Instruments of the Violin Family

By A.H. BENADE

In chapter 23 we learned how the strings and the bow of a violin can work together to maintain a stable oscillation. We also considered the relationships that hold between the vibration amplitude of a string mode, observed at the bowing point, and the corresponding amplitude of the driving force component which is exerted on the bridge. In the present chapter we will follow the consequences of these excitatory forces through the resulting vibrations of the violin body and thence out into the room.

24.1. The Body and the Bridge of Instruments of the Violin Family

It is customary to think about instruments of the violin family as being made up of three reasonably distinct parts: (1) the sound-generating portion of the instrument, consisting of the bow and the strings working cooperatively; (2) the body, whose resonances strongly influence the way the sound is radiated into the room; and (3) the bridge, which mediates between the oscillating strings and the body. Having devoted chapter 23 to a discussion of the bow and strings, we should now acquaint ourselves with some of the acoustical properties of the body and the bridge.

Figure 24.1 shows top and side views of a violin, along with the names of various parts of the structure that will be of particular interest to us. Each of the violin-family instruments consists of carefully arched top and back plates joined at their perimeters by thin strips of wood called the ribs. These combine to form an eggshell-like box whose shape is remarkably well adapted to support the direct pull of four strings as well as a rather significant downbearing force that is exerted on the bridge. On a violin the total tension of the strings is around 25 kg (55 Ibs); the strings' downbearing amounts to about 8 kg (18 Ibs).

While outwardly the violin body looks quite symmetrical, its inner structure reveals some departure from symmetry. The foot of the bridge on the side carrying the treble strings is supported by a soundpost that is lightly wedged between the top and back plates; its placement serves not only to give mechanical strength but also to couple the vibrations of one plate directly to the other. Under the bridge foot on the bass side a long strip of wood known as the bars bar is glued onto the inner surface of the top plate, running more or less parallel to the direction of the strings. This reinforcement serves structurally as a means for distributing the downbearing force from the bridge over the surface of the top plate. In the simplest of acoustic terms, the bass bar also serves to couple the bridge vibrations effectively to both rounded portions of the top plate: these two areas are otherwise somewhat isolated from one another by the nipped-in waist section which contains two cutouts of graceful shape known as the j-holes. The f-holes not only influence the vibration properties of the top plate in a direct way, they also serve as a passageway through which the enclosed Air can communicate its oscillations to the room as part of the total radiation process.

My brief description so far of the structure and function of the various parts of the violin body makes it seem as though these parts somehow maintain their acoustical identity when the instrument is played. Nothing could be farther from the truth. The similarity of the wave impedances of the various wooden parts guarantees that these parts all act as a single vibrating system whose overall behavior cannot be determined by a nave adding up of the characteristic vibration properties of the separate parts (see the second digression in sec. 17.1).

A. The Bridge as a Coupling Lever between Strings and Body.

Despite the general warnings of the preceding paragraph, it is possible for us to introduce ourselves to the gross features of the coupling between bridge and body by making use of the fact that at frequencies well below the first-mode resonance of the bridge (as measured with its feet standing on a rigid support), it is correct to treat the bridge as a rigid object that can act as a simple lever. This means that for violins the validity of our simplified viewpoint is restricted to frequencies well below 3000 Hz (F7#), while for the cello the corresponding resonance frequency is near 1000 Hz (B5), exactly in proportion to its lower musical pitch range.[1]

To the extent that it is permissible to treat the bridge as a simple lever, we see from figure 24.2 that the soundpost (which is placed very nearly under the treble foot) acts as a fulcrum about which the bridge can rock, so that it can exert a twisting force on the part of the front plate that lies between the f-holes. Notice that each of the string notches on a rocking bridge moves along an obliquely curving path. If it is permissible as well to treat the bass bar as rigid (a much riskier undertaking), the bridge also appears to exert up-and-down forces on the plate sections lying at its two ends. Whatever validity the simple lever and brace functions attributed to the bridge, soundpost, and bass bar have is limited to their action at low frequencies. The overall musical behavior of a violin depends on much more, however. The determination of the exact placement of a soundpost, for example, is one of the chalanges to a goon instrument maker - a misplaced soundpost can ruin the tune of the finest instrument.

The bowed string has two very different ways of exerting a driving force on the bridge. The most obvious one comes about directly from the side-to-side oscillation of the string in a direction parallel to the motion of the bow. We discussed the recipe for this sort of driving force earlier with the help of figure 23.6. This excitatory force, which we shall refer to as direct excitation of the bridge, is parallel to the surface of the top plate; a lever like action of the bridge is required to convert it into a force at right angles to the plate surface that can effectively drive the body of the instrument.

The second means whereby the string vibrations are able to drive the top plate is somewhat more subtle. As we have already noticed, the tension of the string goes through two cycles of variation during every cycle of the vibration, reaching maxima when the string moves to its extreme positions on either side of the rest position. The fact that a fiddle string has a great deal of downbearing means that oscillatory changes in string tension give rise to corresponding changes in the downward force exerted by the string on the bridge, a force which is ultimately applied to the top plate. Notice that the frequency of this indirect excitation, as we shall call it, takes place at twice the vibration frequency of the string. This means, for example, that mode 1 of a violin A-string produces direct action on the bridge at 440 Hz, whereas this mode acts by the indirect process to excite the bridge ac 880 Hz. Similarly, mode 2 acting by itself produces direct and indirect driving force excitations at 880 and 1760 Hz. For the sake of brevity we will refer to the two kinds of driving force as Fn dir [for direct] and Fn ind [ for indirect].

