic pitch of the string is immediately apparent; so is the use of the tuning pegs in the machine head to increase the tension in and hence the pitch of the string. Further, even from a casual glance, it is noticed that the open bass strings E, A and D are of larger diameter and mass per unit

301

a b

Figure 2 Human analogues of a, strong vibrator-weak resonator system and b, weak vibrator-strong resonator system

length than the open treble strings G, B and E. Thus for the same tension and length the former have lower natural resonant frequencies. A stroboscope may be usefully employed to 'freeze' the fundamental, second and higher harmonics. The harmonics are stimulated by suppressing the fundamental with a lightly applied index finger, in the best of classical guitar playing tradition.

Why should the fundamental be a preferred oscil- lation for a string instrument? The reason may be quickly deduced from energy considerations. As the higher harmonies correspond to higher frequencies and thus also higher string and/or air velocities, they also correspond, for a given string and air mass, to much higher kinetic energies. Thus, having the lowest kinetic energy, the fundamental is usually the easiest to excite. It also has an antinode at the centre of the string close to where it is commonly plucked or bowed.

String-resonator system Why does a guitar sound different from a violin, even though the violin is plucked and generating the same note? Why does a violin tuned to viola pitch not sound as mellow as a viola? The answer emerges from the observation that a vibrating string itself emits very little acoustic energy and can barely be heard. The bulk of the acoustic energy is emitted by the

302

resonator, the body of the instrument, and it is this which gives each instrument its individual character.

The stringed instruments consist of a strong exciter-weak resonator system in that the ultimate pitch heard is determined by the string vibration, not the natural frequency of vibration of the resonator. An anthropomorphic example is shown in figure 2a. A more scientific analogue is the coupled pendulum system consisting of one light bob and one much heavier bob, the latter being on a string approxi- mately twice the length of the first. They are coupled by a length of wire, typically 0.5 mm diameter, bent in a right angle so as to retain some flexibility. A little ingenuity enables the coupling to be slipped off and the two natural frequencies displayed. When coupled, the light pendulum is found to vibrate at the natural frequency of the heavy one.

This works fine provided that the natural frequencies of the two pendulums are not nearly equal. The classic demonstration with two identical bobs on nearly equal length strings, weakly coupled by an elastic band, is well known. If one pendulum is set in oscillation, energy is transferred to the other pendulum whose amplitude gradually increases. The amplitude of the first pendulum decreases until almost stationary from whence it again builds up its amplitude as the energy is transferred back. Just such a situation occurs when the note played on a stringed instrument corresponds to the natural frequency of

A m p l l f t e r xy p l o t t e r I 1 Figure 3 Schematic diagram of apparatus for investigation of the frequency response of the guitar

vibration of the air in the resonator cavity. Here there is a constant interchange of energy between string and resonator and the note is very unstable. This note is known as the 'wolf note' and is notoriously difficult to play satisfactorily. All the stringed instruments are prone to this effect, which is most serious in the bowed violin family.

The air resonance is not the only mode of import- ance in the instrument body. Various resonances associated with front and back plates and the frame are all important and lead to a non-uniform response as a function of frequency. In an article in this magazine Gough (1979) has described an investigation into the frequency response of the violin which he incorporated into undergraduate project work. We examined the response of the guitar as this is acoustically a more primitive and simpler instrument and, as has been shown, is an excellent pedagogic instrument.

The apparatus used is shown schematically in figure 3, the pick-up being placed on the back plate off-centre. The strings were damped so as to exclude the natural open string resonances and their harmonics. Figure 4a shows the response over a limited audio frequency range. Figure 4b shows a similar plot, taken under identical driving conditions, where the gramophone pick-up was replaced by a microphone above the air hole. Although similar, the traces show significant differences. Most notable is the strong resonance A in figure 4b which, though heard, cannot be seen in figure 4a. This corresponds to the fundamental air resonance with no vibration of the back plate and is equivalent to the wolf note.

