Acoustics.

A term that can embrace all aspects of the science of sound and hearing, but is here treated in two specific senses, that of room acoustics, considered only with reference to the performance of music, and that of sound-source acoustics, limited to various classes of musical instruments and the voice. For other acoustical matters see Hearing and psychoacoustics and Sound; for the history of the subject see Physics of music.

RONALD LEWCOCK, RIJN PIRN (with JÜRGEN MEYER) (I), CARLEEN M. HUTCHINS (II, 1–6 (8 with JOHN C. SCHELLENG and BERNARD RICHARDSON), 9), J. WOODHOUSE (II, 7), DANIEL W. MARTIN/R (III), ARTHUR H. BENADE/MURRAY CAMPBELL (IV), THOMAS D. ROSSING (V), JOHAN SUNDBERG (VI)

Acoustics

I. Room acoustics

1. Introduction.

2. Reflection.

3. Resonance, reverberation and absorption.

4. Insulation against noise.

5. Radio and television studios.

6. Introduction to the history of acoustics.

7. Classical times.

8. Medieval times.

9. Renaissance and Baroque periods.

10. 18th and 19th centuries.

11. The science of acoustics.

12. The contemporary performance of early music.

BIBLIOGRAPHY

Acoustics, §I: Room acoustics

1. Introduction.

A room that has good acoustics is one in which it is possible to hear each sound clearly in all parts of the room; or, in other words, a room in which the sound is adequately loud and evenly distributed. In addition, it is normally required that the quality of sound being listened to in the room should match the type of sound being produced by the source. Room acoustics are relied on in some cases to sustain the sound in the room after the original source has stopped producing it, thus masking unevennesses in the ensemble, while in other cases sound too much sustained would mask the clarity of individual instruments or small groups. Acoustical problems are further complicated if opera is to be performed, for here every syllable is expected to be clearly heard and understood, and therefore only moderate sustained sound is desirable, yet the large ensemble demands sustained sound. Although scientific study permits a certain degree of accuracy in acoustical design, great difficulty is still experienced in determining the correct specification of the acoustics that ought to be provided.

Acoustics, §I: Room acoustics

2. Reflection.

Sound travels across a room in the form of vibrations in the air. Inevitably the amount of energy is diminished as the sound waves spread across the room, which means that there is a limit to the distance an average sound will travel without becoming faint. For increased loudness one normally relies on reflections from walls, ceiling and floor to augment the direct sound arriving at the ears. These reflections are also the source of the reverberation of sound in the room (see §3 below).

Sound can conveniently be thought of as spreading out from its source along straight paths and, like light, casting a shadow when it meets an obstruction (fig.1a). But the nature of this shadow depends on the relationship between two quantities, the wavelength and the dimension of the obstruction. The waves ‘bend’ at the edges of the obstruction, so that if the wavelength of a sound is large compared with the width of the obstruction, practically no shadow is formed (fig.1b). This condition is easily achieved with low-pitched sounds and the objects or screens in a normal room.

Sounds can be focussed to a point by concave reflectors, in the same way as a headlight beam is focussed, or spread out by convex reflectors so that their effect is diminished (fig.2a and b). Because the wavelengths of sounds are so much longer than those of light, the sizes of the reflectors needed to perform these tasks are quite large. For middle C an adequate size would be about 2·5 m.

Similarly, sound can be reflected by a plane wall in just the way light is reflected by a plane mirror. There is an ‘image’ formed behind the wall, which acts as the imaginary source for all sound reflected from the wall. As with light, the angle of incidence is equal to the angle of reflection (fig.2c). Very small reflectors do not work effectively for fundamental sounds, and reflectors for even the higher instrumental sounds need to be relatively wide; for the lowest range of notes reflectors more than 6 m wide are necessary.

If there is too much reflection in a room the sound may be loud and reverberant, and the endless reflections produce booming effects. Or concave reflectors may focus sounds so that some areas of the room receive little or none. Examples are the use of a curved wall behind an orchestra, which produces ‘sound foci’ in parts of the audience (fig.3a), and the use of a hard domed surface on the ceiling of a ballroom; in fig.3b the curved ceiling of the section produces the maximum focussing effect and by corollary the maximum area of diminished sound, sometimes known as a ‘dead spot’. A further example is that of the curved rear wall in an auditorium, which may concentrate sound back on the source or on the people in the front rows (fig.3c).

Echo is one of the most serious problems introduced by reflections. Fortunately it occurs only when there is a pronounced audible gap between the direct sound and the first reflection (or between two reflections). In other words, an echo is a discrete reflection that stands out over and above the other reflections. Ability to hear echoes varies with the individual, but a time interval of 0·08 seconds can be perceived by most people as an echo in music (as compared to less than 0·04 seconds in speech), and has therefore to be avoided. The distances travelled by the two sound paths would have to differ by 27 m before this time interval would occur in music (13·5 m in speech) (fig.4c). This means that any reflector behind the source or behind the listener and more than 13.5 m away is potentially likely to produce an echo (fig.4a and b). Reflectors in side walls or ceilings can generally be further away before they produce echoes, as the difference in length between direct and reflected sound paths is less than in the former case.

A complicated echo occurs when two reflective walls, or a reflective floor and ceiling, are exactly parallel and opposite each other. The difference in paths of travel of the sounds necessary to produce the echo is then formed again and again, resulting in a multiple or ‘flutter’ echo (fig.4d). This is particularly disturbing to the person producing the sound, but may also be heard by members of the audience. For this reason it is normal practice to ensure that reflectors are not exactly parallel; the deviations from the true parallel need not be so much that they are seen.

Two of the problems in room acoustics may be solved with the aid of properly designed reflecting surfaces. The first is the transmission of sound from the front to the back of the room so that it may be heard with reasonable loudness (yet without introducing any artificial coloration, as would almost inevitably happen with electronic amplification). The second is the problem of uneven distribution of sound, which is dealt with by ‘diffusion’, with the aim of producing a ‘diffuse sound field’.

(i) Transmission and the design of reflectors.

The transmission of sound from the front to the back of a room is normally aided by specially designed reflecting surfaces. As an example, consider a recital room of small size, with a flat floor, fairly low ceiling and a raised podium for the performer. It is shown in fig.5a before the reflectors are designed, in fig.5b after reflectors on walls and ceilings have been calculated and in fig.5c after additional angled reflectors have been added to strengthen the sound. The remaining surfaces are not useful areas for strengthening the loudness of the sound, and indeed may be dangerous if they are left as flat reflectors, introducing echoes or sounds that are too prolonged. For this reason the remaining areas of wall and ceiling are usually treated as absorbent or diffusing surfaces (see below).

In larger rooms the shape of the walls, of the ceilings, and even of the floor, may be determined by acoustic needs. Shaping the floor is usually thought necessary when the audience numbers more than 100, and desirable even when it is only 50. The audience seats are raised on tiers so that sound can travel unobstructed to the ears, passing over the heads of the people in front; this compensates to some extent for the greater distance sound has to travel. As an added improvement the musicians may also be raised on tiers so that they are unobstructed by performers in front. It is an old adage among designers that ‘if one can see well one can hear well’. Fig.6c shows a floor shape thus determined, and fig.6d shows the plan of the seating so that the entire audience has a clear view of the whole source of sound. Three sources are shown (S, S₁, S₂), together with their images (I, I₁, I₂) produced by reflection from the various reflecting surfaces, the images being constructed here geometrically.

Electronic amplification may be used when reflection is insufficient to produce a suitable volume, but the argument that there is concomitant coloration and distortion has tended to discourage its use except in the special case of electronic music or when quiet instruments (e.g. harpsichord, guitar) are required to sound well in a large hall.

(ii) Reflectors as diffusing surfaces.

Any rough surface will scatter sound waves, and hence ‘diffuse’ the sound field. Unless the roughness is pronounced, however, the sounds affected will be limited to those at the extreme upper end of the frequency scale. In order to affect sounds over the whole of the frequency range the roughness of the wall has to be of the order of at least 0·75 m and generally it is designed even larger. In the design of diffusing surfaces curved surfaces are often favoured, whether in concave sections, in convex sections or undulating (fig.7). Research has suggested that diffusing surfaces made up of rectangular parallelepipeds are equally efficient, but diffusion can also be achieved in quite different ways, by alternating small areas of absorbing and reflecting materials, or by the use of so-called stepped or profile diffusers, which consist of wells of unequal depth.

It is an ideal in acoustics to produce a ‘diffuse sound field’, so that the sounds reaching the audience are coming from every direction at equal strength. This ideal is never attained, but its approximation is important in producing predictable acoustical behaviour in a room.

Acoustics, §I: Room acoustics

3. Resonance, reverberation and absorption.

The property of sympathetic vibration is encountered in its direct form in room acoustics in the rattling of window panes, light shades and movable panels in the presence of very loud sounds, such as may occasionally be produced by a full organ. As these things rattle (or even if they do not audibly rattle) sound energy is being converted into mechanical energy, and so the sound is absorbed. Wood panelling and anything else that is lightweight and relatively unrestrained have the same effect. Absorptivity is at its highest at the resonant frequency, usually near or below 100 Hz.

Volume resonance occurs when standing waves are created by correspondences between the wavelengths of a fundamental sound and the dimensions of the room, and may result in uneven distribution of sound. This effect is at its worst in small rooms and becomes decreasingly serious in large volumes, where the dimensions are so great that they exceed the fundamental wavelengths of the lowest audible sounds.

A sound that is prolonged by multiple reflections around walls, floor and ceiling is said to have reverberated. The time of reverberation can be used as a simple yardstick to compare the capacities of different rooms for prolonging sound but for the yardstick to be practically serviceable, all the variables have to be specified. These include the frequency at which the reverberation is tested, and the range of loudness over which the decay is measured. Thus, for practical purposes, the ‘reverberation time’ is defined as the time taken for the sound in a room to die from 60 decibels to inaudibility (fig.8). It is customary to compare the reverberation times of rooms at ‘mid-frequency’ (an average of values measured at 500 Hz – just below c'' – and 1000 Hz), but for fuller comparisons reverberation time at 125, 250, 2000 and 4000 Hz are also used, to provide a composite picture of the prolonging characteristics of each room throughout the musical spectrum.

Reverberation is determined by the ability of sounds to bounce around a room for some time, that is, by the number and area of reflecting surfaces. A larger room naturally has sounds travelling for a longer period and the reverberation is more prolonged, though it can be reduced by replacing reflecting surfaces with absorbing ones. A reverberant room offers less clarity but is louder than a non-reverberant room, and vice versa. Analyses have been made of the acoustic characteristics of many concert halls throughout the world that are thought to have ‘good’ acoustics so that they may be compared and a synthesis of the optimum acoustic characteristics determined. The reverberation characteristics are summed up in the graph in fig.9. Using this it is possible to compare the reverberation of a projected room (calculated in advance by means of a standard formula) with the accepted aggregate norm. However, there are more recent additional criteria for satisfactory acoustics, which are discussed below.

Audience size affects reverberation markedly. In concert and recital halls where most surfaces are reflective, people are often the main absorbers of sound. In an endeavour to reduce the effect of this inevitably variable function, the seating is usually designed to provide a maximum of absorption when empty; it is covered with softly padded fibrous material, and the underneath surfaces perforated. But this is only a partial solution to the problem of varying audience size, for at middle frequencies the absorption of the seat is little more than half the absorption provided when a person is sitting in it.

The absorbing surfaces in a room vary in efficiency with the pitch of the sounds reaching them. High frequencies are normally absorbed by fibrous materials – woollen curtains or carpets, or specially designed surfaces incorporating fibrous materials. Sometimes cheap wood fibre blankets are placed behind perforated surfaces to achieve the same end. Glass fibre or slag wool blankets may also be used in this way, or wood fibre may be pressed into boards or tiles (‘acoustic tiles’) that are drilled or otherwise roughened to allow sound to penetrate into the material. Low frequencies are absorbed by using the capacity of resonant surface materials to absorb energy in the manner previously described. Resonant surfaces of this type usually depend partly on a trapped air space behind them; in other words, they are rather like sounding boxes, which, though never activated by enough sound energy to produce audible sounds, continue to resonate whenever small amounts of energy impinge on them. The resonating surface is usually wood or some flexible panel material.

An invention applying this resonating principle to absorb low-frequency sounds is the Helmholtz resonator, which uses the principle of sympathetic vibration of an organ pipe or an open bottle. A container, generally made of concrete or fibrous cement, is fixed behind the ceiling or walls, and connected to the room only by a small opening, or ‘neck’ (fig.10a). Helmholtz resonators are more frequency selective than resonant panels, and a series of them are used to correct specific peaks in the low-frequency spectrum. For this purpose holes are often left in some surfaces in a room when it is being built, enabling Helmholtz resonators to be inserted to correct unevenness in the acoustic spectrum, should that be necessary when the room is completed.

The Helmholtz resonator principle has been used in the design of a special panelled surface that combines the advantages of all three types of absorbent discussed above, absorbing sound over a wide frequency range. The Helmholtz resonator panel surface, or perforated resonating panel, has a dense surface material (compressed hardboard or asbestos cement) perforated with holes usually 3 mm in diameter spaced approximately 25 mm apart; to the volume behind it each perforation acts as the neck of a single Helmholtz resonator (fig.10b). The frictional resistance of each hole is often increased by gluing hessian across the back of the board. Whether this is provided or not, a layer of fibrous material (slag wool or glass wool, usually 2·5 cm in thickness) behind the holes provides considerable frictional resistance and absorbs resonant vibrations as they are set up in the air space and the panel. An important factor is the size of the air space (i.e. the distance between the panel and the wall behind it); this is approximately 13 cm ideally, and the absorption reduces in efficiency as it is decreased.

Another absorbing surface, which has the advantage of improved appearance though it is less efficient, is the strip panel resonator (fig.10c). This is made by fastening narrow strips of wood side by side, leaving small air gaps between them that act as the necks of individual Helmholtz resonators, although these operate only in one dimension of the surface (i.e. at right angles to the direction of the strips). The important dimensions here are those of the width of the strips (not much more than 2·5 cm), the width of the gaps between them, which should be between one fifth and one tenth of the width of the strips, and the depth of the air space, optimally 13 cm as above. Hessian and porous materials are fixed as with perforated resonating panels. The final appearance of the wood strips may be varied considerably; they may be shaped, patterned, painted or varnished without affecting the acoustic absorbent properties of the surface. Also, using the same Helmholtz resonating surface principle, many other absorbent devices are possible. A valuable derivative is the suspended absorbent cone, made of perforated hardboard or fibrous cement, which may be hung in rooms in which the walls and ceiling are difficult to render absorbent. Today, fibrous absorbers of varying thicknesses are widely used.

Besides absorption provided by the walls, floor, ceiling and furnishings of a room, and by the audience, reverberation is also affected by the absorption of the air in the room; in particular, high frequencies are absorbed if they travel considerable distances through air. But the effect of such absorption is only really noticeable in large spaces, especially when the air is dry.

Since an acoustical experience depends not only on the reverberation time, which tends to be fairly uniform throughout a room, but also on the strength, the timing and the direction of arrival of individual reflections, the sound in no two seats, let alone two widely separated areas of an auditorium, is exactly the same. The discerning listener knows this and selects a seat accordingly. Thus ‘perfect acoustics in every seat’ is unlikely to be achieved, yet halls that are known for their good acoustics tend to be accepted as such wherever people sit.

Achieving uniformity is often most difficult in small rooms. Great pains have to be taken to create the diffuse sound field necessary for good acoustics, by scattering the absorbing surfaces so that they alternate with reflecting surfaces in relatively small areas, and by the provision of broken, concave or convex diffusing surfaces. Two spaces connected to each other by an opening, such as the stage tower volume connected to an auditorium by the proscenium opening, may produce curious acoustical affects. Coupled volumes that are more reverberant than the auditorium have been designed to enhance the acoustics of halls in which adequate reverberation is lacking. On the other hand, openings to spaces that are less reverberant than the auditorium act as absorbers.

Reverberation can be measured in completed rooms by a number of methods using physical recording equipment, or by the subjective tests of trained observers. In models it can be measured reasonably accurately, provided care is taken to duplicate materials and surfaces at a smaller scale, or to make allowances for their omission. The patterns of wave distribution may be studied by the use of wave patterns on the surface of water when model sections of the room are placed in test tanks, or by spectrum photographs of the behaviour of sound inside a model placed in a smoke chamber. These time-tested modelling methods are now being replaced by computer programs that allow the designer to study complex reflection patterns and determine parameters, the calculation of which by hand would take many hours.

In severe cases of lack of reverberation, artificial reverberation may be introduced by distributing loudspeakers around the walls, floors and ceiling of a room, and relaying suitably delayed recorded sounds through them into the room. In the analogue era a specially devised tape recorder was used which allowed delays of a fraction of a second to be achieved between recording and replay. Modern digital audio systems not only create a longer and non-colorating reverberation in dry rooms, but add controlled early reflections to enhance the clarity or spaciousness of the sound using complex digital signal processing systems and numerous microphones and loudspeakers. While the result achieved is often a great improvement if too small a room is being used for orchestral, choral or organ music, its use tends to be confined to recording and broadcasting studios, because of the resistance of performers and audiences to artificial alterations of the natural sound in live performance.

At the Royal Festival Hall, London, unexpected deficiencies were found in the frequency spectrum of sound as a result of excessive absorption at certain frequencies. To correct this, artificial resonance was introduced by placing loudspeakers in resonant cavities closely resembling Helmholtz resonators, designed to resonate at the deficient frequencies, and activated by specially placed microphones. In this way the reverberation at many frequencies was increased without audible artificiality or coloration of sound. The many parameters which create an acoustically pleasing environment are constantly revalued. Concert halls and music rooms in general cannot be rank-ordered on an absolute scale of acoustical merit. Like the art for which they are built, they evoke responses that vary from person to person. Some prefer clarity, others a ‘large’ sound that can only be had at the expense of clarity. Musicians expect the room to respond, but they also want to hear each other.

Until recently, the main room-acoustics parameter used by acousticians was reverberation time. While reverberation time is still an important consideration, other, newly developed acoustical measures are thought to be at least as important. These include the clarity index C (the ratio of ‘early’ to ‘late’ energy, in which the boundary between early and late is 0·08 seconds after the arrival of the direct sound), the loudness descriptor G and a quality called ‘spatial impression’ which depends on the directional distribution of the energy. There is continuing discussion among acoustic researchers and consultants as to which measurements are most significant. Besides these objective measures, there are some purely subjective parameters to consider. A good example is the reverence for wood among musicians – a notion that finds limited scientific support, but on which musicians assessing room performance still rely heavily, perhaps through analogy with the rich resonances associated with wood in many musical instruments.

Acoustics, §I: Room acoustics

4. Insulation against noise.

Satisfactory listening is only possible in a room which is relatively quiet. Music rooms should have noise levels approaching inaudibility. Nevertheless, a very low noise level is not always ideal. In offices, for instance, some noise (typically that of the air-conditioning system) is often felt to be preferable to no noise, because in masking other unwanted sounds, such as voices from the next room, the sound contributes to a sense of acoustical privacy. For optimum acoustics in a room for listening to music, it is essential to achieve the lowest possible noise level. As well as quietening the ventilation system, this means insulating the room against two quite different types of noise. The fist is noise reaching the room through the air, whether from inside or outside the building, usually called ‘air-borne sound’. Ideally, external sound should be eliminated, but this is often both difficult and expensive, especially as it involves sealing windows and doors, which in turn necessitates air-conditioning the room. The second is sound generated in the solid material of the building, or in the ground or solid material of neighbouring buildings, known as ‘impact sound’. Of these two, the latter is by far the more difficult to cope with, and treatment of impact sound normally deals with many of the problems of air-borne noise.

Impact sound ranges from the noise of water falling in drainage pipes to slamming doors, footsteps, and the vibration of passing trains or buses. Internally all the floors of the building are usually designed to ‘float’ on insulating pads, so that impact noises are not carried into the structure. Alternatively, the floor material itself becomes the insulator, being soft and resilient, and thick enough to absorb vibrations; materials like thick cork or pile carpeting can be used in this way. Slamming doors and plumbing and drainage noises are either eliminated by careful design, or kept from the structure by thorough insulation. External vibration travelling through the ground and then up into the building through the foundations is much more difficult to deal with. Thick fibrous or rubber pads (when compressed remaining 15 cm thick, or even thicker) may be placed under the foundations to provide some improvement. In the case of the Royal Festival Hall, the whole auditorium was raised three storeys above the ground on tall slender columns that are thought to attenuate some of the solid-borne vibrations within their height (fig.11).

Air-borne noise ranges from sounds made by telephones and instruments playing in other parts of the building to chiming clocks, traffic noises and aeroplanes. In order to insulate the auditorium from all these noises it is sometimes found best to surround it with two completely separate skins of construction supported on separate foundations at ground level (see fig.11); it is thus difficult for any vibrations set up in the outer skin to pass into the inner skin, which serves as the envelope of the room. This insulation is effective only if the gap between the two skins is at least 30 cm, and its effectiveness is sometimes slightly improved by introducing into the gap a fibrous insulating material, such as fibreglass blanket. The points of weakness in such a design are clearly the windows and doors; windows may be sealed and double-glazed with a large gap between the panes (preferably no less than 10 cm), but this is not possible for doors, so they should be made airtight using gaskets, and a ‘sound trap’ in the form of a small sound-absorbent lobby must be provided at all points of access. As a final precaution the whole of the auditorium volume is surrounded by a blanket of other rooms and foyers, all treated with absorbent surfaces so that sounds passing through them are absorbed before reaching the auditorium, as in fig.11. Even with all these precautions, complete sound insulation is never achieved, and the principle of masking referred to above has to be relied on to disguise some external noise and air-conditioning hum. A large audience usually provides a natural masking level of between 20 and 30 decibels.