The positions of the various resonances are extremely important in determining the playing quality of the instruments. Hutchins (1962) demon- strated that a good violin had a rather flat average

l

L

n

l l1

I l

300 6 00 Frequency I Hz +

Figure 4 a, Response curve from gramophone pick-up on the back plate; b, Response curve from a microphone located above the sound hole. Note that resonance A in b is absent from a and resonance B in a is absent in b

response over a wide frequency range with resonances associated with modes of the front and back plates occurring alternately and spaced by about a semitone. The complex structure of resonances in the violin arises from the complexity of the modal patterns which, unlike the guitar, are asymmetric. This asymmetry arises through the off-centre positioning of the sound post wedged between the front and back

303

plates of the violin and visible through one of the ‘f‘ holes. In more primitive instruments such as the guitar, which do not have the sound post, opposite sides of the belly vibrate in opposite phase and destructively interfere, thus reducing the sound intensity radiated. An example of a strong back plate resonance which destructively interferes and is not detected by the microphone is shown as B in figure 4a.

Like the violin, the guitar has a strip of wood called the bass bar attached to one of the plates, and visible through the circular sound hole. Unlike the violin, however, which has a bass bar running down the length of the instrument, the guitar bass bar is set at right angles to the strings, parallel to the bridge. The presence of the bar suppresses certain symmetric vibrations which interfere destructively and provides a certain asymmetry to the system. Fascinating holograms of the modes in both the violin and guitar can be found in Taylor’s (1979b) article.

Before leaving this section it is worth noting that wood is an extremely anisotropic material, variable in composition, and it is impossible to predict the exact resonant characteristics of an instrument. In her development of a new family of violins with optimum volume resonators, Hutchins used a similar apparatus to that described here to measure the frequency response of each section of the instrument prior to construction. By painstaking modification of each component a violin with the desired resonant properties can be manufactured.

Wind instruments After a little reflection one cannot but notice that there are far more wind instruments than string instruments. Compare the total number of instru- ments included in the wind organ, wood wind and brass families to those of the string and piano families. The great prevalence of wind instruments may well be related to two factors. Firstly the human lungs, wind tract and mouth are admirably suited-indeed designed-to supply the necessary air stream. Secondly wind instruments, in contrast to string instruments, are strong resonators (see figure 2b!). That is to say a small disturbance ‘tickles’ the whole air column into resonant vibration. Consequently strong resonators are, at least in the first instance, easier to fabricate, witness garden hosepipe bands.

The oscillation in a wind instrument may be stim- ulated either by blowing across an edge, as in the recorder, or by blowing through or under a reed, as in the oboe or clarinet. When air is blown over an edge, eddies or vortices are formed, effectively creating a noise source which stimulates some of the allowed resonances of the pipe above. The reed performs a similar function by vibrating as air is blown through it. The reed’s vibration is a conse-

304

quence of the Bernoulli effect due to air being forced through its restrictive dimensions, and the springy nature of its composite material. Figure 5 shows the allowed standing wave oscillations in closed pipes. It is apparent that if standing waves are to be generated, a displacement node and antinode must occur at the closed and open ends respectively. Tuning the pipe simply involves sliding the plunger or stop, forming the closed end, in or out. As a result, however, it is difficult to play a scale on a given closed pipe, and apart from bird whistles, the closed pipe appears to be confined to the wind organ alone.

Without exception all orchestral wind instru- ments-reed, wood and brass-are generically open pipes. It is seen from figure 6 that, for the same resonator length, an open pipe has a higher funda- mental frequency than a closed pipe. The simplest form of open pipe is that found in the organ. Its resonant frequency is determined principally by its length, but due to end effects this may be decreased, and hence tuned, by a tone or so by decreasing the size of the upper open end using a tuning flap. In the limit when the flap is completely closed, the resonant wavelength would double, corresponding to a closed pipe.

Reed pipes, oboes and brass instruments use brass, organic reed and the players’ lips respectively for the vibrators. The vibrator also doubles up as the bottom open end of the pipe and thus it is only possible to terminate the instrument as an open pipe. The flared bell on the lower end of brass instruments, as well as increasing the instrument’s volume, reduces the criticality of the resonant frequencies upon end effects. This is achieved by virtue of the bell partially matching the impedance of the narrow instrument bore to the surrounding air. Thus a trumpet player is able to insert a conical ‘mute’ into the instrument to provide a mellow sound without completely pulling the instrument off tune, as would occur if an organ pipe were treated in the same manner. Although the woodwind instruments are generically open pipe types, their harmonic behaviour is complicated by the interaction of the various holes in the pipe body with the vibrating air column. This interaction occurs whether the holes are open or closed.

It has already been stated that the pitch of a pipe is determined by its length. However, if the air in the pipe were replaced by say helium gas, then as the wave velocity in this new medium is greater than that in air, the frequency of the pipe should increase correspondingly. This change in pitch was clearly demonstrated in the Christmas lectures by one of US

inhaling helium and subsequently blowing it through a selection of organ pipes, whilst the other blew them normally for comparison. Of course, inhaling helium has a similar and hilarious Mickey Mouse effect upon the human voice. The reason for this is itself

Fundamental

A - 4 1 1 3

13f,l

1s t ove r tone

2 n d ; c r :... . , Figure 5 Standing waves in closed pipes

interesting. For if the pitch of the human voice is produced by changing the tension of the vocal chords, then changing the gaseous medium in the lungs and throat should not alter their resonant frequency. That is, the vocal chords are pitch generators. The reason that the voice increases in pitch is that the throat, mouth, larynx, etc, all act as resonant chambers, which are stimulated by the vocal chords. Thus the helium changes only the allowed resonances in these body cavities. The human voice is then produced by a strong resonator mechanism except, unlike the organ or reed pipes, the resonance is stimulated by vibrating strings, the vocal chords. Temperature and humidity also affect the velocity of sound and thus the pitch of a wind instrument. This explains the need to warm up all player-blown instruments or fully warm a concert hall prior to any musical performance.

Living in the north east of England, the authors can not leave this section on wind instruments without a special mention of brass bands. As bands often march should it not be possible to detect a slight Doppler

Fundamenta l

1 s t o v e r t o n e

2nd over tone

Figure 6 Standing waves in (cylindrical) open pipes

shift? The answer is in fact yes. According to the calculations of Kelly (1974), if a brass band marches towards an observer at a medium walking speed of about 3 mph (1.3 ms- l ) this should cause an increase in pitch of about 7 cents, i.e. 0.07 of a semitone. This is about twice the minimum pitch change discernible by the ear. By extension, but on a rather less peda- gogic note, if an orchestra playing in the key of F travels in an open top bus at 84 mph (37 ms") towards an observer by the roadside, the observer will conclude that they are playing in G as they approach him but that they dexterously change to E after they pass him and recede into the distance.

Harmonic content The differences in sound quality between various instruments playing the same note arise primarily because no musical instrument generates a pure sine wave at the frequency corresponding to the pitch of the note sounded. Real instruments generate wave-

305

forms that are a subtle mixture of the fundamental and higher harmonics, resulting in some cases in very complex waveforms.

Organists actually perform a Fourier synthesis of complex waveforms when registering organ music. The waveform from a stopped flue pipe is almost purely sinuosodial (figure 7a), much more so than an equivalent open flue pipe (figure 7b). (This arises because the first harmonic of the closed pipe is at the musical interval of a twelfth (3v) rather than the octave (2v) and the second harmonic is at the nineteenth (5v) rather than the twelfth-see figures 5 and 6. As the amplitude of excitation of the harmonics falls with frequency, more kinetic energy being required as the frequency rises, the closed pipe gives the purer note.) A rank of pipes corresponding to one stop on the organ consists of one such pipe for each note on the keyboard. This would produce a perfectly playable chamber organ but larger organs have many ranks of pipes, i.e. the possibility of sounding several pipes at once with the same key. In some of these ranks, known as 4 ft ranks (1 ft z 30 cm), the length of each pipe is half that of the fundamental rank (known as an 8 ft rank because this is the length of the largest pipe). The note emitted is thus twice the frequency of the fundamental (figure 7c), i.e. at the octave. Similarly, in the 2 ft rank each pipe is a quarter of the length of the fundamental and the emitted note is at four times the fundamental (figure 7d), i.e. the super octave. If the organist draws all three stops at once, on pressing any key, three pipes sound at once at frequencies v, 2v and 4v. The resulting waveform is shown in figure 7c. The ear recognises it as the fundamental note but the sound appears brighter than when the fundamental is sounded alone.