Acoustics, §I: Room acoustics

5. Radio and television studios.

The acoustics of radio and television studios follow the same general principles as normal room acoustics, with more rigid standards necessitated by: the special problems of recording and transmitting sounds via microphones; the double reverberation problem introduced by having one reverberation in the room of the broadcast or recording and another in the receiving room; and the outside noise, which has an even more serious effect on broadcast than on live sound.

Reverberation becomes a major concern in studio acoustics because too much of it reduces the definition and clarity of broadcast sound. On the other hand, elimination of reverberation would weaken the ‘character’ of the sound, and is, of course, inconceivable for the performance of music in ensemble. Of particular importance is the variation of acoustic behaviour in different parts of the room, which must be avoided if several microphones are to be used at the same time. For these reasons the reverberation is reduced but not eliminated, and great pains are taken to achieve even diffusion of sound.

The commonest type of studio in a radio broadcasting centre is the small announcing studio with a listening room alongside equipped for control, editing or playback. Of domestic room size, such studios provide spoken programmes of all kinds, and from them drama and music programmes are announced and monitored. A broadcasting centre has larger studios for drama, as well as a range of studios of different sizes for the performance of music; the largest are comparable in size with concert halls.

Stereophonic transmission does not greatly alter the acoustic requirements of studios, and as a rule the same studios are used, with different microphone placings. However, noise interference must be guarded against even more stringently, and precautions taken to avoid strong reflections altering the apparent position of a sound source; the latter usually involves increasing the areas of absorption at microphone level.

The design of small studios is affected by their tendency to add colorations to the sound; these disappear if the studio is made larger. A great deal has been learnt about the acoustic correction of this and other defects in small rooms (see Gilford, 1972). Briefly, this is done in two ways: by improving diffusion, ensuring that all walls, the floor and the ceiling have approximately the same absorption; and by careful measurement of the dimensions of a rectangular room so that prominent standing waves are separated from each other by intervals of 20 Hz or more.

Since all the available evidence suggests that the radio audience prefers music with the same balance and character as under live conditions, experiments with microphone placing have gradually ceased in favour of a placing that duplicates the ears of a single listener in the body of a reasonably sized concert room, and at a distance that produces a good blend and a natural reverberation. It has been found that the ratio of intensity of reverberant sound to direct sound should be approximately 5:8, and that the absence of an audience can be compensated for by moving the microphone closer to the source than it would be under normal conditions. With these assumptions, broadcast studios for music generally have the same design problems and solutions as concert halls. Care has to be taken to achieve adequate diffusion by using irregularities to scatter the wavefronts during reflection, and to avoid masking the bass by allowing too long a bass reverberation relative to middle and upper frequency reverberation. Predominant bass is sometimes avoided by placing special absorbers near loud brass and percussion instruments. For live audiences, experience suggests that the early energy, which includes the direct sound, should be a decibel or two lower than the late (reverberant) sound.

Acoustics, §I: Room acoustics

6. Introduction to the history of acoustics.

Although a good deal was known about the propagation of sounds and the acoustics of musical instruments from ancient Greek times onwards, the acoustics of rooms was imperfectly understood until well into the 20th century, and many traditional concepts and remedies are now known to have been false. Acoustic designs were empirical, successful in the evolution of efficient shapes but permitting the growth of a good deal of mystique where ‘resonance’ and ‘absorption’ were concerned. A particular misconception was the nature of resonance in room acoustics; it was thought that the principle of the sounding box of a lute or lyre could be transferred to architectural design, not taking into account the large amounts of mechanical energy needed if such ‘resonant chambers’ were to operate as amplifiers and enrichers of sound, energy far in excess of any that could be transmitted by the travel of sound through air. The result was that in many cases the ‘resonant chambers’ acted instead as absorbers of lower-frequency sounds, quite the opposite of the intended function. Equally fallacious were many of the attempts to absorb sound; it was thought by some designers until quite recently that any material soft to the touch would absorb sound energy, whereas many resilient materials are in fact poor absorbers (e.g. cork and rubber). Wires suspended around a room above head height were used to absorb excess sound energy, although they are now known to have had negligible effect because of their small cross-sectional area. The properties of wood panelling and large volumes of air as absorbers of sound energy were not generally known. Viewed with hindsight, therefore, the acoustics of the rooms for which much great music and opera was written must be expected to vary a great deal from what would now be considered satisfactory conditions. Nevertheless, the acoustics in those rooms that were considered excellent in earlier periods often approximate closely to those that would now be chosen for the performance of the same music. This is becoming clearer as famous churches and music rooms undergo thorough testing with recently developed techniques.

Acoustics, §I: Room acoustics

7. Classical times.

Before classical times there are only the vaguest references to buildings designed to allow music or speech to be clearly heard (e.g. royal courtyards for audiences or theatre in India were to be ‘built so that each sound [svara] and letter [aksara] should be audible’, Manasara, xxxiv, 506).

The basis of the modern science of acoustics was formulated in Greece by the Pythagoreans in the 6th century bce; Aristotle and his followers, the Peripatetics, continued its development as an empirical science. Aristoxenus of Tarentum (4th century bce), whom Horace called musicus idemque philosophus, examined the study of musical sounds from a physical–acoustical point of view, going beyond the origin and propagation of sound to consider the problems of perception by the human ear. The work concerning acoustics of Vitruvius (1st century bce), the Roman writer on architecture, is largely based on Aristoxenus’s writings (see below).

Greater understanding of acoustics at the beginning of the 5th century bce is exemplifed in the design of the first of the great Greek theatres, that of Dionysos at Athens (c498 bce), and its successor, the theatre at Syracuse (475 bce). The seats were arranged in curved rows round the circular orchestra, which provided a large horizontal reflecting surface to transmit sounds made by the chorus and by actors on the raised platform that served as a stage behind. Further high-frequency reflections were provided by the scenes, painted on skins, which are thought to have served as backdrop and wings to the stage. Because of the shape of the seating the furthest distance between the stage and the back row was only about 30 m. The propagation of sound to the audience was aided by the megaphone effect of the masks worn by the actors; besides providing facial expressions of nobility, anger or mirth, these masks improved the mechanical coupling between the vibrations within the actors’ heads and the surrounding air, thus enabling more of the available vocal energy to emerge in the form of sound waves and directing it towards the audience. A notable feature of later masks was the large opening of the mouth, which concealed megaphone-shaped cavities within the lips. Cassiodorus (51st epistle) remarked that the actors’ voices were so strengthened by ‘concavities’ that it was difficult to believe they could issue from the chest of a man. Acoustical devices were used in the design of the theatre to enhance vocal effects. For example, the space under the wooden stage platform, open to the audience and partly enclosed by removable wooden panels, appears to have been intended to act as a kind of resonant chamber. A similar but smaller volume in the middle of the orchestra housed an altar that was raised for thymelic spectacles but also played an acoustical role when the chorus chose to call down into it. It is thought that the narrowness of the stage was necessary to prevent the actors stepping so far back that the orchestra floor ceased to function as the main reflector, or so far forward that the backdrop lost its reinforcing effect on the higher frequencies (fig.12).

Later developments in the shape of theatres are all thought to have been due to attempts at improving the acoustics. In the second half of the 5th century bce, notably at Catania and Magnesia, theatres became more exaggerated in shape, with side walls converging not to the orchestra but to the stage. In the 4th century bce this trend was reversed with the erection of the great semicircular theatre at Epidaurus (fig.13a). The increased length of the seating area brought more of the audience close to the stage and thus improved the acoustics, especially as less of the sound could escape at the sides of the orchestra, and consequently more was retained in the volume of the auditorium. On the other hand, the direction in which the actors were facing became of greater importance, and the height of the stage building was increased and made of stone to provide more reflection from behind and improve the distribution of the sound. In the following centuries the older theatres were enlarged, both by extending the seating at the sides and by building more seats beyond the back rows to emulate and surpass Epidaurus. Eventually the distance from the stage to the rear seats became 70 m at Syracuse and as much as 100 m at Athens. The stages had to be increased in height and further framed to contain the sound.

In the late Hellenistic drama the chorus assumed a reduced role, while action and scenery grew in importance; the stage was enlarged and began to encroach on the orchestra, so that it no longer formed a complete circle. This was the model for the Roman theatre (fig.13b), in which the stage building was for the first time completely joined to the seating. Thus the whole building became one compact form, with a very high skēnē carrying a large slanting roof over the stage, which deflected sound towards the audience. The stage could now become much deeper, since the reflecting properties of the back wall were not of such importance for the clarity of speech. Similarly, a gallery was built over the back seats to deflect sound back to them. There were even examples of parabolic ceiling reflectors, apparently for directing the sound more accurately back into the auditorium (e.g. the theatre at Orange, 1st century ce).

In De architectura (1st century bce) Vitruvius discussed both the fundamental principles of acoustics and their application to the design of theatres (bk v, chaps. 3–8). In chapter 5 he referred for the first time to acoustic vases (echea):

Now in accordance with these researches bronze vases should be made in mathematical proportions [to each other], taking into account the size of the theatre: and they should be designed so that when they are excited they sound a series of notes at intervals of a 4th, 5th, and so on up to two octaves.
Then cubicles should be built among the auditorium seats on the basis of music theory, and the vases placed in them in such a way that they are not in contact with any of the upright stonework, and have a free space around and above; they should he placed upside-down, with wedges not less than [15 cm] high under them, on the side facing the stage. And in line with these cubicles openings should be left [in] the slabs of the lower rows, [62 cm] wide and [15 cm] deep.
The method of marking out the positions in which the jars are to be placed is as follows. If the theatre is not very large, a horizontal line should be marked out, halfway up the slope [of the auditorium], and 13 vaulted cubicles built, with 12 equal intervals between them: then the sounding jars as described above are placed in them [fig.14].
So by this arrangement, the voice, radiating from the stage as from a centre, spreads itself around [the auditorium]: and, by exciting resonance in particular vases, produces an increased clarity and a series of notes which harmonize with itself.

Chapter 5 elucidates how such a system of acoustic vases might be extended in a larger theatre, with three horizontal rows of cubicles, one for the harmonia, a second for the chromatic and a third for the diatonic (fig.15).

The term for ‘clarity’, claritas, is probably an equivalent for the Greek term lamprotēs, defined by Aristotle in De audibilibus, implying, besides distinctness, loudness and purity; and the context almost certainly implies a singing rather than a speaking voice. The function of the vases would have been to make some sounds louder than others, and to make them purer by stressing their fundamentals and suppressing their harmonics or overtones. The ‘series of notes which harmonize’ with the voice seems to refer to the fact that each vase would resonate and then re-radiate sound after the voice had ceased singing its fundamental note, so that if the concordant scale were sung, a number of the vases might be heard sounding together. In this way a kind of artificial reverberation time (estimated as 0·2–0·5 seconds) of particular quality would be produced in an open-air theatre that otherwise had none.

Although some 20th-century scientists (e.g. Knudsen) have dismissed the efficacy of such acoustic vases, others (e.g. Brüel) have attempted to duplicate their behaviour by direct experiment. As Vitruvius’s original diagrams illustrating the size and shape of the bronze vases were lost in late antiquity, these experimental vases were made in a wine-beaker shape of hard-burnt baked clay and with wider mouths than would be necessary to absorb low frequencies. The experiments showed that they enhanced reverberation at the resonant frequency, although they were not tested in an open-air theatre. There is a clear need here for further practical research. That resonance can be used to augment reverberation is confirmed by acoustical theory (see Gilford, 1972, p.155) although the amount of augmentation is limited. No remains of bronze resonators from antiquity have survived, which is scarcely surprising, since throughout the Middle Ages and the Renaissance ancient bronze was melted down for metal.

12 pairs of compartments corresponding to those described by Vitruvius have been found in the supporting wall of the uppermost row of seats of the Greek theatre at Aizani in Phrygia, eight in the podium of the Roman theatre at Nicopolis, and seven in the Greek theatre at Scythopolis in Syria. There are 20 niches in the upper part of the Greek theatre of Gerasa in Jordan; at Ierapetra and Gortyn in Crete the theatres have 13 niches each; and at Lyttos, also in Crete, there are three rows of 13 niches each (fig.16; Belli, 1854, Müller, 1886).

Vitruvius was at some pains to explain why acoustic vases were not used in the theatres of Rome built in his day. He said it was because of their wooden construction. Singers who wished to sound a loud note could direct their voices towards the scene doors (valvae) and ‘receive help’ from them; when the theatre was to be built of solid materials, such as stone or concrete, which do not resonate, then it should be equipped with the sounding jars described. According to Vitruvius there were many examples of theatres that used them in Greek cities and in the provincial towns. The theatre of Corinth was cited as a classic example; when Lucius Mummius sacked the town (146 bce) he carried off the bronze jars as part of his spoil, and dedicated them at the temple of Luna in Rome, where Vitruvius had seen three of them. Finally, Vitruvius mentioned that ‘many clever architects who have built theatres in small cities, from the want of others have made use of earthen vessels, yielding the proper tones, and have introduced them with considerable advantage’.

Besides bringing a reverberant response into the Greek theatre, it is possible that the acoustic jars helped an unaccompanied singer to keep to proper pitch for long periods. The vases resonating in various parts of the auditorium may also have served to disguise inferior musicianship by giving emphasis to musically important pitches. At the beginning of some early editions of Terence there is a short treatise in which the commentator, whose name is unknown, spoke of brass vases. He assigned to them the same use as Vitruvius, and then added:

I hear that there exists to this day something very like them, in some ancient temples, which have been preserved in their integrity down to our time. At the lower and upper parts of the roof are to be seen holes distributed on both sides, and corresponding diametrically with each other. In these holes are set vases of brass, the opening of which is smaller than the body, and is turned outwards, without projecting. The voices of those who sing in the temple, reverberating in these vases, grow more distinct and harmonious.

Acoustics, §I: Room acoustics

8. Medieval times.

With medieval acoustics one can turn from speculation about acoustic vases to the consideration of their tangible effects, for many examples survive in church buildings throughout Europe, from Russia to Britain (see Harrison, 1967–8); terms for them exist in most European languages. Although little modern research has been done on their effectiveness it appears that earthenware jars were used both as absorbers (see Brüel, 1947) and as resonators.

The methods of use of medieval acoustic jars may be roughly summarized as follows: (a) Areas of spaced jars in two or three rows inserted in the stone walls of the interior above ground level, usually about 2·5 m from the floor, with their mouths opening inwards to the nave or choir (e.g. Fairwell, near Lichfield; and St Clement’s, Sandwich, Kent). According to Viollet le Duc, in his Dictionnaire, many examples in France are placed near angles in the walls. (b) A single or double row of jars inserted in the stone walls just below the ceiling, trusses or vault, often extending down the full length of both side walls (e.g. St Nicholas, Leeds, near Maidstone, Kent – 48 or 52 vases). (c) Jars inserted at regular intervals across the stone barrel vaulting of the choir (e.g. St Martin, Angers; Bjeresjoe, Sweden – about 45 in five rows; and Montréale, France, according to Viollet le Duc). (d) Jars inserted in the sleeper walls below the choir stalls or in pits or cavities (e.g. St Peter Mancroft, Norwich – 40 jars; Fountains Abbey, York – seven jars). These are often separated from the volume of the church by the wooden flooring (e.g. St Peter Permountergate, Norwich – 16 jars) but in other cases a gap is left so that the jars are acoustically coupled to the air in the church (Church of the Cordeliers, Amiens).

The jars used were either specially manufactured (e.g. Leeds, Kent – about 50 jars with their bottoms perforated; and Luppitt, Devon – about six jars flattened on one side; fig.17), or else jars of ordinary domestic type, greatly varying in shape. Most of the jars were between 20 cm and 30 cm in length, and probably resonated at fundamental frequencies of between 90 and 350 Hz. They were 13–15 cm wide at the mouth; the mouths of the wider jars were often reduced in aperture by being placed behind perforated stone or wooden screens (Denford, Northamptonshire; and St Mary-le-Tower, Ipswich) or partly plugged with a wooden block (fig.18), but a number of vases appear never to have had any such constriction of the opening. The former seem to have been intended to act as absorbing resonators to reduce echoes in corners or in the vaults, the latter as resonators to enhance or assist sound. Some of the unconstricted vases in Scandinavia had peat or ashes in them, and it has been thought that they were intended to absorb sound by damping instead of re-radiating resonant sounds. It should be noted that many of the medieval vases were cemented into the masonry, and not placed loose in an air cavity like the Greek vases described by Vitruvius. This would not have prevented them acting as resonators, but it may have reduced their efficacy.

Archaeologists have frequently been sceptical about the function of these vases in spite of the existence of records testifying that they have been known as ‘acoustic’ or ‘sound vases’ since medieval times; they have often been considered relic vases, or their purpose was thought to be the drying of walls to protect fresco paintings, or they were assumed to have some structural purpose. However, such sources as the Metz Chronicle (1432) establish their function clearly: ‘il fit et ordonnoit de mettre les pots au cuer de l’église et pensant qu’il y fesoit milleur chanter et que il ly resonneroit plus fort’. Acoustic jars are therefore a valuable indication of the attitude of medieval designers to acoustics. At least some of the vases were clearly intended to add resonance and amplification to speech and music, although the Chronicler commented, after recording the Metz example: ‘je ne seay si on chante miez que on ne fasoit’.

Vitruvius was certainly the source from which the medieval use of these vases originated. Harrison cited 12 copies of his works known to have existed in England during the Middle Ages, and there were many copies available elsewhere in Europe. They appear to have been used throughout the medieval period and up to the 17th century. In a satire by Claude Pithoys, published in 1662 at Saint Léger, Luxembourg, he reproved the clergy for negligence of their duties: ‘Of 50 singing men that the public maintain in such a house, there are sometimes not more than six present at the service; the choirs are so fitted with jars in the vaults and in the walls that six voices there make as much noise as 40 elsewhere’. Little scientific research has been done on the actual effects of the vases in these churches.

The reverberation times of some important medieval churches have been carefully tested, and this has led to a better understanding of their acoustical characteristics. S Paolo fuori le Mura, Rome (386 ce), an example of a large early Christian basilica with double aisles, an open trussed roof and a transverse bema at the east end (fig.19), has a reverberation time at mid-frequency of 9·1 seconds in the nave. The walls and columns have hard smooth surfaces, leading to close acoustical coupling of all parts of the building, so that it functions to some extent as a single air volume. Nevertheless, some absorption is provided by the depth of the aisles, which scatter the sound so that it does not return to the nave; for this reason there is no echoing from the side walls. The acoustic result is a sustained sound in the church, but also a relatively clear one, with no confusion, echo or fluctuation in intensity. The low-frequency reverberation is more pronounced than reverberation at other frequencies, though the decrease to the higher-pitched sounds is a gradual one (R.S. and H.K. Shankland, 1971). It is likely that in its original form, before its rebuilding after a fire, the diffusion caused by ornament and fluting would have improved the acoustics even further.

S Paolo is an exceptionally large church subdivided by screens of columns, and some of its special acoustical quality is due to its size. Other large basilicas with similar screens that have been tested, such as S Maria Maggiore, Rome (352 ce), have been found to have even better acoustics, with shorter reverberation times, providing almost ideal listening conditions for choral and organ music when full. S Maria Maggiore has only single aisles, but there are chapels beyond the side walls, and a chancel arch separating the apse from the nave. The large volume is therefore broken up into a number of separate volumes coupled together, with the surrounding volumes absorbing sounds made in the nave. The measured reverberation times are 4·9 seconds when empty and 2·5 seconds when full (R.S. and H.K. Shankland, 1971). It must be expected that smaller basilicas did not achieve the same degree of separation between the volumes of nave, aisles, bema and chapels, and therefore the reverberation times were longer, and the acoustics less satisfactory.

With the arrival of the Romanesque style the height of the nave was increased, and stone vaulted ceilings were introduced to protect the interiors against fire. These ceilings increased reflections and reduced diffusion, leading to a significant change in acoustic quality; not merely was the reverberation time lengthened, but the focussing effect of the ceilings brought fluctuations in the reverberant sound, with a resultant decline in clarity. Sounds appeared to pile on top of one another to produce an effect of surging confusion. Gothic cathedrals suffered from the same kind of unruly acoustics, but conditions were often better in large buildings, where the great height of the ceilings reduced the interference of sounds reflected from them. Tests in Durham, Canterbury, Salisbury and York cathedrals have shown that they have remarkably similar acoustics, the reverberation times falling from an average of 8 seconds at low frequencies to 5·5 seconds at mid-frequencies, and continuing to decline as high frequencies are reached. All have volumes of more than 30,000 m³; in such conditions the presence or absence of the congregation has little effect on the acoustic quality (Purkis, 1963). Furthermore, the acoustics of very large volumes are often more satisfactory because there is seldom enough energy produced in them to excite the room.

There seems no doubt that long reverberation times were thought to enhance both music and prayer. Writing in 1535, Francesco Giorgi of Venice recommended that a new church should ‘have all the chapels and the choir vaulted, because the word or song of the priest echoes better from the vault than it would from rafters’. This was still thought to be true for singing in the 17th century (the music of Heinrich Schütz, for example, was carefully written to exploit the long reverberation of the Kapelle in Dresden). But an increased concern with clarity of speech during the sermon led architects as early as the 13th century to omit aisles and transepts altogether, and design churches with single volumes (e.g. S Francesco, Assisi; S Caterina, Barcelona). Even so, it was found that the reverberation continued to be pronounced until the vaults had been lowered and the ceiling flattened. The final improvement was replacing the stone ceiling with a wooden one, which absorbed the predominant low-frequency reverberation, and covering the surface of the ceiling with elaborate decoration of small ribs or coffering, which greatly reduced the fluctuation of sound by increasing uniform diffusion. Giorgi mentioned both these effects in making his recommendations of 1535. ‘In the nave of the church, where there will be sermons, I recommend a ceiling (so that the voice of the preacher may not escape nor re-echo from the vaults). I should like to have it coffered with as many squares as possible … they will be very convenient for preaching: this the experts know and experience will prove it’.