The quality or timbre of a given note thus depends strongly upon the presence and intensity of its various harmonics. Their presence is affected by many para- meters including the air speed over the exciting edge. Overblowing stimulates the second octave above the fundamental on all notes. Another important factor is the pipe resonator material and its thickness. For example, if the pipe is thin walled it can readily resonate like the belly of a guitar, and this conse- quently colours the note. It is of interest to note that organ builders find it impossible to build perfect reproductions of old organs, despite meticulous attention to the metallurgy and geometry of the pipes. The spotty metal used for their fabrication is typically a 50% alloy of lead and tin. This however is found to age over a period of many decades causing the organ to lose its new brightness and become more mellow. The mechanism whereby this occurs is unknown but it has been suggested that weathering of the pipe surface and metallic diffusional effects within the metal are responsible for these acoustic changes.

Fortunately the mellow tone of an old organ is considered by most cognoscenti to be more pleasing to the ear.

Reed pipes produce a waveform rich in harmonics, and having very little fundamental content (figure 70. This is not surprising in view of the abruptly interrupted wind supply produced by the vibrating reed. All members of the woodwind family containing reeds produce very complex waveforms. For the bassoon, in part of the register the fundamental is totally absent, the ear reconstructing this from the total waveform. (The predominant partial in the bassoon always lies in the range 329-523 Hz.)

It is not widely appreciated that the waveform of a given instrument changes very significantly through the register and figure 8 illustrates how the waveform of three adjacent notes on a treble recorder change. Note, in figure 8d, that changing the relative orientation of microphone and instrument leads to a major change in the harmonic quality of the note. Clearly uncovering the holes on a recorder is not simply equivalent to shortening the length of an organ pipe. It is in fact a very complex problem involving the change in acoustic impedance and the nonlinear coupling between excitor and resonator.

Transients One of the major differences between the sound emitted from a signal generator and that from a real instrument lies in the transient behaviour when the note is excited. The note from the signal generator comes on abruptly but in an organ pipe, for example, a significant time is required to build up the vibration of the air in the pipe. An example of this risetime is given in figure 9a. In the recorder, particularly when played by a semi-skilled player (BKT) the velocity of the air on the lip of the pipe is often not initially quite that required to give the correct edge tone frequency. There is complex interaction of pressure waves back and forth giving a period of instability before the vibrations settle down (figure 9b). Underblowing can result in prominent excitation of the first harmonic and this transient phenomenon during the time taken to build up the air velocity can often be detected in the coughing of some Bourdon pipes. The problem was accentuated by the English organ builders who, until recently, tended to use much higher wind pressures than their Continental counterparts, largely to simulate orchestral tones in certain stops. Old organs usually operate on a very low wind pressure and hence have very fast risetimes in setting up the steady condition of oscillation. Rapid passage work in the bass register, lost on many English organs, becomes clear and easy to articulate.

In addition to the initial transient, we notice that there is a low frequency (a few hertz) fluctuation in the

306

Figure 7 Waveforms from organ pipes for the note f': a, Figure 9 a, Initial transients on a B flat trumpet. Scale Stopped flue pipe (Lieblich Gedackt) at 8 ft pitch; b, Open 20 ms/division. Note the surprising asymmetry of the flue pipe (Open Diapason) at 8 ft pitch; c, Stopped flue pipe waveform about zero. b, Transients on the treble recorder (Concert flute) at 4 ft pitch; d, Flue pipe (Piccolo) at 2 ft (playing f'). Scale 100 ms/division. Note that stability is not pitch; e, Combination of 8,4 and 2 ft stops shown in a, c and achieved until about 0.3 S after initiation of the note d; f, Reed pipe (Cornopean) at 8 ft pitch. (Recordings made on the organ of Elvet Methodist Church, Durham)

intensity superimposed on the waveform (figure 9b). This results from the dynamic interaction between the player's lips and lungs and the instrument and is another subtlety difficult to simulate electronically.