Spacious Gothic churches, such as those of the Netherlands and Germany built without transepts and with nave and aisles of equal height, often have excellent acoustics. On the other hand, late polyphonic music was frequently written to exploit the peculiar acoustics of the older churches. S Marco in Venice had two organs and two choirs by the 15th century; the two choirs, with the accompanying organs, were placed facing one another in the tribunes, halfway up the height of the choir, from which position the unusual acoustical effects of the cathedral could be used without too long an initial delay in reflections from the ceilings. The choral and instrumental groups were gradually multiplied, until as many as four choirs and four organs, with instrumental accompaniment, provided the means for achieving a unique kind of polyphonic vocal and instrumental music. Such a disposition of resources still remained in Salzburg cathedral in the late 17th century, when Biber wrote church music utilizing them.

Acoustics, §I: Room acoustics

9. Renaissance and Baroque periods.

Reformation church builders laid great emphasis on acoustic clarity, which suited sermons but necessitated adjustments in church music. Luther arranged his congregation around the sides of his churches, and later Gothic churches were altered by the Protestants to enable similar focussed seating arrangements; an example is the Thomaskirche, Leipzig, to which galleries and tribunes were added, with an especially wide organ loft gallery at the west end used by Bach for the choir and orchestra (fig.20). As the vaults of this spacious Gothic church were low (8 m), the introduction of the galleries created shallow volumes with short paths of reflection. The reverberation time was quite short when the church was crowded (1·6 seconds), with excellent diffusion and absorption of low frequencies due to wooden panelling, hanging draperies and carving; at the same time the high frequencies were bright and clear (Beranek, 1962).

S Pietro in Rome has a remarkably short reverberation time, caused by the combination of its exceptional size and its complex internal structure; in effect the basilica is five large churches interconnected and acoustically coupled, each damped by the air spaces leading into the others (R.S. and H.K. Shankland, 1971). Thus sound travels from the nave into the side spaces, where it undergoes extensive multiple reflection and delay before returning in a markedly attenuated form. The result is a reverberation time in the nave of 5 seconds at mid-frequencies when a large congregation is present.

During the Counter-Reformation one of the main aims was the design of churches in which every word of the service might be clearly heard. Vignola, the architect of the Gesù in Rome, was instructed to design the church with a nave as wide and short as possible, and without aisles, clearly with the intention of improving the acoustics. This was a quality that does not seem particularly to have concerned the Church of England, for Wren’s St Paul’s Cathedral in London has pronounced reverberation due to its relatively long low vaults and high central dome, and the fact that the nave, choir and dome do not function as acoustically separate volumes. When nearly empty the reverberation time is approximately 12 seconds at mid-frequencies, but it improves steadily as the congregation size increases, until it is 6·5 seconds at maximum capacity (Purkis, 1963).

The earliest Renaissance theatres maintained the forms, and therefore presumably some of the acoustic qualities, of the classical Greek and Roman theatres as described by Vitruvius; examples include Serlio’s Vicenza theatre (1539) and Palladio’s Teatro Olimpico, Vicenza (1588). An orchestra was added to the latter, seated on either side of the proscenium between the actors and the audience, but it apparently played only an occasional accompaniment or interlude. The extensive use of wood in the construction of these theatres, the use of elaborate decoration, and in particular the addition of wooden coffered ceilings, must have ensured good diffusion with brilliant high frequencies and rather dulled low frequencies. Allowing for the dense crowding of audiences which was common, the reverberation times must have been very short.

In the earliest operas, music was subordinated to the clarity of the text. It is known that in the first public performance of Peri’s Euridice (1600) the orchestra was placed behind the scenes. Cavalieri’s instructions for the performance of his Rappresentatione di Anima, et di Corpo in the same year were explicit. It was to be given in a theatre or a hall containing not more than 1000 spectators; the orchestra was to be situated behind the scene and be ‘adapted to the needs of each performance’, the latter presumably referring to acoustic conditions as well as other exigencies.

The masques held in the banqueting hall of Whitehall Palace early in the 17th century used musicians seated on either side of the stage at the front; this position appears to have been a common one, necessitated partly by the fact that the flat floor area between the stage and the raised seats of the audience was used for dance. There is a design for a masque house by Inigo Jones that has the same arrangement with the orchestra partly screened (fig.21). The Duke’s Theatre in Dorset Garden, London (1671), designed by Wren, had a music balcony above Grinling Gibbons’s stage front, proscenium balconies over the stage doors, and galleries for the audience.

The first theatres to be built with ranges of boxes one above the other, the Venetian theatres of S Cassiano and SS Giovanni e Paolo (1637–8) – the latter modified especially for opera with five tiers of boxes in 1654 – were characterized by the crowding of a lay audience into the flat floor area in front of the stage. Boxes were sometimes reserved for the use of musicians on either side of the stage, these boxes being called ‘proscenium boxes’ or ‘trumpet loges’. Often an area in front of the stage was enclosed for the use of other members of the orchestra, later increased in size to become the orchestra pit. Such theatres had surprisingly good acoustics. They were small, with closely packed audiences, were largely made of wood and had flat ceilings. The sound had only a short distance to travel to the audience, and there was little risk of echo because of the large areas of absorption provided by the audience and the boxes. Sound reflected from balcony fronts and ceiling was scattered by decorated surfaces to aid in achieving uniform diffusion. The reverberation time was short, and low frequencies were absorbed by the wooden construction, which reflected high frequencies to preserve brilliance.

Even the great opera houses of the 18th century retained these qualities. The original La Scala in Milan (1778) had about 2300 seats, packed closely together, many with a view of only two thirds of the stage area because of the horseshoe shape of the six tiers of boxes (fig.22). The openings of boxes were relatively small, 1·4 m square, which meant that a greater reflecting surface than usual remained in the box fronts. The acoustics were good for all members of the audience, except those at the rear of the boxes; sound was loud and clear, warm in tone and brilliant. The reverberation time at mid-frequencies was about 1·25 seconds (Beranek, 1962). But it must be remembered that the acoustic quality of the voices fell off markedly if the singers retreated from the forestage into the volume of the scenery behind the proscenium arch.

The literature of acoustics begins with the theatrical treatises of the 17th century. Carini Motta’s study of the design of theatres and stages (1676) mentions the importance of the ceiling as a sound reflector, and recommends that it and the supporting structure should be of wood. Motta believed that rooms used for performances in private palaces should have the same kind of construction.

Acoustics, §I: Room acoustics

10. 18th and 19th centuries.

In 18th-century theatres ingenious methods were often resorted to in order to improve the acoustics. In the theatre in Turin (1740) the architect attempted to overcome the lack of balance between a large chorus and a small string orchestra by constructing a hard-surfaced semi-cylindrical resonant chamber running the full length of the orchestra pit below the wooden floor (fig.23). The dish shape was clearly meant to act as a reflector of sound back to the orchestra, while the floor and the volume of air acted together in resonance. The device was often copied, sometimes, apparently, with grilles opening into the orchestra pit from the resonant chamber. In other cases, as at Turin, there were two tubes connecting the ends of the resonant trough with the front of the stage so that the orchestral sound could be heard better by performers and audience. Patté, in his Essai sur l’architecture théâtrale (1782), stated that the Turin theatre had good acoustics, and attributed this to the housing of the orchestra. The volume of sound from the orchestra was considered so strong in its largely enclosed space that it could activate the floor and the volume of air underneath to cause an amplifying resonance.

The success of these features led to further experiments with shaped sections of masonry. In the Teatro Nuovo in Parma the entire parterre of the theatre was built over an enormous masonry saucer, shallow and semi-elliptical in section, with sound passages entering it from the orchestra pit. It is not clear whether grilles were set into the parterre floor allowing sound to pass into the audience without obstruction by the floor. In the Teatro Argentina, Rome (1732), the acoustic problems of an extremely large house, with six tiers of boxes, are said to have been satisfactorily solved using another original device. Here the problems introduced by the size of the theatre were compounded by the elimination of the forestage, which meant that singers’ voices on the stage could not be clearly heard. The design of the theatre was modified after its opening by the introduction of a channel of water under the parterre running from the stage to the back of the theatre; it appears that sound was reflected from the surface of the water inside a vaulted brick enclosure and thus travelled under the parterre whence it emerged through grilles in the floor.

In his project for the theatre at Besançon (1778), Ledoux proposed both a semi-cylindrical resonant chamber under the orchestra pit floor, and a semi-cylindrical stone dish reflector behind the orchestra (fig.24). This must have had an extraordinary focussing effect on the players themselves, but the result was judged successful, as is proved by the repetition of the same device in other theatres, such as Covent Garden, London (1809). Another part of the theatre considered of great importance for its acoustic effect was the ceiling of the auditorium. Writers continued to recommend that it should be made of wood (Algarotti, 1762, for ‘a full, sonorous and agreeable sound’). The ceiling in the Turin theatre had in addition a ‘resonating’ chamber above it, but here its only effect could have been to increase the absorption of low frequencies. The Bordeaux theatre (1773), which was generally considered to have excellent acoustics, was, like all 18th-century theatres, very compact; the maximum distance from the stage to the boxes was only 19·5 m.

Rooms specifically built for the public performance of music without acting or stage presentation began to appear in the late 17th century. The earliest concert room specifically built as such was probably one erected in York Buildings near the Strand in London in 1675; in this ‘great room’ there were ‘proper decorations for musick’. The music of Purcell was performed there, and its last recorded use was for the first performance of Handel’s Esther in 1732, the first appearance of oratorio in England (Forsyth, 1985, citing North). The oldest music room still in use in Europe is that at Holywell, Oxford, opened in 1748 (fig.25) and designed to satisfy a demand for oratorio and choral works. Before the addition of a curtain to one of the side walls (in 1959) the hall, seating 300, must have had a relatively long reverberation time; the present value is 1·5 seconds at mid-frequencies (Beranek, 1962). That composers of this period considered the reverberation time of theatres and opera houses too short is indicated by, for example, Handel’s remark on hearing on one occasion that his theatre would be half empty: ‘Never mind, the music will sound the better’.

The precise music of the Classical period, particularly, required a predominance of direct over reflected sound. Mozart wrote, after a performance of Die Zauberflöte in 1791: ‘You have no idea how charming the music sounds when you hear it from a box close to the orchestra – it sounds much better than from the gallery’. This implies that a narrow rectangular hall, such as was often used at this time for concerts, enhanced the music better than a wide hall. The narrow Redoutensaal in the Hofburg, Vienna, in which a good deal of Haydn’s, Mozart’s and Beethoven’s music was first performed, is estimated to have had a reverberation time of about 1·4 seconds at mid-frequencies with a full audience of 400. For the study of historical performing practice the acoustical properties of the four halls for which Haydn wrote most of his symphonies, and in which he performed them with the local orchestras (whose exact size is known), are particularly interesting. Measurements have been made in the two existing halls: that of the castle of Eisenstadt (c1700) has a volume of 6800 m³ and a mid-frequency reverberation time of 1·7 seconds with 400 persons; the music room of Esterháza castle (1766) has a volume of 1530 m³ and a reverberation time of 1·2 seconds with 200 persons. Acoustical values have been calculated for two halls in London (both before 1790) that have been destroyed: the Hanover Square Rooms had a volume of 1875 m³ and a reverberation time of 0·95 seconds with 800 persons; the King’s Theatre had a volume of 4550 m³ and a reverberation time of 1·55 seconds with about 900 persons. All these rooms had an increasing reverberation time with low frequencies, as did the concert room in the palace of Prince Lobkowitz in Vienna (where Beethoven’s ‘Eroica’ Symphony was first performed), which has a volume of only 950 m³ and reverberation time of 1·45 seconds with 160 persons.

Meyer (1978) draws a connection between the halls for which Haydn’s symphonies were originally written and their instrumentation and scoring. Many of the middle-period symphonies were first performed in the small Esterháza hall, with a short reverberation time: the orchestral works have a chamber music character, with emphasis on clarity and quick changes in effect. By contrast, the London symphonies were written for the King’s Theatre, which had a longer reverberation time; Haydn allows a bar’s rest between a fortissimo chord and a quiet passage, to permit reverberation to take effect – as in the first movement of Symphony no.102. Further, Haydn appears to have tried to utilize the effect of the space in the same movement: there is a long unison passage, in which a piano section swells to a crescendo followed by a diminuendo; this would have been aided by the large room, a sense of spatial broadening produced by the lateral reflections becoming audible during the crescendo passage, after which they would have receded to inaudibility. (It is significant that the Esterháza works often exist in two versions, one using trumpet and kettledrums for performance in other places, including the open air, and another, without these instruments, apparently the original version written for the intimate acoustics of the music room at Esterháza.)

The much admired Altes Gewandhaus (1780) in Leipzig (fig.26), also a narrow rectangular hall, accommodated an audience of 400 and had a reverberation time of only 1·3 seconds (Beranek, 1962). This building, in which Mendelssohn held his concerts from 1835 to 1846, was entirely constructed of wood, securely jointed, which was thought to lend the hall the quality of an immense musical instrument. But the value of a long narrow hall lay in its inherent ability to combine clarity (due to the quickness of the reflections reaching the ear) with a sense of spaciousness (thanks to the reflections, including the late reflections that are perceived as reverberation, being lateral). Wide halls usually lack the clarity and the reverberance that distinguish the best narrow halls.

The development of the orchestra in the early years of the 19th century seems to have given rise to the desire for more sustained reverberation. When the first large halls were constructed specifically for concerts, in the middle of the century, they had longer reverberation times and a lower ratio of direct to reflected sounds. The old Boston Music Hall (1863) had a reverberation time at mid-frequencies of over 1·8 seconds with an audience of 2400; the Grosser Saal (1870) in the Musikverein in Vienna had a reverberation time of 2 seconds with an audience of 1680. Halls of this type with classical ornament characteristically also had highly diffuse sound fields.

Wagner wished the architects of the Bayreuth Festspielhaus (1876) to create a building that would enhance the orchestral sound but still permit the work to be intelligible. At length the reverberation time of 1·6 seconds at mid-frequencies was arrived at, with a full audience of 1800. An important development was the complete sinking of the orchestra pit so that the musicians could no longer distract the audience by their movements (fig.27). A carefully shaped orchestra chamber projects the sound, but at the same time blends the orchestral tones so that instruments cannot be heard individually. The steeply raked, fan-shaped parterre permits clear vision of the stage and ensures minimum shading of direct sound from the singers by the heads of the audience in front. Also, the paired columns along the sides of the seating towards the stage act as acoustic reflectors, diffusing the sound effectively. The important influence of studies of ancient Greek theatres on the acoustical success of this opera house has been demonstrated by Izenour (1992).

Not all concert halls were acoustically satisfactory. The Royal Albert Hall (1871) in London was regarded as disastrous from its opening when ‘The Prince of Wales’ … welcoming address … in many parts … could be heard twice, a curious echo bringing a repetition of one sentence as the next was begun’. Of immense size, 90,000 m³, and an awkward shape, elliptical in plan with a huge dome, the Albert Hall seats 5000 people. The reverberation time must have exceeded 3 seconds when it was opened, and it remains 2·5 seconds after extensive correction (fig.28). Such a hall only begins to function satisfactorily when orchestral and choral forces are large. Nevertheless the visually unifying shape of the Albert Hall is preferred by many performers and listeners to the more acoustically desirable rectangular shape of other famous concert halls.

The undoubted acoustic success of the Paris Opéra’s building (1869–75) was shrugged off by the architect, Charles Garnier:

I gave myself great pains to master this bizarre science [of acoustics] but … nowhere did I find a positive rule to guide me; on the contrary, nothing but contradictory statements … . I must explain that I have adopted no principle, that my plan has been based on no theory, and that I leave success or failure to chance alone … like the acrobat who closes his eyes and clings to the ropes of an ascending balloon.

Acoustics, §I: Room acoustics

11. The science of acoustics.

The science of acoustics received its name from Sauveur, its first noted exponent, who discovered and studied overtones at the beginning of the 18th century. His work was further developed by Euler, who devised a system of binary logarithms to facilitate musical calculations. Ernst Chladni’s Akustik (1802) contained his studies of the vibration of strings, rods and plates by means of sand figures and his discovery of the modal lines. Charles Delezenne (1776–1866) applied calculus to the solution of acoustic problems, and Félix Savart (1791–1840) made investigations into resonance, especially in string instruments. D.B. Reid of Edinburgh published in 1835 his ‘On the Construction of Public Buildings in Reference to the Communication of Sound’ (Transactions of the British Association). It shows an accurate application of recent discoveries in physics to room acoustics, and contains the earliest clear recognition of reverberation.

Hermann von Helmholtz (1821–94) laid the foundations for much modern physical and physiological research in acoustics. Rudolf Koenig (1832–1901) manufactured instruments for the study of acoustics and conducted extensive research. Others who followed closely behind were John Tyndall (Sound, 1867), Lord Rayleigh (Theory of Sound, 1877–8) and Carl Stumpf (Psychology of Tone, 1883–90). Stumpf’s insistence that the scientific system of music theory depended on the psychological interpretation of acoustic data opened a new discipline and many new avenues for research.

W.C. Sabine pioneered the study of applied acoustics in buildings in the period 1895–1915, publishing his results in a series of important papers. Sound decay was analysed in detail and the prediction of reverberation time in rectangular rooms by calculation was made possible. The impedance method of specifying acoustical materials was developed, and a wide variety of acoustical materials began to be manufactured. At the Bell Telephone Laboratories, Harvey Fletcher studied loudness and masking in the 1920s and 30s, and developed new, more accurate techniques of acoustic analysis and measurement. Other research laboratories, especially those in California, England and Germany, contributed to rapid advances in scientific acoustics.

The first large-scale experiment in the application of the new scientific understanding of acoustics to room design came with the building of the Salle Pleyel in Paris (1927), which seated 3000; it was a ‘notorious disappointment’ (Beranek, 1962). The whole shape of the hall, in plan and section, was designed to send sound to the audience (fig.29). Sound diffusion was poor and uneven, and reverberation short. The result was a room in which a large audience could enjoy recitals and chamber music, but in which orchestral music lacked body and colour. In addition, the seats in the front and middle of the parterre received their first reflections from the ceiling, high at this point, with too long a delay after the direct sound, resulting in the loss of any sense of intimacy (Knudsen, 1932).

After the disappointment of the Salle Pleyel, no major attempt to apply the new scientific knowledge of acoustics to design was made until the building of the Royal Festival Hall in 1948–51 (see London (i), fig.40). Great care was taken to avoid external and internal noise interference (see fig.11); the entire auditorium was raised high in the air and a double construction used to isolate the interior. In this respect the hall was an important and successful experiment. The interior was designed along principles not dissimilar from those of the Salle Pleyel, and with some of the latter’s defects. In particular, concentrating the first reflections at the audience by reflectors over almost the whole ceiling means that a large proportion of the sound energy is absorbed in a much shorter time than the reverberation time of the hall. The effect is to make reverberation much less evident than it is in the older concert halls because it is relatively less loud, and so the hall seems ‘dry’, especially in loud ensemble passages.

Later advances in acoustics have therefore concentrated on finding methods of increasing the amplitude and the length of reverberation while maintaining a high level of direct and first-reflection sound to all the seats. One solution is that adopted with success in the Koussevitzky Music Shed in Lenox, Massachusetts (1959; fig.30); a pattern of suspended ceiling panels reflects a proportion of the sound to the audience from quite a short distance above the orchestra, while the spaces between allow the rest of the sound to travel into the volume above, where it is diffused before returning as prolonged reverberation. A defect of this technique is that sounds of short wavelength are almost completely reflected by the panels, whereas sounds of long wavelength pass almost completely around them, giving an imbalance in first reflections and a further imbalance in reverberation. Such problems led to the initial failure of one of the major concert halls to be built in the 1960s, Philharmonic (now Avery Fisher) Hall at Lincoln Center in New York (1961).

Other recent concert halls have concentrated on relating the volume carefully to the type of music for which the hall is built, achieving ceilings that are more diffuse and providing fewer reflectors of direct sound close to the orchestra. The result has been a considerable increase in the reverberation time and in the relative strength of the late or reverberant sound; the Maltings at Snape, Suffolk (1967; fig.32), for example, has a reverberation time of 2·25 seconds at mid-frequencies, whereas the Royal Festival Hall, with a volume nearly three times as large, has a reverberation time of only 1·47 seconds. Today, many new concert halls are expected both to provide for audiences of varying sizes and to accommodate new demands that far exceed the needs of auditoriums even 50 years ago. Simply referring back to acoustically approved music rooms of earlier times will not deal with these problems. New rooms are being successfully designed which demonstrate the benefit of our increased scientific understanding. Certain new concert halls, such as the Doelen hall in Rotterdam, have more uniform reverberation, with diffusion and reflection on walls and ceiling; all sound paths are within calculated limits, and the balance between low and high frequencies is carefully maintained. Others, like the Segerstrom Hall of the Orange County Performing Arts Center, California (1986), which seats 3000, are multi-purpose halls, yet, by the use of flexible acoustical adjustments, manage to achieve good uniformity and ‘a particularly surprising suitability for solo, chamber and opera performances … an enveloping sound character, but more immediate’ than the normal concert hall (Barron).