The decay time is determined primarily by the acoustics of the building rather than the instrument. Sabine, in a classic experiment, determined that the reverberation time varies as the reciprocal of the acoustic attenuation and it is very straightforward to derive this relation theoretically. It is interesting to note that the reverberation times for buildings varying in volume by a factor of twenty, but all agreed to be ideal for performance, fall on a universal curve, the reverberation time varying roughly as the cube root of the volume of the hall. (For opera, a shorter reverberation time is desirable than for choral music, which in turn is also less than that for orchestral music.)

ours. African music makes extensive use of quarter and eighth tones. Why is much of our music written in the scales as we know them?

Our music has evolved from that of the Greeks, who used a system based on a four note pattern known as a tetrachord. Two tetrachords were then joined together to give the various modes. One of these modes, the Lydian, corresponds to our modern major scale, and proved to be the most versatile. Pythagoras (or at least the philosophers of that school) was the first person to form a scientific theory of the structure of this scale. Using just two intervals, the fifth and the octave, it is possible to arrive at all the notes in the major scale. The importance of these two intervals is that they correspond to simple fractions of the length of a string vibrating at the fundamental frequency v. It is easy to demonstrate with a guitar that placing the finger at exactly the midpoint of the string's length gives the note an octave above the

Scales and temperament fundamental (i.e. at 2v). Similarly by dividing the string in the ratio of 2:l gives the fifth (i.e. at $v). Simply

Have you ever noticed that Scottish folk tunes can be using these geometrically derived intervals one can played solely on the black notes of the piano? The cons&t the-whole major scale by jumping up a fifth, music is pentatonic, that is, it is built from a scale of down an octave, up a fifth again and so on. Starting on five notes. Schonberg's music was written for a scale middle C (denoted c' in the Helmholtz notation) the of 12 notes separated by semitones. Chinese music is sequence progresses c', g', g, d , d, a', a, e, E, b, B, F founded on a scale totally different in intervals from sharp. We now have all the notes corresponding to

307

Table 1 Relative frequencies of notes in major scales

Pythagorean

Natural scale v ;v :v $v :v :v q v 2 v

scale v ;v g v f v :v g v %v 2 v

U U U I I I ~ I I I Semi- Semi-

Tone Tone tone Tone Tone Tone tone

the scale of G major. With respect to the fundamental v, the frequencies of the notes in this scale (ascending) are shown in table 1.

The Pythagorean scale gives very complex fractions when expressed in this manner and, most import- antly, the harmonic partials of the fundamental do not form simple multiples of those of other notes, except for the fifth and fourth. (The fourth partial of the fundamental coincides with the third partial of the fourth, both at 4 v . ) When this harmonic coincidence occurs, the two notes played together sound harmoni- ous and pleasant to the ear. With the development of harmony, it was found that the Pythagorean scale was not satisfactory, particularly for the interval of a third, and in 1561 Zarlino suggested minor adjust- ments to produce simpler fractions in the frequency ratios (table 1). The relative movements of the frequencies of the notes are shown (exaggerated) in figure 10. This scale is now known as Natural pitch. Helmholtz claimed that choirs trained by the Tonic Sol-fa method always sang (when unaccompanied) Natural intervals though subsequent experiments on violinists suggested their intervals were nearer Pythagorean!

The problem with both these scales is that they are unsuitable for keyboard instruments. If one continues on from F sharp, we produce successively C sharp, G sharp, D sharp, A sharp, E sharp and finally B sharp which is very close but not quite equal in pitch to C. Similarly E sharp is not quite equal in pitch to F, produced if we take a fifth downwards from C. If the scale is built up using descending fifths and ascending octaves (instead of vice versa) we end up with the note B flat not quite equal to A sharp, A flat not quite equal to G sharp, etc. This proves no problem to a singer but it is rather difficult to construct a keyboard instrument with different keys for sharps and flats. Accordingly Zarlino suggested that if instead of having two sizes of tone (major and minor tones) one averaged these tones (figure 10) and used a mean tone temperament one could use the same note for sharps and flats. By the 17th century mean tone tempera- ment was in extensive use throughout Britain.

Mean tone temperament is fine if one remains in keys closely related to the original scale in which the instrument is tuned. In remote keys the discrepancies become serious and such discordances probably

explain why organ works by Bach in remote keys such as B minor always call for extensive use of mutation stops resulting in a very complex harmonic content of the waveforms. This tends to cloak the discord produced between partials of different notes sounded together. If, however, the octave is broken up into 12 equal pitch intervals of a semitone each (100 cents) it proves possible to play in remote keys without an unpleasant sound resulting. Bach wrote the 48 Preludes and Fugues (two in each major and minor key) to demonstrate the versatility of this Equal Temperament (figure 10). Pianos and organs are nowadays tuned in Equal Temperament. The intervals are different from the natural intervals but sufficiently close that the ear will accept them as concordant.

Psycho-acoustical effects We are now able to deduce some of the probable mechanisms whereby the ear-brain system decodes the complex sound of an orchestra into its instru- mental components. Recall what an amazing process this is, for the ear detects solely temporal pressure changes at the ear drum. One of the most important cues that the brain uses to identify an instrument is the finger print of its characteristic transients. Indeed if an instrument is recorded on tape and the transients removed by careful splicing, it can be extremely difficult or even impossible to identify the instrument. This is one reason why some of the simpler electronic organs imitating pipe organs sound so false. Their transient response bears little or no resemblance to that of the wind organ. Thus it would appear that the brain, in recognising an instrument, is performing some form of fast Fourier transform!

Perhaps the simplest example of a psycho- acoustical effect is that of beats. Beats are simply demonstrated by coupling two oscillators of fre- quency v1 and v 2 to an oscilloscope and loudspeaker in parallel. When v1 = v 2 a single trace is displayed and a single note heard. For v1 v2 a beat frequency AV = Iv1-v2( is heard in addition to v1 and v2. This beat frequency can be correlated to the observed modulation of the oscilloscope trace. This is however a modulation in amplitude alone and so is not a true Fourier frequency component of the signal. When AV is small the ear detects the two frequencies present in the spectrum and a periodic fluctuation in their net amplitude, the beat frequency. If AV is increased to around 80 Hz then the observer begins to assign a pitch to AV, now called the difference frequency. There is however no fundamental difference between a beat and a difference frequency, nor is any difference identifiable on the oscilloscope. That is to say the ear is nonlinear in its response. Cine films as detected by the eye form an optical analogy. If the frame speed is

308

U Sernl-tone

~~ -

Pythagorean "1- A

II m m II I m m h

AL h A Ah A AA A

1 N a t u r a l ~~

Mean tone 4 x X X ,~ ~ X x ; I x I I

Equal temperament 1 , 1 , , , L - = = = *

C C # D D # E F F # G G # A A # B C O b Eb Gb Ab Bb

P l t c h ( c e n t s ) - Figure 10 Schematic representation of Pythagorean, Natural, Mean Tone and Equal Tempered Scales. Relative differences are exaggerated

slow enough the picture is observed to flicker, but above a threshold value a continuous stable display is seen.

To demonstrate unequivocally that the ear detects difference frequencies the two oscillators may be set to inaudible frequencies of around 30 and 60 kHz. Nothing is heard until the high frequency is reduced so that the difference frequency is less than - 15 kHz. An audio frequency is now heard although the two oscillators are themselves still generating inaudible frequencies. (It has been suggested that automotive brake squeal is an example of two high frequencies beating to form an audible difference frequency.)

It is also interesting to feed a sine wave signal into a 2 inch ( - 5 cm) diameter transistor radio loudspeaker, slowly decreasing the frequency. At around 300 Hz the loudspeaker's output falls dramatically, to become almost inaudible. As 300 Hz is in the region of d (above middle C), it would seem impossible to listen to any bass melody or bass instru- ment on a small radio. Some hi-fi enthusiasts will at this point be nodding wisely, although not altogether correctly. For it is possible to hear the bass parts of an orchestra using combination tones. That is to say if the ear hears the second and third harmonics of an instrument, it will put in the fundamental of its own accord. The brain may also perform this correction when it is not needed! For example if notes of frequency 320, 420 and 520 Hz are generated simultaneously by three oscillators, then although the true interval for these frequencies is 20 Hz, the ear detects a pseudo-fundamental of 105 Hz and thus four notes of an E chord are heard (Taylor 1979b).