Flexible acoustical conditions can be created by physically manipulating the surfaces, by varying the number of coupled volumes, and even by reducing or increasing the overall volume, as has been introduced at the Jesse Jones Hall, Houston, the experimental workshop L’Espace de Projection at IRCAM in the Centre Pompidou, Paris (1978) and the Theater de Maaspoort in Venlo, the Netherlands (1986).

While reverberation time remains a prime consideration in designing concert halls, the importance of early reflections that create a sense of acoustical intimacy has come to be increasingly recognized. Recently, attention has focussed on the spatial qualities of sound and the factors that control these qualities: the early lateral reflections, which make a source appear wider than it is; the late lateral energy, which provides a sense of being enveloped by sound; and loudness, which further adds to a hall’s spaciousness.

Acoustics, §I: Room acoustics

12. The contemporary performance of early music.

Sensitivity to the acoustics of rooms seems always to have affected the composition and performance of music. This is clear in the character of much early music, but only rather late was it actually written about. Quantz, in his book Versuch einer Anweisung, die Flöte traversiere zu spielen (1752), recommends:

In the choice of pieces in which he wishes to be heard in public, the flautist … must adjust … to the place where he plays … . In a large place, where there is much resonance, and where the accompanying body is very numerous, great speed produces more confusion than pleasure. Thus on such occasions he must choose concertos written in the majestic style, and in which many passages in unison are interspersed, concertos in which the harmonic parts change only at whole or half bars. The echo that constantly arises in large places does not fade quickly and only confuses the notes if they succeed one another too quickly, making both harmony and melody unintelligible.

The practice today is to attempt to relate the size of the orchestra not merely to that originally used, but to the room acoustics – to compensate for a weak bass response in a hall, for example, by increasing the bass section relative to the upper parts. Where the volume of the room is considerably larger, and the impact of the original small orchestra would be substantially reduced, some increase in orchestral size is necessary to compensate for differences in loudness and spatial impression. Changes in the composition and style of an orchestra are therefore often judged necessary in order to respond to altered acoustical conditions in particular halls. Where possible, conductors often prefer to select rooms that are naturally appropriate to particular types of early music, but this usually restricts audience size, and may therefore be uneconomic.

The acoustician’s contribution to the problem of matching concert halls to the style and type of performance has been to provide, in a number of recent examples, designs allowing considerable variation of the acoustics of concert halls at will (see §11 above).

Acoustics, §I: Room acoustics

BIBLIOGRAPHY

O. Belli: ‘History of Candia’, Museum of Classical Antiquities (London, 1854)

G.M. Hills: ‘Earthenware Pots (Built into Churches) which have been called Acoustic Vases’, Transactions of the Royal Institute of British Architects (1882), 65

A. Müller: Lehrbuch der griechischen Bühnenalterthümer (Freiburg, 1886), 46

H. Bagenal and A. Wood: Planning for Good Acoustics (London, 1931)

V.O. Knudsen: Architectural Acoustics (New York, 1932)

P.V. Brüel: ‘Panel Absorbants of the Helmholtz Type’, First Summer Symposium of the Acoustics Group: Papers on Resonant Absorbers (London, 1947)

F. Giorgi: ‘Memorandum for S. Francesco della Vigna’, in R. Wittkower: Architectural Principles in the Age of Humanism (London, 1949, 4/1988)

P.H. Parkin and H.R. Humphreys: Acoustics, Noise and Buildings (London, 1958, 4/1979)

L.L. Beranek: Music, Acoustics & Architecture (New York, 1962)

H.J. Purkis: ‘The Reverberation Times of some English Cathedrals’, Bulletin of the Institute of Physics and the Physical Society (1963), no.14, p.8

J.G. Landels: ‘Assisted Resonance in Ancient Theatres’, Greece and Rome, 2nd ser., xiv (1967), 80

K. Harrison: ‘Vitruvius and Acoustic Jars in England during the Middle Ages’, Transactions of the Ancient Monuments Society, new ser., xv (1967–8), 49–58

R. Taylor: Noise (London, 1970)

R.S. and H.K. Shankland: ‘Acoustics of St Peter’s and Patriarchal Basilicas in Rome’, JASA, l (1971), 389–96

C. Gilford: Acoustics for Radio and Television Studios (London, 1972)

A.H. Benade: Fundamentals of Musical Acoustics (New York, 1976, 2/1990)

G.C. Izenour: Theater Design (New York, 1977/R)

J. Meyer: ‘Raumakustik und Orchesterklang in den Konzertsälen Joseph Haydn’, Acustica, xli (1978), 145–62

A. Trochidis: ‘Reverberation Time of Byzantine Churches of Thessaloniki’, Acustica, li (1982), 299–301

M. Forsyth: Buildings for Music (Cambridge, MA, 1985)

G.C. Izenour: Theater Technology (New York, 1988/R)

G.C. Izenour: Roofed Theaters of Classical Antiquity (New Haven, CT, 1992)

M. Barron: Auditorium Acoustics and Architectural Design (London, 1993)

J. Meyer: Akustik und musikalische Aufführungspraxis (Frankfurt, 3/1995)

L.L. Beranek: Concert and Opera Halls: how they Sound (Woodbury, NY, 1996)

Acoustics

II. String instruments

1. Foundations.

2. Differences between viols and violins.

3. Wood and varnish.

4. Acoustical findings important for violin tone.

5. The mute and vibrato.

6. Sound radiation.

7. Bowing.

8. The plucked string.

9. Current research.

BIBLIOGRAPHY

Acoustics, §II: String instruments

4. Acoustical findings important for violin tone.

Modern acoustical science has been applied to violin making with significant results. The normal bending modes of unattached violin plates have been revealed by hologram interferometry and other vibration methods. These show the frequency sequence and configuration of the normal resonance modes basic to the sounds and stiffnesses in free violin plates that violin makers have long known how to assess by flexing, feeling, tapping and listening. The frequencies and damping characteristics of modes 1, 2 and 5 in both top and back free plates are critical to the sound of the violin in spite of the fact that they are not transferred intact into the finished instrument (fig.36). It has been found that the best instruments result when modes 2 and 5 are tuned to matching frequencies in both top and back free plates and the two pairs are an octave apart, with mode 1 in the top an octave below that of mode 2 (C.M. Hutchins, 1981). This finding has been corroborated by many violin makers. However, its successful application depends on other factors such as wood quality, and the arching and thickness characteristics basic to good violin making. Current thinking is that the free-plate modes provide the violin maker with clues as to the desired local stiffnesses in the plates, which then share in the bending modes of the entire instrument (C.M. Hutchins, 1991). Finite element analyses to quantify free-plate tuning show the effects of wood stiffness, local thickness changes, plate arching and local thickness patterns on free-plate frequencies (Rodgers, 1988, 1990, 1993; and Molin, Lindgren and Jansson, 1988).

The body and cavity modes of the whole violin have been mapped in various ways since the early 19th century (Savart, 1819; Backhaus, 1931; Meinel, 1957; Reinicke and Cremer, 1970; and Stetson, 1970). More recent studies by means of modal analysis (Marshall, 1985) and finite element analysis (Knott, 1987) have elucidated the unsuspected and bewildering body vibrations, up to about 1300 Hz, not only in the top and back of the violin, but also in the ribs, neck, fingerboard and scroll. Fig.37 gives a three-dimensional representation of the three important body modes (B−1, c 145–190 Hz; B0, c 250–300 Hz; and B1, c 500–570 Hz), as well as the two important cavity (air) modes (A0, c 260–290 Hz; and A1, c 430–490 Hz) in the first 1000 Hz of the violin range.

For centuries the ‘Helmholtz’ or breathing mode (A0) of the violin cavity was thought to be the only cavity mode. Research since the 1970s has shown that there are many cavity modes whose frequencies depend on the geometry of the box, the stiffness of its walls and the special characteristics of the f-hole openings.

The interrelations of wood and cavity modes are critical in the final tuning of an instrument in playing condition. Very desirable effects in tone and playing qualities result when the Helmholtz mode and the nearby body mode (B0) are at the same frequency (C.M. Hutchins, 1985). The A0 mode frequency is essentially fixed in the finished violin by the dimensions of the cavity, the compliance of its walls and the characteristics of the f-holes. The B0 mode frequency can be moved up or down by altering the mass and stiffness of the neck, fingerboard, pegs and chinrest. With practice the pitch of the A0 mode can be heard by blowing across an f-hole. The pitch of the B0 mode can be heard by holding the violin at the nut (a node of B0) and tapping on the end of the fingerboard or on the chinrest.

When one hums into an f-hole, a slightly lower pitch can be heard; this is shown in fig.37 as the wood prime bowed tone. Good tone and playing qualities also result when Bo and wood prime are at the same frequency. Wood prime is not a fundamental vibration; it is a strong bowed tone, due to reinforcement from the main wood bowed tone (c 440–570 Hz), which is an octave higher and acts as its second harmonic. These two bowed tones, main wood and wood prime, do not exist in a single frequency (sine-wave) excitation of the violin; however, they are two of the strongest tones in the bowed violin.

The main wood bowed tone usually lies just above the open A string of the violin, and has been found to be a combination of the A1 and B1 modes when the instrument is bowed. The frequency spacing between the A1 and the B1 modes indicates whether the tone qualities of a violin are very harsh (over 100 Hz), brilliant for solos (65–80 Hz), or suitable for orchestral work (45–65 Hz), chamber music in small halls (25–45 Hz) or more intimate Hausmusik (0–25 Hz) (C.M. Hutchins, 1989).

Controlled vibration tests of violins in playing conditions show that over an extended period (more than 1500 hours) the frequency of the B1 mode goes down about 25 Hz, resulting in smoother tone and playing qualities. After a rest period of several months the B1 frequency rises 10–15 Hz, but never regains its original frequency. This effect is due to the fracturing of the microfibrils in the cell walls, as well as to the decoupling of the long-chain cellulose polymers under stress and vibration. During a rest period variations of moisture and temperature slowly reverse these changes. Thus after many years of use the tone of a violin becomes smoother and the instrument becomes easier to play. The Hill brothers estimated that it takes 20–80 years of playing to bring a violin to optimum condition (W.H., A.F. and A.E. Hill, 1931). Experiments show that the B1 mode frequency can be raised or lowered some 10–20 Hz by means of various structural changes (Hutchins and Rodgers, 1992). This process indicates (1) that violins played over many years slowly change tone qualities, losing brilliance and power, and eventually wear out in spite of restorations by excellent violin makers; (2) that violins left unplayed for some time need to be ‘played in’ again, since during the rest period the wood cells tend to reform partially under temperature and moisture changes, thus altering the tone qualities; and (3) that new violins which seem harsh and stiff will improve with playing or some form of vibration, a fact that expert violin makers have known for many years (Otto, 1817).

Acoustics, §II: String instruments

5. The mute and vibrato.

The mute, which consists merely of a suitable mass attached to the top of the bridge, changes both the volume and quality of the sound. Its tendency to immobilize the top of the bridge increases with frequency, so that higher tones are reduced and timbre becomes softer and less brilliant; the loudness of sound is correspondingly reduced. The low partial notes of the instrument are not greatly affected, but the loudness of the low notes is indirectly reduced by virtue of the ‘residue effect’. According to this, the subjective sensation of fundamental pitch produced by the higher harmonics is somewhat reduced.

Of vibrato, much has been written from a musical point of view. Here only the physical characteristics will be considered, namely the changes in frequency level (recognized by the ear as pitch changes) and intensity level, and variations in harmonic structure of the sound. The changes in pitch as the finger moves back and forth on the string are quite familiar. This motion causes all the harmonics to have the same rate of pitch variation as the vibrato rate, typically four to six per second. The intensity level of each harmonic also varies at this rate, but is different for each harmonic, some having a high intensity level and some a very low. Also, for some of the harmonics the intensity level is increasing, while at the same time it is decreasing for others. These variations cause the aurally pleasing changes in the quality of the sound of notes played with vibrato. For further details see Fletcher and Sanders (1967).

Acoustics, §II: String instruments

6. Sound radiation.

The mechanical subsystems of the violin discussed thus far all contribute to how the violin functions as a musical instrument in the hands of the player. The sound waves radiated by a violin activated by the broad-band input from the bowed string contain all the partials of the note played, which are essentially in simple multiples of the fundamental: 1, 2, 3, 4 etc. If any part of the instrument, including the air of the cavity, resonates at one of the partial frequencies, the amplitude and sound quality of that partial are enhanced. The quality of the sound heard depends largely on the strength of the various partials as they are affected by the resonance modes of the instrument.

Modal analysis shows that there are some 41 modes in the first 1000 Hz of the vibrating violin (Marshall, 1985). However, a reciprocal method of vibrating the violin with sound and measuring the velocity of vibration at the bridge shows that of the many modes in the violin below 1000 Hz only six to eight are important for sound radiation (Weinreich, 1983).

Much research on the radiation of violins has been done using a single-frequency (sine-wave) input sweeping across the entire spectrum. The response is picked up by one or more microphones and plotted on a chart against a bowed ‘loudness curve’ (fig.38). By comparing many such charts with the tone and playing qualities of the violins tested, investigators have built up a body of knowledge enabling them to assess the tone qualities of an instrument (Moral and Jansson, 1982; Dünnwald, 1990; and Meyer, 1993). However, variations in methods make it difficult to compare tests carried out by different investigators.

The acoustic radiation field surrounding the violin has been measured in various ways. Fig.39 shows the main directions of radiation in the horizontal plane around the player. Fig.40 gives the measured radiation characteristics of different frequency ranges around the violin in a plane parallel to the bridge. Fig.40a shows wavelength large in comparison to source dimensions (290 and 517 Hz respectively); fig.40b shows wavelength comparable to source dimensions (922 and 950 Hz); and fig.40c shows wavelength small in comparison to source dimensions (2323 and 2630 Hz). A method capable of measuring the entire spherical output of a violin, giving phase and amplitude information, has been developed by Weinreich and Arnold (1980). Weinreich (1997) has also found that the radiation pattern of a violin varies rapidly, not only with direction but also with frequency, typically changing drastically from one semitone to the next. In a concert hall this can produce an illusion that each note played by a solo violin comes from a different direction, endowing fast passages with a special flashing brilliance. This finding has important consequences for the perception and reproduction of violin tone.

Acoustics, §II: String instruments

7. Bowing.

When a bow is drawn across a string, the string appears as a lens-shaped blur. Within this blur the string is vibrating in a remarkable manner, first described by Helmholtz in 1877. If the motion is frozen in a series of snapshots (fig.41) the string is found to take the form of a sharply bent straight line. The ‘corner’ shuttles back and forth along the length of the string at the frequency of the note being played, for example 440 times per second for the open A string of a violin. While the corner travels from the bow to the player's finger and back the string is sticking to the bow hairs, and while it travels the shorter distance from bow to bridge and back the string is slipping rapidly across the hairs of the bow. The arrival of the corner at the bow triggers the transitions between these two states. This is what distinguishes the bowed string from other ‘stick-slip’ vibrations such as squealing brakes: the accurate timekeeping provided by the shuttling Helmholtz corner makes the pitch of the note very stable. When reversing from an up-bow to a down-bow stroke, the Helmholtz corner must be made to run in the opposite direction, one reason why it is hard to perform a completely inaudible bow change in a long note.

The player controls four quantities when bowing: the bow speed, the bow ‘pressure’ (more properly ‘force’), the position of the bow on the string and the degree to which the ribbon of bow hair is tilted relative to the string. A consequence of the ‘Helmholtz motion’ is that, to a first approximation, the steady vibratory motion of the string is the same, regardless of these details of bowing. For ideal Helmholtz motion the waveform of oscillating force exerted by the string on the instrument's bridge, which is ultimately responsible for the sound, takes the form of a ‘sawtooth wave’ . Three factors affect the player's ability to influence the sound quality of a note. First, if the bowing parameters fall outside a certain range the Helmholtz motion is not possible at all, and something else (usually undesirable) happens, Second, within the range for which Helmholtz motion occurs, the fine details of the string motion do depend somewhat on the bowing parameters, enough for audible variations in sound quality to be produced for musical effect. Third, our perception of sound quality is influenced by the length and detailed nature of the transients – the brief sounds made by the way an individual note is started or stopped – and different bowings produce different transients.

The range of bowing parameters for which Helmholtz motion is possible can best be appreciated with the aid of the diagram shown in fig.43. For a given bow position and speed the bow force must lie within a certain range. Below the lower limit the Helmholtz motion gives way to one in which the string slips relative to the bow more than once per cycle, producing what is usually described as ‘surface sound’. Above the upper limit the arrival of the Helmholtz corner is insufficient to ‘unstick’ the string from the bow, the note ceases to be exactly repetitive from cycle to cycle, and the result is no longer a musical tone but a raucous ‘crunch’. To bow nearer to the bridge (sul ponticello) the player must press harder and control the force more accurately: both limits rise but they become closer together. Beginners often fail to control the bow position on the string, and so may inadvertently leave the Helmholtz range by moving horizontally in the diagram.

The loudness of a bowed note is influenced by bow position, force and speed, while the brilliance is influenced mainly by bow force alone. The player must keep all three parameters in mind in order to produce the desired combination of volume and tone quality. Loudness is governed mainly by the amplitude of the string motion, brilliance by the exact shape of the Helmholtz corner: a rounded corner gives a more mellow sound, a sharper one a brilliant sound. The precise details of string motion following a particular bowing gesture are influenced by the material and construction of the string, and by its interaction with the body of the instrument and with the player's finger. For more details of the physics of bowing, see Cremer (1984). For an introduction to more recent research using computer simulation, see Schumacher and Woodhouse (1995).

Bowed string instruments frequently display a disconcerting phenomenon known as the ‘wolf’ note, a narrow range of frequency within which the response tends to stutter or warble. This generally occurs around the frequency of the most prominent resonance of the instrument (typically around f' on a viola G or C string, or f on a cello G or C string). The wood of the body vibrates so vigorously that the bridge does not provide a sufficiently solid support to the string, especially the heavier lower strings. The viola and cello are more prone to this problem than the violin because they have bodies which (for ergonomic reasons) are smaller than one would obtain by scaling up the violin in proportion to the tuning. To compensate, they have flimsier bodies and heavier strings, which exacerbate the wolf. The best way to ameliorate a wolf is to fit a ‘tuned absorber’ to the instrument. This takes the form either of a small resonant cantilever which can be installed inside the instrument by a repairer, or of a weight (a commercial ‘wolf-eliminator’ or a piece of plasticene) attached to one of the strings between bridge and tailpiece. It must be accurately tuned: fit a heavy mute to the bridge and adjust the mass and position of the eliminator until the short string, when plucked, rings at the pitch of the wolf.

Acoustics, §II: String instruments

8. The plucked string.

When the string is plucked, the pull of the finger creates a kink, or discontinuity, that divides the string into two straight sections. On release, a dynamic condition is set up in which two discontinuities travel in opposite directions, one towards the bridge and one towards the nut. These are identical to the modes of motion described for the up-bow and down-bow action in the bowed string (see §7 above). Since they are now both present at the same time, however, the wave shape of the force exerted on the bridge is radically different from that of the bowed string. In the bowed string, the Helmholtz motion indicates a sawtooth wave in which reversal is instantaneous regardless of the position of the bow on the string (fig.44a). With the plucked string, on the other hand, the force at the bridge has a rectangular shape that depends on the point of plucking. If plucking occurs at the middle of the string, the shape is that of a square wave with a minimal content of the higher overtones (fig.44b). If the pluck is near the bridge or nut, a sharp rectangular wave is produced which is exceedingly rich in high-frequency components (fig.44c). Thus a wider range of timbre is possible by changing the point of plucking than by changing the point of bowing.

The plucked-string waveforms of fig.44 are highly idealized. Since energy is supplied once only at the start of the note, the string vibrations die away with time. Decay mechanisms include air damping, internal damping in the string material (dominant in gut and nylon strings) and energy loss to the body of the instrument. Only a small fraction of the string’s vibrational energy is converted to sound. High-frequency string components tend to die out more rapidly than the low frequencies, though the decay rates depend on the precise dynamic response of the body. Thus the force waveforms, and the spectral content of the sound, vary considerably not only with time but also from note to note. In practice the player can exert further control over the spectral content of the sound by varying the plucking position or the angle of release of the string (which modifies the coupling between the string and body), or by modifying the shape and frictional characteristics of the plectrum or fingertip.

An important difference between bowing and plucking is that in the former the phenonemon is periodic, so that the overtones are kept in a strictly harmonic relationship to the fundamental. In a plucked string, which involves ‘free’ rather than ‘forced’ vibrations, stiffness in the string and coupling to the body create inharmonic partials. String stiffness causes higher partials to become slightly sharper than integral multiples of the fundamental. Strong coupling between the string and body, the result of an over-compliant soundboard (as commonly found in the guitar), can cause some partials to become sharp or flat. A little inharmonicity can add interest to the sound, but too much causes notes to sound out of tune or generally imparts poor sound quality.

Acoustics, §II: String instruments

9. Current research.

It is apparent that, in respect to quantitative analysis and the predetermining of the effects of structure, wind instruments possess two important advantages over strings: their shapes are amenable to simpler mathematical description, and the resonating material, air, is homogeneous, with the same elasticity in all directions. By contrast, the shapes of string instruments, while a delight to the connoisseur, are forbidding to the mathematician, and the resonating material, wood, is neither homogeneous nor isotropic, and cannot be standardized. This uncertain property of wood is not a serious difficulty in woodwind instruments because their massive walls do not share vitally in the resonance of the instrument. The result has been that designers of wind instruments have had the possibility, which they have brilliantly used, of forecasting the effects of changes in design, while scientific luthiers have been far more dependent on a series of steps in carving the plates of their instruments, each step guided by the best means at their disposal.

Since the 1950s experimenters around the world have applied modern electronic and optical technologies to the acoustical problems of the violin. Efforts by individuals and by groups such as the Catgut Acoustical Society to relate these technical findings to the actual construction of fine violins are making it possible to build consistently fine-sounding violin-family instruments of any size and tuning. Modern violin makers have a ready market for their violins, violas, cellos and double basses, which are beginning to replace the fine early instruments that are gradually losing tone quality. However, we are still far from understanding fully how these apparently simple yet amazingly complicated instruments can produce such beautiful music.

Acoustics, §II: String instruments

BIBLIOGRAPHY

CASN

Catgut Acoustical Society Newsletter

JCAS

Journal of the Catgut Acoustical Society

C.M. Hutchins, ed.: Musical Acoustics (Stroudsburg, PA, 1976) [MusA]

C.M. Hutchins, ed.: Research Papers in Violin Acoustics 1975–1993 (Woodbury, NY, 1996) [RP]

PraetoriusSM, ii

F. Savart: Memoire sur la construction des instruments à cordes et à archet (Paris, 1819/R)

J.A. Otto: Über den Bau und die Erhaltung der Geige (Halle, 1817; Eng. trans., enlarged, 1848)

G. Fry: The Varnishes of the Italian Violin Makers of the Sixteenth, Seventeenth and Eighteenth Centuries: their Influence on Tone (London, 1904)

W.H., A.F. and A.E. Hill: The Violin Makers of the Guarneri Family (1626–1762) (London, 1931/R)

J. Michelman: ‘Composition of Old Italian Varnishes’, Organic Finishing, xi/5 (1950), 22–3

H.F. Meinel: ‘Regarding the Sound Quality of Violins and a Scientific Basis for Construction’, JASA, xxix (1957), 817–22 [MusA]

C.M. Hutchins: ‘The Physics of Violins’, Scientific American, ccvii/5 (1962), 78–84, 87–92 [MusA]

H. Fletcher and L.C. Sanders: ‘Quality of Violin Vibrato Tones’, JASA, xli (1967), 1534–44

C.M. Hutchins: ‘Founding a Family of Fiddles’, Physics Today, xx (1967), 23–37 [MusA, RP]

L.W. Condax: ‘Examination of the Ground Layer of the Italian Violin Varnish’, CASN, no.10 (1968), 12–13

J.C. Schelleng: ‘Acoustical Effects of Violin Varnish’, JASA, xliv (1968), 1175–83 [MusA, RP]

W. Reinicke and L. Cremer: ‘Application of Holographic Interferometry to the Bodies of String Instruments’, JASA, xlviii (1970), 988–92 [MusA]

K.A. Stetson: ‘A Progress Report on Holographic Testing of String Instruments’, CASN, no.14 (1970), 19–22

J. Meyer: ‘Directivity of the Bowed String Instruments and its Effect on Orchestral Sound in Concert Halls’, JASA, li (1972), 1994–2009 [MusA]

W. Reinicke: ‘Übertragungseigenschaften des Streichinstrumentenstegs’, CASN, no.19 (1973), 26–36

J.C. Schelleng: ‘The Bowed String and the Player’, JASA, liii (1973), 26–41

C.M. Hutchins: ‘A Note on the Function of the Soundpost’, CASN, no.21 (1974), 27–8

W.M. Fulton: ‘Old Italian Varnish’, The Strad, lxxxv (1974–5), 491–501

G. Bissinger: ‘Tuning the Bassbar in a Violin Plate’, CASN, no.26 (1976), 10–12; no.30 (1978), 23–7

D.W. Haines: ‘On Musical Instrument Wood’, CASN, no.31 (1979), 23–32 [RP]

H.A. Müller: ‘The Function of the Violin Bridge’, CASN, no.31 (1979), 19–22

G. Weinreich and E.B. Arnold: ‘Method of Measuring Acoustic Radiation Fields’, JASA, lxviii (1980), 404–11 [RP]

C.M. Hutchins: ‘The Acoustics of Violin Plates’, Scientific American (1981), 170–86 [RP]

M.A. Hutchins: ‘Acoustical Parameters for Violin and Viola Back Wood’, CASN, no.35 (1981), 29–31 [RP]

L.W. Condax: ‘Final Summary Report of Violin Varnish Research Project’, CASN, no.36 (1982), 31–6

J.A. Moral and E.V. Jansson: ‘Eigenmodes, Input Admittance, and the Function of the Violin’, Acustica, l (1982), 329–37 [RP]

M.A. Hutchins: ‘Physical Measurements on Sampling of European Spruce and Maple for Violin Top and Back Wood’, CASN, no.40 (1983), 28–30

G. Weinreich: ‘Violin Radiativity: Concepts and Measurements’, SMAC 83: Proceedings of the Stockholm Music Acoustics Conference: Stockholm 1983, ii, 99–109 [RP]

V. Bucur and R. Archer: ‘Elastic Constants of Wood by an Ultrasonic Method’, Wood Science and Technology, no.18 (1984), 255–65

L. Cremer: The Physics of the Violin (Cambridge, MA, 1984)

M.E. McIntyre and J. Woodhouse: ‘On Measuring Wood Properties’, JCAS, no.42 (1984), 11–15; no.43 (1985), 18–24; no.45 (1986), 14–24

C.M. Hutchins: ‘Effects of an Air-Body Coupling on the Tone and Playing Qualities of Violins’, JCAS, no.44 (1985), 12–15 [RP]

K.D. Marshall: ‘Modal Analysis of a Violin’, JASA, lxxvii (1985), 695–709 [RP]

V. Bucur: ‘Varieties of Resonance Wood and their Elastic Constants’, JCAS, no.47 (1987), 42–8 [RP]

C.M. Hutchins: ‘The Effects of Five Years of Filler and Varnish Seasonings on the Eigenmodes in Four Pairs of Viola Plates’, JCAS, no.48 (1987), 25–6 [RP]

G.A. Knott: A Modal Analysis of the Violin using MSC/NASTRAN and PATRAN (thesis, Naval Postgraduate School, Monterey, CA, 1987) [RP]

W.J. Trott: ‘The Violin and its Bridge’, JASA, lxxxi (1987), 148–54 [RP]

N.-E. Molin, L.-E. Lindgren and E.V. Jansson: ‘Parameters of Violin Plates and their Influence on the Plate Modes’, JASA, lxxxiii (1988), 281–90 [RP]

O.E. Rodgers: ‘The Effect of the Elements of Wood Stiffness on Violin Plate Vibrations’, JCAS, 2nd ser., i/1 (1988), 2–8 [RP]

C.Y. Barlow and J. Woodhouse: ‘Of Old Wood and Varnish: Peering into a Can of Worms’, JCAS, 2nd ser., i/4 (1989), 2–9 [RP]

J.I. Dunlop: ‘The Acoustic Properties of Wood in Relation to Stringed Musical Instruments’, Acoustics Australia, xvii/2 (1989), 37–40 [RP]

C.M. Hutchins: ‘A Measurable Controlling Factor in the Tone and Playing Qualities of Violins’, JCAS, 2nd ser., i/4 (1989), 10–15 [RP]

H. Dünnwald: ‘Ein erweitertes Verfahren zur objectiven Bestimmung der Klangqualität von Violinen’, Acustica, lxxi (1990), 269–76 [RP]

O.E. Rodgers: ‘Influence of Local Thickness Changes on Violin Free Plate Frequencies’, JCAS, 2nd ser., i/5 (1990), 13–16 [RP]

O.E. Rodgers: ‘Relative Influence of Plate Arching and Plate Thickness Patterns on Violin Back Free Plate Tuning’, JCAS, 2nd ser., i/6 (1990), 29–33 [RP]

O.E. Rodgers and T.R. Masino: ‘The Effect of Wood Removal on Bridge Frequencies’, JCAS, 2nd ser., i/6 (1990), 6–10

N.H. Fletcher and T.D. Rossing: The Physics of Musical Instruments (New York, 1991)

C.M. Hutchins: ‘A Rationale for Bi-Tri Octave Plate Tuning’, JCAS, 2nd ser., i/8 (1991), 36–9 [RP]

M.A. Hutchins: ‘Effects on Spruce Test Strips of Four-Year Application of Four Different Sealers plus Oil Varnish’, JCAS, 2nd ser., i/7 (1991), 11–16 [RP]

C.M. Hutchins: ‘A 30-Year Experiment on the Acoustics and Musical Development of Violin-Family Instruments’, JASA, xcii (1992), 639–49 [RP]

C.M. Hutchins and O.E. Rodgers: ‘Methods of Changing the Frequency Spacing (Delta) between the A1 and B1 Modes of the Violin’, JCAS, 2nd ser., ii/1 (1992), 13–19 [RP]

N.J.-L. Fang and O.E. Rodgers: ‘Violin Soundpost Elastic Vibration’, JCAS, 2nd ser., ii/1 (1992), 39–40 [RP]

J.E. Miller: ‘Spectral Measurements of Violins’, JCAS, 2nd ser., ii/4 (1992), 1–4 [RP]

C.M. Hutchins: ‘The Effect of Relating the Tailpiece Frequency to that of Other Violin Modes’, JCAS, 2nd ser., ii/3 (1993), 5–8 [RP]

J. Meyer: ‘The Sound of the Orchestra’, Journal of the Audio Engineering Society, xli (1993), 203–13 [RP]

O.E. Rodgers: ‘Influence of Local Thickness Changes on Violin Top Plate Frequencies’, JCAS, 2nd ser., ii/3 (1993), 14–16 [RP]

V. Bucur: Acoustics of Wood (Boca Raton, FL, 1995)

R.T. Schumacher and J. Woodhouse: ‘Computer Modelling of Violin Playing’, Contemporary Physics, xxxvi (1995), 79–92

H.A. Müller: ‘How Violin Makers choose their Wood and what the Procedure means from a Physical Point of View’, Research Papers in Violin Acoustics 1975–1993, ed. C.M. Hutchins (Woodbury, NY, 1996)

B.E. Richardson: ‘Stringed Instruments: Plucked’, Encyclopedia of Acoustics, ed. M.J. Crocker (New York, 1997)

G. Weinreich: ‘Directional Tone Colour’, JASA, ci (1997), 23–38

J. Woodhouse: ‘Stringed Instruments: Bowed’, Encyclopedia of Acoustics, ed. M.J. Crocker (New York, 1997)

C.M. Hutchins: ‘The Air and Wood Modes of the Violin’, Journal of the Audio Engineering Society, xlvi (1998), 751–65 [RP]

Acoustics

III. Keyboard string instruments

Acoustics, §III: Keyboard string instruments

1. Foundations.

Keyboard string instruments are distinguished by key-operated mechanical devices which are used initially to shock the strings into free vibration, and finally to damp these free vibrations. Differences in the shape, size, material and function of these percussive and damping devices create most of the characteristic variations at the beginnings and ends of the notes. These differences also influence the timbre while the strings are vibrating freely, because the initial shock determines the waveform of the ensuing vibrations. There are other acoustical factors, such as the stiffness of the strings, the wrapping of the bass strings, the gradual damping of tone at the string supports, the number of strings per key, the acoustical properties of the structure supporting the strings and, in some cases, the effect of passive sympathetic strings.

The clavichord, harpsichord and piano all have systems of individually tuned strings lying approximately in a plane. Each string is under tension between a hitch-pin, secured in the string-supporting structure, and a tuning-pin which during tuning is rotated to and fro in a wooden pinblock. Between these two pins and generally near each of them the string bears against a point of sliding contact. The distance between the two intermediate contact points defines the length of the tuned vibrating string segment. Each string can vibrate freely at a series of frequencies which would be a harmonic series if the string were perfectly flexible. Because wire strings have some stiffness (remotely resembling a bar) the series of natural frequencies for string vibration is slightly, but audibly, inharmonic. Research has shown that this minute degree of inharmonicity is an important characteristic of the timbre. The inharmonicity also causes the optimum tuning of the instrument scale to be slightly stretched (flat on low notes, sharp on high notes) from the strictly mathematical scale of 2:1 octaves.

In keyboard string instruments the playing-key action causes a dynamic string excitation event. Some impact sound occurs immediately at the point of impact, and more comes through the string vibration system soon after. This is the ‘cause’ sound. The nature of both the impact and the resulting string excitation differs for each of the three major types of instrument, but in each case the shape of at least a portion of the tuned string segment is suddenly and momentarily changed. This string shape change is the pulse wave which travels up and down the string between the two contact points after the event occurs. This is the ‘effect’ sound.

The pulse wave is partially reflected back and forth repeatedly at the string contact points, causing the string vibration and the resulting note to sustain. However, some of the stored wave energy is removed during each reflection, eventually causing the note to die away. This vibratory energy, removed at the end of each round trip of the wave, vibrates the supporting structure, particularly the soundboard from which most of the sound energy enters the air. Only a small part of the sound reaches the air directly from the motion of the strings. The string typically vibrates at many frequencies simultaneously. Because these frequencies lie almost in a harmonic series, the listener hears the combination as a single note having a pitch corresponding to the lowest frequency in the series. The relative amount of sound produced at each frequency within the complex tone largely determines the timbre during the ‘effect’ portion of the tone. This depends on the shape of the pulse given to the string at impact, the response of the contact point of the supporting structure at each frequency of string vibration and the efficiency of the structure at each frequency in the transformation of vibration into sound.

In Table 1 the structural and mechanical differences between the three major types of keyboard string instrument are shown; the acoustical effects are explained below. For diagrams of the mechanisms described below, see articles on individual instruments.

Factor		Clavichord	Harpsichord	Piano


strings:	number per key	1 or less	1 per footage	2/3
	material	brass/steel	brass/steel	steel
	diameter stiffness	smaller	smaller	larger
	tension	low	medium	high
	bass wrapping	–	fine gauge	heavier gauge



structure:	string terminations	tangent, bridge	bridge, bridge	cast iron, bridge
	soundboard size	small	small to medium	medium to large
	structure material	wood	wood	metal and wood
	structure mass	light	medium	heavy



excitation:	mode	strike-hold	pluck	strike-rebound
	exciter material	metal	quill, leather	felt
	exciter shape	edge	tapered	curved
	exciter size	small	small	large to medium



damping:	mode	threaded through strings	jack weight/spring	damper head
	damper material	cloth	felt/cloth	felt
	damper shape	strip	flat	V/flat
	damper size	continuous	small	large

Acoustics, §III: Keyboard string instruments

2. Clavichord.

The clavichord is typically a rectangular box with the keyboard occupying the left two-thirds of the long side. The strings are stretched in a horizontal plane from hitch-pins near the left end to tuning-pins near the right end. In the right end there is a very small horizontal soundboard supporting a bridge (sometimes segmented) which provides one of the downbearing contact points for each string. The longer bass strings are closest to the keyboard; the shorter treble strings are near the rear. Each playing key is pivoted on a balance-rail, and the key is guided by a rear tongue which rides up and down in a vertical slot. The unique mechanical feature of the clavichord action is the tangent, a wedge-shaped piece of iron or brass borne upon the rear key extension. As the key is depressed, the tangent rises and strikes the string directly, remaining in contact with the string until the key is released. The tangent thus serves two functions: its impact creates the string excitation without the aid of any intermediate action mechanism; and it provides the second downbearing contact point for its associated string, thus defining the length of the vibrating string segment and controlling its pitch frequency. In other words, the striking point and the termination point of the tuned segment are the same.

In many clavichords the same string is used for several notes that would seldom be played in combination (e.g. C and C), by positioning the tangents at different points along the same string. Such instruments are known as fretted clavichords. Some others employ individual strings for greater musical versatility.

The sustained contact between the key-borne tangent and the struck string gives the clavichordist a ‘Bebung’ tonal effect not obtainable with other keyboard string instruments. When the player varies the key force periodically (after striking) the string tension fluctuates, producing a pitch vibrato.

Between the tangent strike points and the hitch-pins there are strips of cloth interwoven among the strings. They quickly damp string vibrations to the left of the strike point where the string segment would not be properly tuned. They also damp the entire string vibration when the tangent contact with the string is released, making a separate damper action unnecessary.

Clavichords have several inherent acoustical advantages and disadvantages. The metal edge striking at the string termination excites a complete frequency series. The pulse shape sharpens with harder blows, giving dynamic range in brilliance. However, both the maximum tangent velocity and the area of the soundboard are so small that the clavichord is limited in output. The direct connection through the key from the player’s finger to the string permits ‘Bebung’ vibrato, but it also provides mechanical damping which shortens note duration.

Acoustics, §III: Keyboard string instruments

3. Harpsichord.

Members of the harpsichord family have one of three orientations for the usually horizontally stretched strings. In the small rectangular virginals the strings run crosswise as in the clavichord; in the small wing-shaped spinet the strings extend obliquely from tuning-pins just behind the keyboard to hitch-pins in a row curving away to the right of the keyboard; and in the large harpsichord the strings extend directly away from the keyboard. This shape difference and the additional long bass strings make the harpsichord a larger instrument. Larger soundboards can provide greater tonal efficiency and fuller timbre.

In all three the strings are excited by the upward plucking action of a quill or plectrum which engages with, then releases the string. The plectrum projects from the central tongue portion of a narrow vertical jack which slides within guides, and is supported on the rear of the pivoted playing key. In large harpsichords there are two or more sets of strings and two or more manuals of keys. Thus different plectra borne by the same key can simultaneously pluck strings related by octave in the different string sets. The same string can also be plucked at different points along its length by plectra from different manuals. The closer the plectrum is to the string termination, the stronger are the higher, brighter partials of the note. The farther away, the stronger are the lower partials. Plectra may be quill, leather, wood, metal or plastic. The harder materials and the sharpest edges give the brightest timbre.

The most important acoustical difference between harpsichords and other keyboard string instruments is the plucking initiation of notes. The smallness and the sharpness of the plectrum edge, and the suddenness of string release as the plectrum passes on, produce a string waveform with ample high-pitched partials, giving the characteristic brilliance of harpsichord tone. In contrast with clavichord action, the string exciter leaves the string immediately and cannot absorb string vibration and so shorten note duration. The return of the plectrum on release of the playing key often produces a weak second excitation of the string. This stroke is minimized by the pivoting of the returning jack tongue. Although the resulting sound is a subtle characteristic of harpsichord tone, it happens so immediately before the jack-borne damper reaches the string that it sounds like part of the damping action.

Acoustics, §III: Keyboard string instruments

4. Piano.

The first Cristofori piano was a harpsichord with the string-plucking action replaced by an upstriking hammer action with escapement. This freed the leather-covered hammer before impact, allowing it to strike the string and bounce back, in contrast with the tangent action of the clavichord. The hammer action gave both piano and forte levels with fully controlled gradations in between, introducing to a concert situation the keyboard dynamic expression previously possessed only by the clavichord, which is too quiet to be played with other instruments. Cristofori’s instrument was the predecessor of the grand piano. The piano hammers were larger and more rounded than the previous tangents and plectra, imparting a larger, rounder pulse shape to the strings. This gave more power in the low partials and less in the higher ones, making piano tone fuller than that of the harpsichord. Hammer sizes and shapes were smaller and more pointed for high notes, where brilliance is needed.

Later a smaller piano was developed by combining a simple, upstriking hammer action with a string vibration system derived from that of the clavichord. Because it was small, this piano was acoustically weak. Many different action mechanisms were invented for this popular instrument, the rectangular or ‘square’ piano.

The substitution of hammers for plectra allowed multiple coplanar unison strings for each note, which had not been feasible in harpsichords. This increased tonal loudness and produced other advantages described below. Forward-striking hammer actions were developed for vertical piano string systems, requiring less floor space than horizontal pianos. Cast iron plates to support string forces and steel piano wire of higher tensile limits permitted larger wire diameters and higher-tension strings, increasing tonal power and brilliance. Compressed felt replaced leather hammer covering, softening the tone and increasing hammer control and durability. Crossing bass strings over tenor strings reduced piano size, and more oblique cross stringing and drop actions led to the small vertical piano.

Multiple strings for each individual note contribute significantly to the distinctiveness of piano tone, adding a choral effect to each tone because of slight differences in string tuning. Research has shown that slight detuning of unison strings (less than 0·1%) is aurally preferred to mathematically exact tuning; experienced piano tuners typically leave this tuning margin. Immediately after piano hammer impact the multiple strings vibrate in close synchronism. At this time the rate of power transfer to the bridge and soundboard is maximal, causing rapid tone diminution initially. Later the strings gradually become asynchronous because their frequencies are slightly different, and the note dies away more slowly. This characteristic dual decay rate in well-tuned pianos lets successive notes stand out clearly over recently played, sustained notes, and this has influenced the development of composition for the piano. The vibrations of struck strings travel along the piano bridge to other strings, which can vibrate sympathetically when the sustaining pedal lifts the dampers, producing a stronger choral effect. In grand pianos the soft pedal moves the action transversely, reducing the number of strings the hammer strikes.

Except in the treble range, each piano note contains many partials, with the strongest lying between 100 and 1000 Hz. Hammer-string impact sound, which spreads throughout the pitch range, is an important characteristic of piano tone but is noticed only in the treble, where the partials are too sparsely spaced to conceal it.