The notion of combination tones also explains why pitch and frequency are only clearly related for pure isolated frequencies. Once more than one is present they interact in a complex manner and the brain may become deceived. Further by recourse to both com- bination tones and the minimum pitch difference discernible by the ear-brain system it is apparent why the even tempered scale of the piano is an acceptable compromise to the ear.

An additional effect of interest is information loss due to high note repetition rates. This may be demonstrated by slowing down an apparently plain but fast recording of bird song. At half or even quarter speed as many as four or eight times as many notes may now be recognised. This represents a high frequency nonlinear corollary of the beat to difference frequency transition.

Memory, apart from the synthesis of combination tones, can also give very important cues to a listener. For example radio reception of a play or music can become so poor under adverse conditions that a listener can make no sense of it at all. If however he is told that it is his favourite play or concerto, the broadcast can become immediately intelligible. In the same manner almost disastrously worn and scratched 78 records may be tolerated and even enjoyed! Memory must also play a vital role for persons who possess perfect pitch. As has been shown, musical scales are really quite arbitrary, differing even between ethnic groups. Thus perfect pitch cannot be genetically determined. Related to perfect pitch is its antithesis, tone deafness. Musicologists confirm that memory is important here too by demonstrating that

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Figure 11 Waveforms from a Copeman-Hart electronic chamber organ for the note f‘. a, Stopped flute (8 ft pitch); b, Recorder (4 ft pitch); c, Block flute (2 ft pitch); d, Combination of the three stops shown in a-c. (Relative amplitudes maintained throughout)

tone deafness can be corrected by suitable music learning programmes.

Physiologists and medical physicists have often been recorded as saying that the ear is ridiculously sensitive and the brain used to but a fraction of its potential. Could it be that the sense of hearing and comprehending a complex musical work uses their capacities to a far greater extend than is currently realised?

Electronic music Certainly the greatest impact that electronics has had on music is in the recording and broadcasting of music. It has made great performances by great artists available to millions and created a popular music industry based on sales of recorded music. However, in synthesis of musical sounds, as opposed to simple reproduction, electronics has also had great impact.

The first application was in the electronic or ‘pipeless’ organ. Pipe organs are large and expensive to build and maintain and very substantial savings were to be made if organ pipe tone could be syn- thesised. The problem is one of synthesising the complex waveforms formed by small admixtures of harmonics. As a limited number of oscillators are available only an approximation to the pipe wave- form is possible. In the Hammond organ the first six and eighth partials alone are used, the relative intensities being adjusted to give the various types of organ pipe tone. Waveforms corresponding to some of the organ stops illustrated in figure 7, produced from a Copeman Hart chamber organ, are displayed in figure 11. Unfortunately in electric organs the subtle changes in harmonic content through the register are not synthesised and the tone appears uniformly flat on going from bass to treble registers. Further, the synthesis of transients is extremely

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difficult and has not been satisfactorily solved, though the Allen Digital Computer organ does provide ‘chiff which can be added to all stops, and the Copeman Hart organ has it on the stopped flute stop.

In addition to simulating other instruments, electronic synthesisers are used in their own right to produce a totally new range of sounds, many of which can be heard on popular television programmes. A typical small synthesiser will have three or four oscillators, filters, pulse shaping circuitry, white noise generator and mixers. These can be interconnected so that, for example, an oscillator can be coupled to the voltage control of a filter, allowing an oscillatory varying bandwidth of white noise to be let through. Most synthesisers are analogue devices but there is increasing interest in the synthesis of musical sound digitally. A number of projects are under way, includ- ing one at the Durham University Music Depart- ment. In this project a dedicated minicomputer is being used to convert, via a fast 16 bit digital- analogue converter, rapid trains of digital pulses into a varying waveform. There are very serious problems involving the volume of information to be stored and in reading it out as an unbroken train.

The Commodore Pet microcomputer can be programmed to play quite elaborate tunes if the CB2 (high current) output line on the parallel user port is fed through an external amplifier. The simplest of these is to feed the output CB2 via a 10 kR resistor to the base of a TIP 121 Darlington pair of transistors. The emitter is grounded and an 8 R speaker is inserted between the collector and an external voltage supply of typically +3-5 V. (The ground line of this supply must be earthed to the Pet’s ground line.) When CB2 is at digital ground, i.e. 0, the TIP 121 is turned off and no current flows through the speaker coil. When CB2 is outputting a 1, the TIP 121 turns on and current flows in the speaker coil, displacing the speaker cone. By rapid switching of the Pet output line one can produce a square wave of the appropriate frequency and quite complex sounds built up which can be quite audially spectacular.