Standard procedures used by piano tuners for tuning octaves produce a stretched scale. Upper notes are higher and lower notes lower than strict equal temperament by about a third of a semitone at each extreme, an effect resulting from slight inharmonicity of the partials of string tone. Research has shown that pianists and listeners prefer the piano scale stretched this way, and attempts to synthesize true piano tone from a strictly harmonic series of partials have had limited success.

The science, engineering and art that combine in the evolution of the piano comprise what is known as ‘scale design’, the configuration of tonally related major structural parts. These include the strings, hammers, dampers, bridges, soundboard, plate and, to some extent, the case, along with their material properties.

Acoustics, §III: Keyboard string instruments

BIBLIOGRAPHY

F. Trendelenburg, E. Thienhaus and E. Franz: ‘Zur Klangwirkung von Klavichord, Cembalo und Flügel’, Akustische Zeitschrift, v (1940), 309

D. Martin: ‘Decay Rates of Piano Tones’, JASA, xix (1947), 535–41

R.E. Kirk: ‘Tuning Preferences for Piano Unison Groups’, JASA, xxxi (1959), 1644–8

D. Martin and W.D. Ward: ‘Subjective Evaluation of Musical Scale Temperament in Pianos’, JASA, xxxiii (1961), 582–5

D. Droysen: ‘Akustische Untersuchungen an Tasteninstrumenten des 18.–20. Jahrhunderts’, GfMKB: Leipzig 1966, 416–23

S. Tomek-Schumann: ‘Akustische Untersuchungen an Hammerflügeln’, JbSIM (1975), 127–72

R.-D. Weyer: ‘Time-Varying Amplitude-Frequency-Structures in the Attack Transients of Piano and Harpsichord Sounds’, Acustica, xxxv (1976), 232–52; xxxvi (1976), 241–58

E.L. Kent, ed.: Musical Acoustics: Piano and Wind Instruments (Stroudsburg, PA, 1977)

V.G. Porvienkov, M.V. Gridněv and N. Čelnokov: ‘Hodnocení hlasitosti a barvy zvuku klávesového nástroje’ [Measuring the volume and colour of the sound of keyboard instruments], Hudební nástroje, xv/1 (1978), 14–17

U.R. Müller: ‘Influence of Ribs on the Acoustic Behaviour of Piano Resonant Plates’, Archives of Acoustics, v (1980), 147–55

K. Wogram: ‘Die Bedeutung nichtstationärer Schwingungsvorgänge für die Bewertung von Musikinstrumenten’, Qualitätsaspekte bei Musikinstrumenten, ed. J. Meyer (Celle, 1988), 23–34

J. Meyer and K. Wogram: ‘Perspektiven der Musikinstrumentenakustik’, Acustica, lxix (1989), 1–12

M. Podlesak and A.R. Lee: ‘Effect of Inharmonicity on the Aural Perception of Initial Transients in Low Bass Tones’, Acustica, lxviii (1989), 61–6

A. Askenfelt, ed.: Five Lectures on the Acoustics of the Piano (Stockholm, 1990)

A. Askenfelt and E. Jansson: ‘From Touch to String Vibrations’, JASA, lxxxviii (1990), 52–62; xc (1991), 2383–93

A.I. de La Campa: Aproximación analítica a la interpretación en el piano (Madrid, 1990)

H. Suzuki and I. Nakamura: ‘Acoustics of Pianos’, Applied Acoustics, no.30 (1990), 147–205

X. Boutillon: Aperçu général sur les modèles physiques de piano (Paris, 1991)

A. Chaigne and A. Askenfelt: ‘Numerical Simulations of Piano Strings’ JASA, xcv (1994), 1112–18, 1631–40

I. Bork, H. Marshall and J. Meyer: ‘On the Radiation of Impact Noises from a Grand Piano’, Acustica, lxxxi (1995), 300–08

H. Conklin: ‘Design and Tone in the Mechanoacoustic Piano’, JASA, xcix (1996), 3268–96; c (1996), 695–708, 1286–98

Acoustics

IV. Wind instruments

1. Introduction.

2. Modes of oscillation of an air column.

3. Maintenance of oscillation.

4. Musically useful air column shapes.

5. Brass instruments.

6. Reed instruments.

7. Flute.

8. Wind instrument tone-colour.

9. Early wind instruments.

BIBLIOGRAPHY

Acoustics, §IV: Wind instruments

1. Introduction.

Every wind instrument consists of a long and carefully shaped duct coupled to an airflow control system that converts the steady wind supply from the player’s lungs, or from the wind chest of a pipe organ, into oscillations of the instrument’s air column. The mechanism for controlling the airflow can be the reed of a clarinet or bassoon, the vibrating lips of a trumpet player, or the less easily visualized steering of a jet of air from the flautist’s lips as it travels across the embouchure hole. In each of these the flow control device sends puffs of air in a regularly varying sequence into the instrument’s mouthpiece to keep the air column oscillating in its longitudinal vibratory motion. The nature and timing of these puffs are in turn controlled by acoustical variations taking place within the mouthpiece, these being a manifestation of the air column’s own oscillations. In order to make this two-part oscillating system useful for musical purposes, the performer must be able to select one or another of the possible sounds that it can generate. A bugler who plays the notes of a call is making such a selection, as is a woodwind player who uses a single fingering to sound notes in the bottom and second registers of the instrument. A musician is able to fill in the remaining gaps left in the musical scale by transforming the original air column of the instrument into a longer or shorter one. These will give alternative sets of sounds from which he can choose according to his needs. In modern brass instruments, changes in air column length are produced by the addition of various lengths of tubing, as exemplified by the slide of a trombone or the valve loops of a french horn. In woodwind instruments these length changes are accomplished by opening a greater or lesser number of tone holes arranged to pierce the air column wall at various points along it.

To be successful the design of a wind instrument must achieve several relationships between the air column and its flow control device: the two must be able to work together to permit the prompt and stable production of each one of the various notes in the scale, and these must be under good control by the player; the pitches of these notes must lie close to those belonging to the musical scale; and the tone-colour of the generated sounds must conform to aesthetic standards, which may vary from period to period and from nation to nation. The nature of these relationships and the way they may be attained is briefly outlined below.

Acoustics, §IV: Wind instruments

2. Modes of oscillation of an air column.

The sloshing of water in a length of rain gutter is made up of oscillatory motions which provide some insight into analogous motions that take place in the air column of a wind instrument. Fig.45 shows a water-filled trough and the first three modes of oscillatory motion that are possible within it; these modes are those having the lowest frequencies of oscillation. Observation of the motion of the water in such a trough makes it quickly apparent that the vertical motion of the water level is very different from the water’s back-and-forth flowing motion: points of small horizontal motion lie at points of maximum vertical motion; conversely, the vertical motion displays its nodes at those points where the horizontal motion is large. Vertical and horizontal motions are inextricably coupled to one another by the fact that the only way to change the water level at some point is to have water flowing towards or away from it.

There are important differences between the sloshing motion of water in a gutter and the oscillatory flow of air in a musical wind instrument: the first is a wave on the surface of an incompressible liquid, while the second is a longitudinal wave in a compressible gas. Nevertheless, a useful analogy can be drawn between the two types of wave motion. The vertical motion of the water (a consequence of a localized inflow or outflow of water) is the cognate of the rise and fall of air pressure at some point within the duct; obviously such changes in air pressure arise from the flow of air. Because woodwind reeds and brass players’ lips are ultimately controlled by pressure variations acting on them within the mouthpiece, and also because eardrums respond directly to the acoustic pressure variations exerted on them by the surrounding air, it is generally convenient to describe the behaviour of air columns in terms of the pressure aspect of their vibrations.

Every air column, regardless of its shape and the way in which its ends are terminated, has its own characteristic collection of vibration modes; each of these modes has its own pattern of flow and pressure and its own frequency of oscillation. Air columns of different shapes not only have different frequencies for their various modes of oscillation but also different ratios between them. It is a straightforward (though sometimes tedious) business to calculate the frequencies of an air column regardless of its shape, or, conversely, to calculate the shape needed to give a specified set of frequencies. As discussed in §4 below, there are only a very few basic air column shapes that can be coupled to a flow control device in order to make a useful note generator. It is a fortunate circumstance that these same shapes are also compatible with the requirements for proper tuning.

Acoustics, §IV: Wind instruments

3. Maintenance of oscillation.

A water analogue to a musical instrument can help one to visualize the way a flow control device maintains a steady oscillation. Fig.46 shows a device that could be called a water trumpet, in which a water supply valve is arranged to open or close progressively in response to the rise and fall of the water level at the shallow end of a tapering channel containing water. The end at which the valve is located is analogous to the mouthpiece end of a trumpet, and the valve itself replaces the player’s lips. A water valve arranged to open and close in this manner acts as a flow controller worked by variations in water height as it maintains oscillations in the duct. In a similar way, the single and double reeds of orchestral woodwind instruments and the lips of brass players function as flow controllers of the pressure-operated type, in that they open and close under the predominating influence of acoustic pressure oscillations within the mouthpiece cavity of the instrument. The generic name ‘reed-valve’ will be used for all these pressure-operated controllers, including those associated with brass instruments.

Any acoustic disturbance has a pattern of flows intertwined with a corresponding pattern of pressure variations. If attention is directed to the flow variations in a wind instrument air column, they should suggest the possibility of another kind of controller that operates by the flow aspect rather than the pressure aspect of the disturbance within the mouthpiece; this other type is found in the flute family.

In 1877 Helmholtz presented the simple theory of the maintenance of oscillation by means of a reed-valve, showing that such oscillations tend to occur at one or other of the natural frequencies of the air column to which the reed-valve is attached (see also Backus, 1963). Wilhelm Weber had already in 1830 elucidated the influence of the reed’s own elasticity on these natural frequencies (see Bouasse, 1929–30). It remained for Bouasse in the late 1920s to recognize that under certain conditions several modes of air column oscillation can act simultaneously on the reed-valve to facilitate the maintenance of a note. The implications of Bouasse’s observations were worked out and given practical application by Worman (1971) and Benade (1973).

The nature of the collaborative effect of several resonances can be summarized in the following terms. The various characteristic air column modes influence one another via the shared excitatory airflow. The valve must therefore come to terms with the oscillatory preferences of these modes to produce a self-consistent oscillation that includes several harmonically related frequency components in setting up what is known as a ‘regime of oscillation’. The name is chosen deliberately to draw attention to what can metaphorically be considered as political negotiations taking place between the air column’s own set of vibrational tendencies and those of the reed-valve, with the alliances changing as varying musical conditions give dominance to different members of the regime. Oscillation is particularly favoured when the air column has two or more natural frequencies arranged to coincide with the harmonics of the note being produced.

The operation of the reed-valve depends crucially on the fact that the relationship between the pressure on the downstream side of the valve and the rate at which air flows through it is non-linear (that is, a change in pressure does not simply produce a proportional change in air flow). It is this non-linearity which allows an initially unidirectional flow of air from the player’s mouth to destabilize the reed, resulting in the development of a periodic oscillation of the coupled system of reed and air column. The non-linearity is also an essential ingredient in the locking together of different air column modes into an oscillation regime with stable phase relationships. In recent years the theory of non-linear dynamics has been fruitfully employed to investigate such issues as the minimum blowing pressure necessary to make a wind instrument sound, and the changes in pitch, loudness and timbre which take place as the blowing pressure is increased beyond this threshold (see McIntyre and others, 1983; Fletcher, 1992; Kergomard, 1995; and Grand and others, 1996).

Acoustics, §IV: Wind instruments

4. Musically useful air column shapes.

We have seen that a stable regime of oscillation is more easily achieved in a wind instrument if it has a set of air column modes whose frequencies are in a harmonic relationship. No real wind instrument air column has exactly harmonic mode frequencies, but a conical tube complete almost to the apex comes quite close to this ideal. For this reason many woodwind instruments have approximately conical tubes: the oboe, bassoon and saxophone come into this category.

A cylindrical tube open at both ends also has a set of modes whose frequencies are, to a good approximation, members of a complete harmonic series. This is the acoustical basis of the flute, which has an effectively open embouchure hole at its upper end. A cylindrical tube closed at one end and open at the other has a set of harmonically related modes, but the even members of the series are missing. This is the case with the clarinet, which has a cylindrical tube effectively closed at the upper end by a cane reed.

In the brass instrument family, the player’s lips form a reed-type control valve which effectively closes the upper end of the tube. The bugle and the Swiss alphorn can be classed as conical instruments; so can the flugelhorn, saxhorn and many types of tuba, at least when the valves are not depressed. A trumpet or trombone, on the other hand, has cylindrical tubing over a substantial fraction of its length, with a final section which flares more and more rapidly into a pronounced bell. If a trombone consisted only of the cylindrical slide-section, it would have an odd-member-only harmonic series of modes, like the clarinet. The addition of the flaring section, together with the cup-shaped mouthpiece at the entrance, not only lowers all the mode frequencies but also reduces the pitch intervals between the modes. On a well-designed trombone or trumpet, the modes from the second upwards have frequencies close to a complete harmonic series, although the first mode frequency is much too low to fit the series.

Extending the slide on a trombone, or depressing valves on a trumpet or horn, increases the length of cylindrical tubing in the instrument. This lowers all the mode frequencies, but also increases the pitch intervals between the modes. It is thus impossible to achieve an ideal harmonic mode relationship for all slide positions or valve combinations. Similarly, opening side holes on a woodwind instrument does not merely shorten the effective length of the tube; it also changes the pitch intervals between the modes of the air column. There is thus no simple solution to the problem of designing a successful wind instrument with a large chromatic compass. Fortunately the non-linear coupling between air column and control valve can provide a strong oscillation regime even when only two or three modes have an approximately harmonic relationship, so the problems are less severe than they might at first appear.

Acoustics, §IV: Wind instruments

5. Brass instruments.

The vibration of a brass player’s lips is controlled by the oscillatory pressure present in the mouthpiece: this pressure is an aspect of the air column’s own oscillation. It is convenient to characterize the air column itself with the help of measurements carried out with an electronically operated pump (a special type of miniature loudspeaker) that produces a sinusoidally varying flow of air in and out of the mouthpiece cavity at any desired frequency. A tiny microphone placed inside the mouthpiece measures the amplitude of the resulting pressure variations. This microphone gives the desired air column response information, which can be displayed on a graph as a function of the pump driving frequency (see Benade, 1973). Such a graph will be called a ‘pressure response curve’; formally it is known to acousticians as an input impedance curve. Fig.47 shows an example of such a curve for a modern B trumpet; its general nature is typical of the pressure response curves of all brass instruments. Each of the peaks on this curve indicates a large pressure variation within the mouthpiece cavity, and each peak corresponds to excitation at the frequency of one of the modes of the air column.

As an illustration of the usefulness of a pressure response curve, consider what fig.47 shows for the playing of the written note c'. The figure indicates that a regime of oscillation is set up for this note involving response peaks 2, 4, 6 and 8 of the air column, which collaborate with the player’s lips to generate a steady oscillation containing many harmonically related partials. The lowest four of these partials get their major sustenance from the peaks named. When the trumpeter plays very softly, peak 2 dominates the oscillation and, because it is not very tall (i.e. the given excitation produces only a mild oscillation in this mode), the note is not well stabilized. As the musician plays louder the other peaks become influential and the note is steadier and better defined. The regime of oscillation for the written note g' is dominated by peak 3 with the cooperation of peaks 6 and 9. Since peak 3 is taller than peak 2, at pianissimo playing levels g' will be steadier than c'. During a crescendo the tall sixth peak enters the regime for g' and greatly stabilizes the oscillation, which gains some help also from peak 9. These are the acoustical reasons why g' is one of the easiest notes to play on a trumpet.

Further examination of fig.47 shows why the notes become increasingly hard to play as one moves up the scale. For example, g'' is still fairly easy to play softly because it is fed by the tall sixth peak; but during a crescendo it becomes progressively more ‘stuffy’, because the increasing dissipation of acoustic energy via the generation of higher partials in the note is inadequately offset by the contributions made by the small 12th peak. The note c''' is difficult at all levels, since it is sustained only by the eighth peak, which is not particularly tall and which, moreover, has no assisting peak in the neighbourhood of its second harmonic (near 1864 Hz).

In every case described above, the tuning of each note is determined not only by the resonance peak closest to the nominal frequency of the note, but also by any other peaks that lie near whole-number multiples of this frequency. If errors in the shape of the air column lead to resonance peaks that are not exactly in harmonic relationship, not only is the steadiness and clarity of the note spoiled by the less than perfect cooperation at forte levels, but also the player must compensate for pitch shifts that take place during crescendos and diminuendos as the misplaced resonances gain or lose their votes in the regime. The fact that misplaced resonances lead to changes of playing pitch during a crescendo provides the basis for an extremely sensitive technique for the adjustment of the proportions of brass instruments.

It has already been shown that the positions of the various resonance peaks are crucial to the proper speech of a wind instrument, and there have been hints that their frequencies may systematically be adjusted by suitable modifications of the air column shape. For example, if a mouthpiece is to work properly on any given brass instrument, there is a certain critical relationship that must be maintained between its total volume (cup plus backbore) and its ‘popping frequency’, which is the frequency of the sound made by slapping the rim closed against the heel of the hand. Similar adjustments can be made at the other end of the instrument. Skilled horn players develop great sensitivity in moving their right hand as they go from note to note. The placement of the hand in the bell ekes out the last bit of perfection in the alignment of the resonances of the instrument. On other brass instruments it is left to the maker to carry out similar but fixed adjustments of the bells.

In an air column’s resonance curve, the positions and alignment of the peaks are important, but so also are their heights. For example, the mouthpiece of a brass instrument has an acoustical duty beyond the simple one of helping to achieve suitable frequency relationships between the peaks: it is responsible also for the increased height of the middle four or five peaks relative to their low- and high-frequency neighbours, the maximum peak height lying roughly in the region of the mouthpiece popping frequency itself. Without this area of added height in the response peaks, even a perfectly aligned brass instrument tends to be difficult to play.

The bell also plays an important role in influencing the height of the resonance peaks, in that it causes the disappearance of the peaks above a certain frequency, determined by its rate of flare. The presence of the horn player’s hand in the bell raises the frequency above which there are no resonances, which allows the pressure response curve to have half a dozen additional resonance peaks. If these additional peaks are properly aligned, they will join with the other peaks to stabilize various regimes of oscillation and will also raise the upper limit of the player’s range. Much of the confusion surrounding the phenomenon of handstopping is resolved when proper account is taken not only of the fact that moving the hand rearranges the peaks which collaborate in producing the note, but also of the fact that handstopping makes additional peaks available to the collaboration.

Acoustics, §IV: Wind instruments

6. Reed instruments.

It was mentioned in §4 above that many reed woodwind instruments are based on the conical air column shape. In such instruments the apical segment of the prototypical cone is replaced by a reed cavity (or mouthpiece) with a staple, neck or bocal, while the lower, large end of the active bore extends down to the first of a row of open tone holes. It is important to recognize that the presence of closed tone holes on the bore significantly alters its acoustical behaviour.

Fig.48 shows the pressure response curves measured on an oboe for the air columns (including staple and reed cavity) used in playing the low-register notes b', f' and b (the curves for intermediate notes follow the trend implied in the figure). These curves are closely similar to those found for notes having similar fingering on the other conical woodwind instruments.

In reed instruments each low-register note is produced by a regime of oscillation involving the first (lowest-frequency) resonance peak, along with one or more other peaks whose frequencies match those of the next higher partials of the note being played. For the note b' there are only two peaks that participate in the regime, the higher-frequency peaks being much less tall besides being inharmonically positioned. On a good instrument f' is a much more stable note, being based on a negotiated agreement between three accurately positioned resonance peaks. Once again it is notable that above about 1300 Hz the peaks are not very tall, and they are irregular in their placement. The note b near the bottom of the oboe’s scale is produced by a regime of oscillation involving four accurately harmonic and fairly tall peaks, with one less tall peak whose position is a little below the frequency of the tone’s fifth partial. The size, spacing and chimney length of the tone holes determine the frequency above which the air column resonance peaks become less tall and more irregular in their position. The behaviour is reminiscent of the manner in which the bell of a brass instrument puts an upper limit on the number of resonance peaks. This explains why the bell of a woodwind instrument (even that of an english horn) can be replaced by an extension of the main bore, if this is provided with a suitably designed set of additional tone holes.

The second register of a conical woodwind instrument’s playing range is produced by regimes of oscillation involving response peak 2, along with peak 4 if it exists. The question arises as to how the reed can be persuaded to operate in such a regime. As a general principle, when one plays pianissimo on a reed instrument, oscillation is favoured at the frequency of the tallest air column response peak, and intermode cooperative effects are relatively unimportant. Fig.48 shows that the fingerings for b' and f' give air columns that favour low-register playing under these conditions, whereas pianissimo playing using the b fingering favours sound production an octave higher at b' because of the deficient height of the first peak (a deficiency characteristic of all nearly complete conical air columns). The tendency of the lowest notes to jump an octave in soft playing plagues every saxophonist, and also causes problems of harshness and instability for the player of double-reed instruments. When the b fingering is used to play loudly, the reed prefers the low-register regime, based on all four peaks, to the two-part regime, based on peaks 2 and 4. The different behaviour of the regimes under loud and soft playing conditions explains the functioning of the register hole of a woodwind instrument. This hole must produce two changes: it must cause peak 1 to become less tall than peak 2 in order to assure second-register playing at a pianissimo level; and it must shift the frequency of peak 1, giving it an inharmonic relationship with the other peaks (an inharmonicity chosen to produce the maximum possible disruption of any cooperative effects) in order to assure second-register playing at the forte level. The dotted curves in the middle segment of fig.48 show how opening a register hole alters the heights and positions of peaks 1 and 3 for the f' fingering, leaving peak 2 unscathed and able to cooperate in the production of the note f'' at all dynamic levels.

As in the case of brass instrument mouthpieces, there is a fixed relationship between the proportions of the air column of a conical instrument and the reed cavity and neck or bocal with which it can function. The active volume of the reed cavity (under playing conditions), with that of the associated tube, must closely approximate the volume of the missing apical segment of the basic cone. Furthermore, the playing frequency of the reed with its cavity and neck must (when sounded using a normal embouchure) agree with that of a doubly open cylindrical pipe whose length is that of the missing part of the cone. Systematic methods are available for the mutual adjustment of the air column, tone holes, reed cavity and neck of conical woodwind instruments. These can be as helpful in the fitting of a proper reed to an ancient instrument as they are in the construction of a modern one. The best instruments of all eras show great consistency with the principles outlined above.

The clarinet family (which uses a basically cylindrical air column) has properties remarkably similar to those of the conical woodwind instruments described above. The fingering used to produce the clarinettist’s e' is analogous to that used for b' on the oboe. Once again it gives an air column with only two response peaks, while the lower notes of the scale are played using air columns with an increasing number of active peaks, exactly as before. As discussed in §4, the replacement of the conical air column by a cylindrical one has the effect of shifting the natural frequency ratios from the 1:2:3 etc. harmonic series to a sequence of the type 1:3:5 etc. made up of the odd members of a harmonic series. This has two noteworthy musical consequences. First the clarinet has an enormously large and easily controlled dynamic range in the low register. As one goes from pianissimo playing (peak 1 alone being active) to a mezzo-piano level, the nascent second harmonic partial in the note occurs at a dip in the resonance curve, preventing a tendency for an abrupt growth of tone which would otherwise result from its entry into the regime of oscillation (e.g. as tends to happen on the saxophone). Harder blowing progressively brings in the influence of the third harmonic partial as it cooperates now with response peak 2. The successive entry of cooperative and anticooperative influences as the odd and even partials become important is what makes a crescendo so easily manageable on a clarinet. The second consequence is that the notes of the clarinet’s second register sound a 12th above the corresponding low-register notes, rather than an octave above as among the conical instruments.

Acoustics, §IV: Wind instruments

7. Flute.

As remarked in §2 above the airstream from a flute player’s lips is steered alternately into and out of the instrument’s embouchure hole under the influence of the flow aspect of the air column’s oscillation. It is worthwhile to maintain the analogy with the reed instruments, referring to the controlled airstream informally as an ‘air-reed’ to distinguish it from the aerodynamicist’s ‘edge tone’, with whose action it is often confused. It will suffice to note that an air-reed sets up regimes of oscillation in conjunction with the dips rather than the peaks of the air column response curves. Apart from this the behaviour is strictly analogous to that of reed instruments.

There are three basic shapes of air column that provide adequate cooperation with an air-reed by giving harmonically related response dips: the cylindrical pipe, the contracting cone and the expanding cone. Only the first two shapes are in common use. Certain subtleties of the cooperative action of an air-reed require a contraction of the bore near the blowing end as compared with the trend of the main bore. Thus a cylindrical tube requires a contracted headjoint, as exemplified by the Boehm flute, while the Baroque flute has its conical taper contracted into a cylinder in the headjoint. In both cases the volume of the small cavity existing between the cork and the embouchure hole can be adjusted to match the proportions of the hole itself to the rest of the instrument and to its player.

Acoustics, §IV: Wind instruments

8. Wind instrument tone-colour.

There are four significant influences on the tone-colour of a wind instrument. First, varying the profile of the reed tip and the mouthpiece tip on single-reed instruments changes the relation between the shapes of the puffs of air that come through the reed and the acoustic stimulus (in the mouthpiece) that controls them, thus causing modifications in the strengths of the generated partials. Similar behaviour is observed in brass instruments, flutes and organ pipes. Second, the number of cooperating peaks in the regime of oscillation and their height directly influence the strengths of the partials of a note generated within the instrument. Partials that are directly sustained by the cooperating peaks have strengths roughly proportional to the height of the peaks. The higher partials, lying at frequencies where the peaks are irregular or nonexistent, are weak because they are by-products of the main oscillation. Third, the transmission of sound out of an instrument into the room via the bell of a brass instrument or via the set of open woodwind tone holes also affects the tone-colour. This transmission is small for the lower partials of the internally generated note, rising steadily to a maximum value at the frequency at which the resonance peaks disappear. The resulting ‘treble boost’, characteristic of the emission process, partially offsets the progressively weaker generation of the higher partials. One hears the aggregate result of both effects. Experiment shows that a rise of only 2–3% in the frequency beyond which there are no resonance peaks makes an easily perceived brightening of the tone-colour on any wind instrument. Fourth, misalignments in the resonances will make many changes in the whole sound, but the acceptability of such misalignments is limited by the accompanying deterioration in the responsiveness of the instrument.

In the family of trumpets and trombones, a further factor comes into play which is responsible for a dramatic increase in brightness of timbre in very loud playing. The adjectives frequently used to describe this tone quality – ‘brassy’ or ‘metallic’ (cuivré in French) – reflect a common misconception that the effect arises from vibration of the metal bell of the instrument. In 1996 Hirschberg and his co-workers showed that the cause is in fact the development of shock waves in the cylindrical section of the air column. At the point in the vibration cycle at which the player’s lips open, a large pressure jump is created in the mouthpiece. This pressure rise becomes steeper and steeper as it travels down the tube; by the time it reaches the bell it has become an extremely short and powerful pulse. The form of this shock wave is similar to that created by the passage of a supersonic aircraft, and the sound of a fortissimo g' on a trombone has been graphically described as ‘four hundred sonic bangs per second’ (Gilbert and Petiot, 1997).

Acoustics, §IV: Wind instruments

9. Early wind instruments.

The last quarter of the 20th century saw a remarkable growth of interest in the performance of music on instruments typical of the period in which the music was composed. Acoustical studies of surviving original instruments have helped to clarify some of the important differences between early and modern instruments, and have provided useful guidance to makers engaged in manufacturing reproductions (see Drinker and Bowsher, 1993; Benade, 1994; Campbell, 1994, 1996; Myers, 1995; and Escalas and others, 1998).

Some instruments have undergone a continuous and relatively subtle evolution. The trombone, which appeared after the mid-15th century, remained a very similar instrument acoustically during the following four centuries. In the 20th century the bore diameter of the typical orchestral trombone increased significantly, as did the diameter of the bell. These changes, together with modifications of the mouthpiece, increased the acoustic power of the instrument, and also tended to reduce the brightness of timbre, especially in loud playing.

The most dramatic change in the acoustics of trumpets and horns occurred in the early 19th century, with the invention of the valve. Before this time the air column length of a trumpet or horn was fixed, and the playing technique relied heavily on natural tones corresponding to the frequencies of the air column modes. On the horn the technique of hand-stopping was used to modify the mode frequencies to provide additional notes, although the changes in pitch were inevitably accompanied by some changes in timbre. Natural trumpet technique relied on the ability of the player to alter the pitches of certain natural notes by changes of embouchure, a technique described as ‘lipping’. It has been noted that the pressure response curves of Baroque natural trumpets have less sharply peaked resonances than do those of modern trumpets, making it easier to lip notes on the older instruments (Smithers and others, 1986).

Some wind instruments of the medieval and Renaissance periods fell completely out of use in the 18th and 19th centuries. Although an evolutionary line can be traced from the conical-bored double-reed shawms to the modern oboe and bassoon, the present-day orchestra contains no descendants of the crumhorn or the racket, which were double reed instruments with narrow-bored cylindrical air columns. Nor can modern equivalents be found for the cornett and the serpent, which were conical tubes with side finger holes, normally fitted with cup mouthpieces and sounded by the lips. The 17th-century serpent, although an instrument of great charm, suffered from serious acoustical problems related to the small size and irregular spacing of the fingerholes. These acoustical difficulties were largely overcome in the fully keyed ophicleide, which enjoyed considerable popularity in the 19th century. The cornett had developed an acoustically satisfactory form by the 16th century; its combination of lip excitation, short tube length and side hole fingering gave it a high degree of flexibility and agility. The cornett became the supreme wind instrument of the early Baroque period and has been successfully revived in the late 20th century by a new generation of virtuoso performers.

Acoustics, §IV: Wind instruments

BIBLIOGRAPHY

H. von Helmholtz: Die Lehre von den Tonempfindungen (Brunswick, 1863, 4/1877; Eng. trans., 1875/R, 2/1885/R, 6/1948, as On the Sensations of Tone)

H. Bouasse: Instruments à vent (Paris, 1929–30/R)

A.H. Benade: ‘On the Mathematical Theory of Woodwind Finger Holes’, JASA, xxxii (1960), 1591–1608

J. Backus: ‘Small-Vibration Theory of the Clarinet’, JASA, xxxv (1963), 305–13

A.H. Benade and J. French: ‘Analysis of the Flute Head Joint’, JASA, xxxvii (1965), 679–91

J. Coltman: ‘The Sounding Mechanism of the Flute and Organ Pipe’, JASA, xliv (1968), 983–92

W. Worman: Self-Sustained Nonlinear Oscillations of Medium Amplitude in Clarinet-Like Systems (diss., Case Western Reserve U., Cleveland, 1971)

K. Wogram: Ein Beitrag zur Ermittlung der Stimmung von Blechbläsinstrumenten (diss., Technical U. of Brunswick, 1972)

A.H. Benade: ‘The Physics of Brasses’, Scientific American, ccxxix/1 (1973), 24–35

N.H. Fletcher: ‘Nonlinear Interactions in Organ Flue Pipes’, JASA, lvi (1974), 645–52

E.V. Jansson and A.H. Benade: ‘On Plane and Spherical Waves in Horns of Non-Uniform Flare’, Acustica, xxxi (1974), 80–98, 185–202

A.H. Benade: Fundamentals of Musical Acoustics (New York, 1976, 2/1990)

N.H. Fletcher: ‘Mode Locking in Non-Linearly Excited Inharmonic Musical Oscillators’, JASA, lxiv (1978), 1566–9

J.W. Coltman: ‘Acoustical Analysis of the Boehm Flute’, JASA, lxv (1979), 499–506

S.J. Elliot and J.M. Bowsher: ‘Regeneration in Brass Wind Instruments’, Journal of Sound and Vibration, lxxxiii (1982), 181–217

M.E. McIntyre, R.T. Schumacher and J. Woodhouse: ‘On the Oscillations of Musical Instruments’, JASA, lxxiv (1983), 1325–45

D. Smithers, K. Wogram and J. Bowsher: ‘Playing the Baroque Trumpet’, Scientific American, ccliv/4 (1986), 108–15

M. Campbell and C. Greated: The Musician’s Guide to Acoustics (London, 1987/R)

N.H. Fletcher: ‘Autonomous Vibration of Simple Pressure-Controlled Valves in Gas Flows’, JASA, xciii (1992), 2172–80

P.A. Drinker and J.M. Bowsher: ‘The Application of Noninvasive Acoustic Measurements to the Design, Manufacture and Reproduction of Brass Wind Instruments’, HBSJ, v (1993), 107–31

A. Benade: ‘Woodwinds: the Evolutionary Path since 1700’, GSJ, xlvii (1994), 63–110

D.M. Campbell: ‘The Sackbut, the Cornett and the Serpent’, Acoustics Bulletin (1994), May/June, 10–14

J.P. Dalmont and others: ‘Some Aspects of Tuning and Clean Intonation in Reed Instruments’, Applied Acoustics, xlvi (1995), 19–60

A. Hirschberg, J. Kergomard and G. Weinreich, eds.: Mechanics of Musical Instruments (Vienna, 1995) [incl. J. Kergomard: ‘Elementary Considerations on Reed-Instrument Oscillations’, 229–90; A. Hirschberg: ‘Aeroacoustics of Wind Instruments’, 291–369]

A. Myers: Characterisation and Taxonomy of Historical Brass Musical Instruments from an Acoustical Standpoint (diss., U. of Edinburgh, 1995)

D.M. Campbell: ‘Cornett Acoustics: Some Experimental Studies’, GSJ, xlix (1996), 180–96

N. Grand, J. Gilbert and F. Laloe: ‘Oscillation Threshold of Woodwind Instruments’, Acustica, lxxxii (1996), 137–51

A. Hirschberg and others: ‘Shock Waves in Trombones’, JASA, xcix (1996), 1754–8

J. Gilbert and J.-F. Petiot: ‘Brass Instruments: Some Theoretical and Experimental Results’, Proceedings of the International Symposium on Musical Acoustics: Edinburgh 1997 [Proceedings of the Institute of Acoustics, xix (1997)], 391–400

N.H. Fletcher and T.D. Rossing: Physics of Musical Instruments (New York, 2/1998)

C.J. Nederveen: Acoustical Aspects of Woodwind Instruments (DeKalb, IL, 2/1998)

R. Escalas, A. Barjau and V. Gibiat: ‘Les instruments de ménétriers de la cathédrale de Salamanque’, Acoustique et instruments anciens: Paris 1998

Acoustics

V. Percussion instruments

Percussion instruments generally use one or more of the following basic types of vibrators: bars, membranes, plates, air columns or air chambers. Except for air columns, the frequencies of the modes of vibrations in these components are not related harmonically; therefore percussion instruments are characterized by inharmonic partials in their sound. Another characteristic is the constant change in amplitude of their sounds, rising rapidly at the onset and immediately beginning to die away without reaching a steady state, as do the sounds of most string and wind instruments.

1. Drums.

Drums are probably our oldest musical instruments (with the exception of the human voice). The sounds of some drums, such as kettledrums, tablā and boobams, convey a strong sense of pitch but others do not; in the latter category are the bass drum, snare drum, tenor drum, tom-toms, bongos, congas and many drums of African and East Asian origin. As vibrating systems, drums can be divided into three categories: those consisting of a single membrane coupled to an enclosed air cavity (such as kettledrums), those consisting of a single membrane open to the air on both sides (such as some tom-toms and congas), and those consisting of two membranes coupled by an enclosed air cavity (such as bass drums and snare drums).

The first 12 vibrational modes of an ideal membrane are shown in fig.49. Their frequencies depend upon the radius, the tension and the mass per unit area. The normal mode frequencies of real membranes in a drum may be quite different from those of an ideal membrane, however. The principal effects acting to change the mode frequencies are air mass loading, membrane stiffness and the pressure of air enclosed within the drum, if any. In general air loading lowers the modal frequencies, while the other two effects tend to raise them. In thin membranes air loading is usually the dominant effect.

Although the modes of vibration of an ideal membrane are not harmonically related, a carefully tuned kettledrum is known to sound a strong principal note with two or more harmonic overtones. In the 19th century the physicist Lord Rayleigh recognized the principal note as coming from the (1,1) mode (at frequency f₁) and identified overtones about a perfect 5th (f/f₁ = 1·50), a major 7th (f/f₁ = 1·88) and an octave (f/f₁ = 2·00) above the principal tone. He correctly identified these overtones as originating from the (2,1), (3,1) and (1,2) modes respectively, which in an ideal membrane should have frequencies of 1·34, 1·6 and 1·83 times the frequency of the (1,1) mode. Modern experiments have verified Rayleigh's results (see Rossing, 1982).

Air mass loading, which lowers the low-frequency modes more than those of higher frequency, is mainly responsible for establishing the harmonic relationship of kettledrum modes. Other effects, such as membrane stiffness and the size and shape of the kettle, merely fine-tune the frequencies, although they may have considerable effect on the rate of decay of the sound.

Harmonic mode tuning on Indian drums, such as the tablā and mrdangam, takes place in a different way than in the kettledrum. In these drums, with their small membranes, the effect of air mass loading is quite small, and so many layers of black paste are skilfully applied to load the drumheads by the required amount.

The physicist C.V. Raman studied the acoustical properties of tablā and correctly identified five harmonic partials as originating from nine normal modes of vibration, several of which have the same frequencies. The fundamental tone is from the (0,1) mode; the 2nd harmonic is from the (1,1) mode; the (2,1) and (0,2) modes provide the 3rd harmonic; the (3,1) and (1,2) modes supply the 4th harmonic; and the (4,1), (0,3) and (2,2) modes contribute to the 5th harmonic (see Raman, 1934).

In double-headed drums, such as the snare drum and the bass drum, there is considerable coupling between the two heads as they vibrate, especially at low frequency. This coupling, which takes place mechanically through the drum shell and acoustically through the enclosed air, leads to the formation of mode pairs. In fig.50, mode pairs in a freely suspended snare drum based on the (0,1) and (1,1) modes of each membrane are shown. When the drum is placed on a stand, further mechanical coupling on the modes of the support structure occur (see Rossing, 1992).

2. Mallet instruments.

Marimbas, xylophones, vibraphones and glockenspiels employ tuned bars of wood, metal or synthetic material. These bars can vibrate by bending (transverse modes), twisting (torsional modes) or elongating (longitudinal modes). Although longitudinal and torsional modes in bars or beams of uniform cross section have nearly harmonic frequencies, transverse modes are quite inharmonic. Since transverse modes are mainly used in mallet percussion instruments, harmonic tuning must be accomplished by shaping the bars appropriately.

The modes of transverse vibration in a bar or rod depend upon the end conditions. Three different end conditions are commonly considered: free, simply supported (hinged) and clamped. There are six different combinations of these end conditions, each leading to a different set of vibrational modes. Three of the more common combinations are shown in fig.51.

A deep arch is cut in the underside of marimba bars, particularly in the low register. This arch serves two purposes: it reduces the length of the bar required to reach the low pitches, and it allows tuning of the overtones (the 1st overtone is normally tuned two octaves above the fundamental). Marimba resonators are cylindrical pipes tuned to the fundamental mode of the corresponding bars. A pipe with one closed end and one open end resonates when its acoustical length is a fourth of a wavelength of the sound. The purpose of the tubular resonators is to emphasize the fundamental and also to increase the loudness, which is done at the expense of shortening the decay time of the sound.

Xylophone bars are also cut with an arch on the underside, but the arch is not as deep as that of the marimba, since the first overtone is tuned to a 12th above the fundamental (that is, three times the frequency of the fundamental). Since a pipe closed at one end can also resonate at three times its fundamental resonance frequency, a xylophone resonator reinforces the 12th as well as the fundamental. This overtone boost, plus the hard mallets used to play it, give the xylophone a much crisper, brighter sound than the marimba.

Vibraphones or vibraharps have deeply arched bars, so that the first overtone is two octaves above the fundamental, as in the marimba. The aluminium bars tend to have a much longer decay time than the wood or synthetic bars of the marimba or xylophone, so ‘vibes’ are equipped with pedal-operated dampers. The most distinctive feature of vibe sound is the vibrato introduced by motor-driven discs, known as ‘vanes’, at the top of the resonators, which alternately open and close the tubes. The vibrato produced by these rotating vanes consists of rather substantial fluctuations in amplitude (intensity vibrato) and a barely detectable change in frequency (pitch vibrato).

Chimes, or tubular bells, are usually fabricated from lengths of brass tubing 3–4 cm in diameter. Although they are tubular, the modes of vibration excited by striking with a mallet are essentially those of a beam or bar with two free ends. An interesting acoustical property of chimes is that there is no mode of vibration with a frequency at, or even near, the pitch of the strike tone one hears. Modes 4, 5 and 6 appear to determine the strike tone. These modes are nearly in the ratio 2:3:4, so the ear considers them as overtones of a missing fundamental an octave below mode 4.

3. Cymbals and gongs.

The vibrational modes of a circular plate are similar in shape to those of the circular membrane shown in fig.49, although the frequencies are quite different. Using holographic interferometry, more than 100 plate modes have been observed in an orchestral cymbal.

The level below about 700 Hz shows a rather rapid decrease during the first 0·2 second after striking; this is apparently due to conversion of energy into modes of higher frequency. Sound energy in the range of 3–5 kHz, which gives the cymbal its ‘shimmer’ or aftersound, peaks about 0·05–0·1 seconds after striking, and may become the most prominent feature in the sound spectrum of a cymbal (see Rossing and Shepherd, 1983).

Gongs play a very important role in East Asian as well as Western music. Gongs used in symphony orchestras are usually 0·5–1 metre in diameter, cast of bronze with a deep rim and a protruding dome. When they are struck near the centre with a massive soft mallet, the sound builds up relatively slowly and continues for a long time.

4. Steel drums.

Steel drums or pans are fabricated from 55-gallon oil drums. The first step in making a steel drum is to hammer the end of the oil barrel to the shape of a shallow basin; then a pattern of grooves is cut with a nail punch to define the individual note areas, which may range from 28 in a single tenor to only three in a bass pan. Modern steel bands span five octaves, from around G' to g'''. A skilled pan maker tunes at least one mode of vibration in each note to a harmonic of the fundamental (usually the octave) and, if possible, another mode to the 3rd or 4th harmonic. Additional harmonics in the sound spectrums of steel drum notes result from sympathetic vibration of nearby note areas and from non-sinusoidal motion of the note area itself.

5. Bells.

When struck by its clapper a bell vibrates in a complex way, which can be described in terms of vibrational modes resembling those of a circular plate, with the nodal diameters replaced by nodal meridians. The first five modes of a church bell or carillon bell are shown in fig.52. Dashed lines indicate the locations of the nodes. The numbers (m, n) indicate the numbers of complete nodal meridians extending over the top of the bell (half the number of nodes observed at the mouth) and the numbers of nodal rings respectively. Since there are two modes with m=2 and n=1, one with a nodal ring at the waist and one with a nodal ring near the mouth, we denote the second one as (2,1#); likewise for (3,1#).