Conclusion While not laying claim to great originality we hope that this article may provide a stimulus to further investigation. Despite its venerability, the book by Alexander Wood, originally published in 1944 and revised by Bowsher (1976), is a joy to read and a mine of information. In conjunction with the works already referenced, the enquirer can obtain open access to the literature. It is a field ripe for use in student projects. Much of the apparatus used in our exposition is available in schools, indeed a vast amount of teaching can be done simply with an instrument, a microphone and an oscilloscope. Use of real musical instruments,

Pnys Educ Vol 16. 1981 Prlnted In Great Britain

as opposed to sonometers and resonance boxes, is the key to the fascination. Physicists come to appreciate the subtle intricacies inherent in the art of per- formance, and musicians come to understand the reasons for certain mysterious playing techniques. To those of us fortunate to be both musicians and physi- cists the subject is the sweet synthesis of labour and love.

Acknowledgment The article stems from a lecture which would have been impossible without the help and encouragement of the technical staff of the department to whom our

NOTES ON EXPERIMENTS

'Notes on experiments' enables teachers at both sixth-form and tertiary level to share their ideas with other readers. Physics Education welcomes sub- missions from readers who know of some simple improvement to a commercially made piece of apparatus, or who have designed a new gadget or improved a standard experiment. In particular the Editor would welcome brief descriptions of exper- iments devised or procedures evolved during the course of project work or investigation undertaken by students; such submissions should be made under the joint name of the teacher and the student.

A U N I V E R S A L C U R V E F O R CASSETTE TAPE T IMES D W 0 H E D D L E Department of Physics, Royal Holloway College, Egham, Surrey

There have recently been a number of discussions of the nonlinear relationship between the reading of the index counter of a tape recorder and the elapsed time (Budden 1979, Gottlieb 1981, Jordinson 1980, McKelvey 1981). This nonlinearity is because the index counter records the rotation of the supply spool or the take-up spool rather than that of the capstan. The relationship is a simple one

t = (nxs / c ) (2r, - T X , ) (la)

or t = (nx,/c) (2ri + rx,) (1b)

where T is the thickness of the tape, c the linear speed

0031 -91 20.81 05031 1 +02$01 50 1981 The lnstltute of Physlcs

thanks are expressed. In particular we are grateful to Mr W Spalding for permission to use his illustrations in figure 2.

References Gough C E 1979 Phys. Educ, 14 318-22 Hutchins C M 1962 Sri. Am. November Kelly R E 1974 Am. J . Phys. 43 452-5 Taylor C A 1976 Sounds of Music ( London: BBC) -1979a Phys. Educ. 14 20-5 -1979b Contemp. Phys. 20 515-34 Wood A (revised J M Bowsher) 1976 The Physics of Music

(London: Chapman and Hall)

of the tape, r, and ri are respectively the outer radius of the full spool of tape and of the core on which the tape is wound. x , and x , are the number of rotations of the supply or take-up spool respectively (note that this is not necessarily the reading of the appropriate index counter). Equation (la) refers to the case where the index counter is driven by the supply spool and equation (lb) to the other case. If the total number of turns of the spool is X the total time Tcan obviously be expressed as

T = ( n X / c ) (2r0 - T X ) ( 2 4

or T = ( n X / c ) (2ri + 7 X )

Noting that the total thickness T X of tape on a fully wound spool is equal to ro - ri equations (1) and (2) can be combined to give expressions for the fraction of the time elapsed in terms of the corresponding fraction of the number of turns:

or ~ = 3 [ T X ro+ri 2ri I "(")l X r,+ri

and these equations can be written as

where A, + A, = 2. Because equation (4) involves the ratio of numbers of rotations the gearing between the shaft and the index counter does not matter and the nonlinearity can thus be expressed in terms of one parameter, A. While A can be calculated from measurements of ro and ri it may be easier to measure

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