Handbells are much thinner and lighter than church bells and carillon bells. They have no thickened soundbow, and they employ relatively soft clappers to give a delicate sound. In recent years handbell choirs have become popular in schools and churches; there are an estimated 40,000 handbell choirs in the USA alone.

Generally the first and second vibrational modes of a handbell are tuned to a 3:1 frequency ratio. Each of these modes radiates a double-frequency partial as well, however, so the sound spectrum of a handbell includes a fundamental, a second harmonic, a third harmonic and a sixth harmonic (see Rossing and Sathoff, 1980).

BIBLIOGRAPHY

C.V. Raman: ‘The Indian Musical Drum’, Proceedings of the Indian Academy of Sciences, section A, i (1934), 179–88

T.D. Rossing and H.J. Sathoff: ‘Modes of Vibration and Sound Radiation from Tuned Handbells’, JASA, lxviii (1980), 2225–6

T.D. Rossing: ‘The Physics of Kettledrums’, Scientific American, ccxlvii (1982), 172–8

T.D. Rossing and R.B. Shepherd: ‘Acoustics of Cymbals’, Proceedings of the 11th International Congress on Acoustics: Paris 1983 (Paris, 1983), 329–33

T.D. Rossing: ‘Percussion Instruments’, The Science of Sound (Reading, MA, 2/1990), 257–86

T.D. Rossing: ‘Acoustics of Drums’, Physics Today, xlv/3 (1992), 40–47

C.-R. Schad and G. Frik: ‘Klangfiguren einer Glocke’, Acustica, lxxviii (1993), 46–54

T.D. Rossing: ‘Modes of Vibration and Sound Production in Percussion Instruments’, Modèles physiques, création musicale et ordinateur, no.1 (1994), 75–112

C.-R. Schad and G. Frik: ‘Über den Schlagklang von Glocken’, Acustica, lxxx (1994), 232–7

T.D. Rossing, D.S. Hampton and U.J. Hansen: ‘Music from Oil Drums: the Acoustics of the Steel Pan’, Physics Today, xlix/3 (1996), 24–9

N.H. Fletcher and T.D. Rossing: The Physics of Musical Instruments (New York, 2/1998)

Acoustics

VI. The voice

1. Introduction.

2. Air pressure supply.

3. Oscillator.

4. Resonator.

5. The singing voice.

BIBLIOGRAPHY

Acoustics, §VI: The voice

1. Introduction.

The voice organ can be regarded as a wind instrument consisting of an air pressure supply driving an oscillator, the output signal of which is fed into a resonator from which the sound is radiated to the air outside the instrument (see fig.53). The air pressure supply is the respiratory system (i.e. the lungs and the respiratory muscles). In the case of voiced sounds, the oscillator is the set of vocal folds (earlier also called cords); they convert the airstream from the lungs into a complex sound built up by harmonic partials. For voiceless sounds the oscillator is a narrow slit through which the airstream is forced; the laminar airstream is then converted into a turbulent airstream which generates noise. The sound generated by the oscillator is called the ‘voice source’. It propagates through the resonator constituted by the cavities separating the oscillator from the free air outside the instrument. In resonators the ability to transmit sound varies considerably with the frequency of the transmitted sound. At certain frequencies (the resonance frequencies), this ability reaches maximum. Thus in the case of the voice, those voice source partials that lie closest to a resonance are radiated with higher amplitudes than other partials. In this way the spectral form of the radiated sound mirrors the properties of the resonator. The resonances and the resonance frequencies of the vocal tract are called ‘formants’ and ‘formant frequencies’ respectively.

Acoustics, §VI: The voice

2. Air pressure supply.

In singing, the air pressure is much more carefully regulated than in normal speech, by a skilled control of the inspiratory and expiratory muscles. The air pressure provided by the respiratory system in singing varies with pitch and vocal effort, generally between 5 and 40 cm of water. The resulting air flow depends also on the glottal conditions. Air flow rates of 0.1–0.3 litres per second have been observed in singers. These air pressure and air flow ranges do not appear to deviate appreciably from values observed in untrained speakers.

Acoustics, §VI: The voice

3. Oscillator.

(i) Voiced sounds.

The vocal folds originate at the angle of the thyroid cartilage, course horizontally backwards and are inserted into each of the arytenoid cartilages. By adduction (i.e. drawing these cartilages towards each other), the slit between the folds, called the ‘glottis’, is narrowed, and an airstream can set the folds into vibration. A vibration cycle can be described as follows. When the glottis is slightly open an airstream from the lungs can pass through it. This airstream throws the vocal folds apart and at the same time generates a negative pressure along the edges of the folds. The sucking effect of this negative pressure along with the elasticity and other mechanical properties of the folds closes the glottis again. Then the air pressure difference across the glottis throws the folds apart, thus starting the next vibratory cycle. The frequency of a vibration is determined by the transglottal air pressure difference and the mechanical properties of the folds. A high pressure difference or tense and thin vocal folds, or both, give a high vibration frequency; converse states give a low frequency. The mechanical properties of the folds are regulated by a series of muscles that vary the length and stiffness of the folds by manipulating the positions of the laryngeal cartilages. Thus these muscles are used to regulate the vibration frequency. As the vibration frequency determines the pitch perceived, these muscles are often referred to as the ‘pitch regulating muscles’. An increase of the subglottal pressure raises the amplitude of the sound produced and also increases the vibration frequency, raising the pitch. Thus, in order to perform a crescendo at a constant pitch a singer has to raise the subglottal pressure and simultaneously compensate for the pitch increase by adjusting the pitch-regulating muscles.

By vibrating, the vocal folds repeatedly interrupt the airstream from the respiratory system. Thus they act as a valve oscillating between open and closed positions: the result is a chopped airstream corresponding to a complex sound, the fundamental frequency of which is equal to the vibration frequency of the folds. The glottis is schematically shown as a function of time in fig.54. The horizontal portion of the curve corresponds to the closed phase of the glottal vibration cycle, and the triangular portion is the open phase. As the air flow generally increases more slowly than it decreases, the triangular part of the curve is asymmetrical in the figure. In trained voices the glottal closure is often observed to be more efficient than in untrained voices. Also, the vibration pattern appears to vary considerably less with pitch and vocal intensity in trained voices than in untrained ones (see Sundberg, Andersson and Hultqvist, 1999).

The sound generated by the chopped transglottal airstream is built up by a great number of harmonic partials whose amplitudes generally decrease monotonically with frequency, roughly by 12 dB per octave at neutral loudness. It is noteworthy that this holds as an approximation for all voiced sounds. Partials of measurable amplitude in the source spectrum are generally found up to 4–6 kHz. This means that a tone with a fundamental frequency of 100 Hz may contain between 40 and 60 partials of appreciable amplitude. However, the amplitudes of the source spectrum partials vary with pitch and vocal intensity (see Sundberg, Andersson and Hultqvist, 1999).

(ii) Voiceless sounds.

The sound source in this case is noise generated by a turbulent airstream. The narrow slit required for the noise generation can be formed at various places along the vocal tract, the lowest position being at the glottis itself, which can be kept wide enough to prevent the folds from vibrating and narrow enough to make the airstream turbulent. This is the oscillator used in the ‘h’ sound. Another place used in some languages is the velar region, which can be constricted by the tongue hump. The resulting sound is used as the voice source in the German ‘ach’ sound. In most remaining unvoiced sounds the tongue tip constricts the vocal tract in the palatal, alveolar or dental regions as in the initial phonemes of ‘sheep’, ‘cheap’ and ‘sip’. In the ‘f’ sound the upper incisors and the lower lip provide the slit.

Acoustics, §VI: The voice

4. Resonator.

The frequencies of the formants are determined by the shape of the resonator. In the case of non-nasalized sounds the resonator consists of the pharynx and mouth cavities. In vowels these cavities constitute a tube resonator which may be regarded as closed at the glottal end and open at the lip end. The average vocal tract length for males is generally considered to be 17·5 cm. A tube of that length and having a uniform cross-sectional area would display a series of resonances falling close to the odd multiples of 500 Hz. However, as the cross-sectional area of the vocal tract is not constant, the formants deviate from these frequencies. The vocal tract shape is determined by the positions of the articulators (i.e. the lips, the jaw, the tongue, the velum and the larynx). The positions of these articulators are continuously varied in singing and in speech, so that the formants are tuned to various target frequencies. Thus each vowel sound corresponds to a certain pattern of articulator positions.

The dependence of the formant frequencies on the articulatory configuration is rather complex. Only a few factors have the same type of effect on all formant frequencies; for instance, all formants drop in frequency more or less when the vocal tract length is increased, by protrusion of the lips or lowering of the larynx or both, and when the lip opening area is decreased. Moreover, certain formants are more dependent on the position of a specific articulator than are others. The first formant frequency is particularly sensitive to the jaw opening: the wider the jaw opening, the higher the first formant frequency. The second and third formant frequencies are especially sensitive to the position of the tongue body and tongue tip respectively. The highest frequencies of the second formant (2–3 kHz) are obtained when the tongue body constricts the vocal tract in the palatal region, as in the vowel of ‘keep’. The lowest values of the third formant (around 1500 Hz) are associated with a tongue tip lifted in a retroflex direction. Fig.55 provides examples of articulatory configurations associated with some vowels.

These guidelines apply to oral sounds; in nasalized sounds the dependence of the formants on the articulator positioning becomes considerably more complex. The nasal tract introduces minima in the sound transfer of the vocal tract resonator. The acoustical effect of nasalization varies between vowels, but a general feature is that the lowest partials are emphasized.

For both oral and nasalized sounds the two lowest formant frequencies are generally decisive in the vowel quality perceived. Frequencies typical of male speakers are given in fig.56. Females have shorter vocal tracts and therefore higher formant frequencies. On average for vowels, the three lowest formant frequencies of female voices are 12, 17 and 18% higher, respectively, than those of male voices. Children, having still shorter vocal tracts, possess formant frequencies that are 35–40% higher than those of males (see Fant, 1973).

The amplitudes of the partials emitted from the lip opening depend on the sound transfer ability of the vocal tract. This ability depends not only on the partials’ frequency distance from the closest formant, but also on the frequency distance between formants. Thus a halving of the frequency distance between two formants increases the sound transfer ability by 6 dB at the formant frequencies and by 12 dB midway between the formant frequencies, other things being equal. Another factor important to the amplitudes of the radiated partials is the sound radiation properties of the lip opening, which boosts the entire spectrum envelope by 6 dB per octave. For this reason, the amplitudes of all spectrum partials tend to increase with the pitch even when there is no change in vocal effort.

Acoustics, §VI: The voice

5. The singing voice.

Basically the voice organ seems to be used in the same way in singing as in speech. In both cases the sound produced is entirely determined by the properties of the sound source and the vocal tract resonances. In other words, there seems to be no reason to assume that in non-nasalized vowels, resonance outside the vocal tract, such as in the maxillary sinuses or the lungs, contributes to the acoustic output to any appreciable extent. In singing, however, the possibilities inherent in the normal voice organ are used in quite special ways.

(i) Breathing.

The demands on the breathing apparatus differ significantly between speech and singing. There are two main reasons for this. Firstly, phrases in neutral speech are generally short, typically using no more than 15–20% of lung capacity. In singing, on the other hand, phrases tend to be considerably longer, using twice as much and occasionally nearly 100% of lung capacity. As the recoil forces of the respiratory apparatus vary with lung volume, a singer needs to supply different degrees of respiratory muscle force depending on lung volume. Secondly, the mean over-pressure of air in the lungs, which controls the loudness of phonation, is basically constant in neutral speech, although it is raised for emphasized syllables. In singing, as higher pitches require higher pressures, this air pressure needs to be varied with pitch. As lung pressure affects pitch, failures to reach target pressures result in singing off the pitch. Singers generally use the diaphragm muscle for inhalation, which is reflected in an expansion of the abdominal wall. However, the strategy used for achieving the necessary control of the respiratory apparatus differs between singers. Some contract the abdominal wall, thus raising the level of the diaphragm in the trunk before phonation, while others keep the abdominal wall expanded and thus the diaphragm low in the trunk at the initiation of a phrase. Some even contract both abdominal wall muscles and diaphragm during singing. It is frequently assumed that these different strategies affect the function of the vocal folds and hence the voice timbre (see Thomasson and Sundberg, 1997).

(ii) Vibrato.

One of the typical peculiarities of opera and concert singing is vibrato. In Western operatic singing its acoustical correlate is an undulation of the frequencies and amplitudes of the partials (fig.57). The undulation is almost sinusoidal and has a rate of about 5–7 Hz in good voices. The rate is generally constant within a singer, although it tends to slow down with advanced age. The magnitude of the frequency excursions is of the order of ±70 cents, but greater variation occurs for expressive purposes and at advanced age. Vibrato tends to increase in regularity as voice training proceeds successfully. The frequency and amplitude undulations are synchronous but not necessarily in phase, depending on the frequency distance between the strongest spectrum partial and the nearest formant. If the strongest partial is slightly below the strongest formant, an increase in frequency will cause the amplitude to increase, so that frequency and overall amplitude will vary in phase. The opposite occurs if the partial is slightly higher than the frequency of the strongest formant.

The physiological origin of vibrato is not well understood. EMG (electromyographic) measurements in laryngeal muscles have revealed rhythmical contractions, synchronous with the vibrato undulations, of the pitch-raising cricothyroid muscle. This suggests that the laryngeal muscles produce the vibrato. The neural origin of these rhythmical contractions is unknown. Possibly as a consequence of this, the transglottal air flow varies with the frequency variations, and the resulting vibrato notes tend to consume more air than vibrato-free notes (see Large and Iwata, 1971). In popular singing subglottal pressure seems to be the vibrato-generating mechanism. In some singers the variations in the muscle activity affect the larynx height and even other parts of the voice organ. Pitch seems to be perceived with comparable accuracy regardless of the presence of vibrato for a single note. The perceived pitch agrees within a few cents with the pitch of a vibrato-free note with a fundamental frequency equal to the average frequency of the vibrato note.

(iii) Register.

The term ‘register’ is used for groups of adjacent notes that sound similarly and are felt to be produced in a similar way. However, there are a great number of conflicting terms and definitions in common use. In untrained voices in particular a change from one register to another may be accompanied not only by a marked shift in tone quality but also by a ‘register break’, a sudden jump in pitch. In both male and female adults register shifts typically occur in the range of approximately 300–450 Hz. The register above this shift is mostly referred to as ‘falsetto’ in male voices and ‘middle register’ in female voices, while the register below the shift is known as ‘chest register’ or ‘modal register’. A further shift occurs below 100 Hz; this register is called ‘vocal fry’. Registers are associated with certain vocal fold configurations. Thus, in chest/modal register the folds are thick while in falsetto they are thinner. Acoustically, the lowest spectrum partial, other things being equal, has been found to be more dominating in falsetto than in chest/modal register. Also, the ‘heavy’ register in male and female voices has been reported typically to contain stronger high partials than the ‘light’ register. The physiological origin of register is confined to the voice source. According to some experts, a difference between the falsetto and the normal voice in males is that the vocal folds never reach full contact with each other during the vibration cycle in falsetto. Transitions between registers have been found to be accompanied by changes in the EMG signals from laryngeal muscles, and by changes in transglottal air flow. There is reasonable agreement on the importance of the laryngeal muscles to registers, though it has been suggested that a purely acoustical interaction between the glottal oscillator and the resonator is a contributory factor.

(iv) Singer’s formant.

The ‘singer’s formant’ is a peak in the spectrum envelope typically appearing near 3 kHz in all voiced sounds as sung by Western operatic singers except sopranos. It corresponds acoustically to a high spectrum envelope peak which is present in all vowels and generally centred at a frequency of 2500–3500 Hz. In vocal pedagogy it is often referred to as ‘singing in the mask’, ‘focussing’ etc. Mainly a resonatory phenomenon, the singer’s formant is achieved by clustering formants 3, 4 and 5 into a rather narrow frequency band. This seems to explain why sopranos lack a singer’s formant: they mostly sing at high fundamental frequencies, i.e. the frequency distance between adjacent partials is typically quite wide, equalling the frequency of the fundamental. This means that a partial would fall into the frequency band of the formant cluster producing the singer’s formant only for certain pitches, causing a salient timbre difference between different pitches. If the pharynx is wide enough, the larynx tube can act as a separate resonator, the resonance frequency of which is rather independent of the rest of the vocal tract; it may be tuned to a frequency lying between those of the third and fourth formants in normal speech. The condition of a widening of the pharynx seems to be met when the larynx is lowered, a gesture occurring typically in male professional Western operatic singing. At high pitches the demands on a wide pharynx are increased, and extreme lowering of the larynx is frequently observed when males sing high-pitched notes. In such cases the term ‘covering’ is sometimes used. The widening of the pharynx and the lowering of the larynx affect the frequencies not only of the higher formants, but also those of the lower formants. As an acoustical consequence of such articulatory modifications, the frequency of the second formant drops in front vowels. This alters the vowel quality to some extent, so that, for instance, the vowel in ‘sheep’ is ‘coloured’ towards the German ‘ü’ sound.

The perceptual function of the ‘singer’s formant’ seems to be to make the voice easier to hear above a loud orchestral accompaniment. It has also been suggested that it helps the singer to be more audible in large auditoriums.

(v) High-pitched singing.

Vowel quality is associated with specific combinations of the two lowest formant frequencies, and these frequencies are maintained regardless of the fundamental frequency. Normally the fundamental frequency is lower than the frequency of the first formant, which varies between about 250 Hz (close to c') and 1000 Hz, depending on the vowel. When the fundamental frequency is higher than the normal frequency value of the first formant, singers tend to increase the latter so that it remains higher in frequency than the fundamental. This partial is the strongest in the source spectrum, and, if it coincides with the first formant frequency, its amplitude will be maximized without raising extreme demands on vocal effort. The degrees of tongue constriction and, in particular, of jaw opening represent important articulatory tools for achieving the necessary increases of the first formant frequency. Though this increase affects vowel quality, this disadvantage is limited since in high-pitched singing the vowel quality cannot be maintained even with correct formant frequencies owing to the great frequency distance between the partials as compared with the number of formants (see Sundberg and Skoog, 1997).

(vi) Voice categories.

Male and female voices tend to differ significantly with regard to their formant frequencies as well as pitch range, and this factor seems also to be significant in differentiating tenors, baritones and basses. Thus when singing the same pitch, voices of these types can be distinguished by their vowel formant frequencies. In most vowels a bass is likely to show the lowest formant frequencies and a tenor the highest; and all formants, not only the two lowest, are relevant. The formant frequency differences between male and female voices resemble closely those observed between bass and tenor voices, which suggests that the dimensions of the resonating system are of major importance. In addition, the centre frequency of the singer’s formant seems to be typically higher in voices with a high pitch range than in voices with a lower pitch range. Thus, centre frequencies at about 2400 and 3000 Hz tend to give a bass-baritone-like and a tenor-like voice quality respectively.

(vii) Overtone singing.

In some Inner Asian cultures the voice is used in a rather special manner, in that the tones produced are perceived as possessing two different pitches. This can be explained as follows. If the frequency of a formant coincides with that of a partial, this partial is likely to be much stronger than the adjacent spectrum partials, other things being equal. If two formants are tuned to the near vicinity of a partial, the effect can be greatly enhanced, so that the partial is perceived as a second pitch of the tone along with the fundamental. This strategy of tuning two formants to a partial is applied in overtone singing. The second and third formants (sometimes the first and second) are tuned to closely spaced frequencies, thus enhancing a specific partial. The fundamental frequency is either low, <100 Hz, produced with a growl or vocal fry quality, or is higher, often with a pressed quality. By these means the amplitude of the fundamental is reduced, and hence the dominance of the amplified overtone is enhanced. In tuning formants the lip opening, the position and elevation of the tongue tip and, in some cases, nasalization seem to play important roles (see Bloothooft and others, 1992; see Overtone singing).

Acoustics, §VI: The voice

BIBLIOGRAPHY

G.E. Peterson and H.L. Barney: ‘Control Methods used in the Study of Vowels’, JASA, xxiv (1952), 175–84

G. Fant: Acoustic Theory of Speech Production (The Hague, 1960)

J. Large and S. Iwata: ‘Aerodynamic Study of Vibrato and Voluntary “Straight Tone” Pairs in Singing’, Folia phoniatrica, xxiii (1971), 50–65

G. Fant: Speech Sounds and Features (Cambridge, MA, 1973)

R.T. Sataloff, ed.: The Professional Voice: the Science and Art of Clinical Voice Care (San Diego, 1977)

W. Seidner and J. Wendler: Die Sängerstimme (Wilhelmshaven, 1978, 3/1997)

J. Large, ed.: Contributions of Voice Research to Singing (Houston, 1980)

J. Sundberg: The Science of the Singing Voice (DeKalb, IL, 1987)

G. Bloothooft and others: ‘Acoustics and Perception of Overtone Singing’, JASA, xcii (1992), 1827–36

I. Titze: Principles of Voice Production (Englewood Cliffs, NJ, 1994)

J. Sundberg and J. Skoog: ‘Dependence of Jaw Opening on Pitch and Vowel in Singers’, Journal of Voice, xi (1997), 301–6

M. Thomasson and J. Sundberg: ‘Lung Volume Levels in Professional Classical Singing’, Logopedics, Phoniatrics, Vocology, xxii (1997), 61–70

K.N. Stevens: Acoustic Phonetics (Cambridge, MA, 1998)

J. Sundberg, M. Andersson and C. Hultqvist: ‘Effects of Subglottal Pressure Variation on Professional Baritone Singers’ Voice Sources’, JASA, cv (1999), 1965–71