Let us now compare the driving-force recipes that are produced at the bridge by the direct and indirect excitation processes. To begin with, we almost instinctively recognize that the direct driving force Fn dir produced by the corresponding string mode acting alone has an amplitude that is proportional to the vibrating amplitude An of chat mode, so that Fn dir doubles with every doubling of An, and so on. We also take it for granted that when several modes are in action, the force spectrum can be found by simply listing the actions of the several modes acting independently.

The indirect excitation process behaves quite differently. Here we find that if a single string mode is excited to an amplitude An, the corresponding indirect bridge-force amplitude Fnind is proportional to the square of An, so chat Fn ind grows fourfold for every doubling of An. This tells us right away that under pianissimo playing conditions the indirect excitation process is negligible in comparison with direct excitation, whereas at mezzoforte and higher levels the sound emitted via the indirect process can equal or even exceed the direct contribution.

The change in the sound spectrum arising from the relations between the two kinds of driving force is even more elaborate than is suggested by the discussion so far. When a number of string modes are excited (as in normal playing), the nonlinearity of the relation between An and Fn ind results in a great deal of heterodyne action among the various frequency components. In particular, then, for a bowed string whose frequency components are arranged in the harmonic series 100, 200, 300, 400, . . . Hz, indirect excitation cakes place at 100 Hz by way of heterodyne action between all pairs of adjacent partials (e.g., 500-400=100 Hz; 400 - 300 = 100 Hz; etc.). Similarly, an indirect excitation at 200 Hz takes place because of heterodyne contributions between alternate partials (such as 400-200=200 Hz, 500-300=200 Hz), as well as the double-frequency heterodyne action (100 + 100 = 200 Hz) that was our introduction to this type of excitation I have calculated that the simplest Helmholtz-type vibrational amplitude spectrum (that makes all the Fn dir of equal size; see sec. 23.5 following statement 1) gives rise to an indirect-excitation spectrum whose components have the following sizes:

component number 1 2 3 4 5

indirect driving force Fn ind 1.00 1.25 1.11 0.98 0.87

These figures indicate that the overall spectrum of the force chat drives the bridge is nor drastically altered when one plays loudly enough to make the indirect type of bridge excitation important. Nevertheless, the efficiency of the transfer of oscillatory energy from string to fiddle increases significantly under fortissimo conditions as the indirect processes come into action.

B. The Air Resonance of a Fiddle Body.

In 1937 Frederick Saunders devised an ingenious and straightforward means for studying the sound output of a stringed instrument: one simply plays a chromatic scale on the instrument at a force level in a room of reasonable size and for each note writes down the readings of a sound level meter.' The reverberant properties of the room, the moving-around of the player and his helper (if one is present to record the data), and the effects of any vibrato all conspire to give a good average of the statistical properties of the room and of the radiation behavior of the instrument. What Saunders called loudness curves are obtained by plotting the sound level readings against the note names of the corresponding tones. Such curves show certain stable features that are characteristic of good instruments of each category. Even though each reading on the sound level meter indicates the aggregate effect of all the partials of the tone being played, it will show a certain increase if one of these partials happens to be unusually strong. This is the main reason that loudness curves of this type and some of their more recent descendants prove valuable in the study of stringed instruments.

One of the first things we can see in a violin loudness curve is evidence for a strong peak in the sound output whenever a partial of the played tone matches a well-defined frequency that is found in the neighborhood of 290 Hz. This peak, which is known as the plain air resonance of the instrument, is a consequence of the resonant excitation of the lowest characteristic mode of vibration of the air within the violin body. In the introductory remarks about the excitation mechanism of a flute, we learned of the way the slug of air in the neck of a bottle can bounce sinusoidally on the springiness provided by the air within the bottle (see sec. 22.6). The air within a violin body acts in exactly similar fashion as a spring upon which the mass of air in the f-holes can oscillate. The natural frequency of such a bottle-shaped air resonator will be lowered if the volume of enclosed air is increased, and is will be raised if the area of the f-holes is increased. If the walls of our cavity are elastically yielding, the natural frequency of its air resonance will be lowered (see sec. 22.7). The thin walls of violin-family instruments make this effect particularly pronounced. However, the soundpost and strings contribute significantly to the re-stiffening of the body, as is shown by the following simplified figures for a violin air-resonance frequency, which are based on measurements by Carleen Hutchins: [3]

without soundpost or strings ........................ 227 Hz

with soundpost, without strings .................... 282 Hz

normal conditions ......................................... 290 Hz

rigid-walled cavity of same proportions ......... 350 Hz

Let us see how the bridge can excite this air resonance of the fiddle body, and how the excitation is then communicated to the air. To begin with, we see chat the rocking of the bridge on its soundpost at low frequency alternately contracts and expands the volume of air contained within the body, so that air is alternately exhaled and inhaled by the f-(roles in a manner exactly reminiscent of the breathing behavior produced when a plastic squeeze bottle is pressed periodically between the fingers. This indicates that the t-holes themselves are able to function as a simple acoustic source of the kind defined in section 11.2 However, not every transfer of air through the f-holes will give rise to a sound. It is fairly obvious that denting the violin body by the local pressure of a bridge foot gives rise to a flow of room air into the region of the dent, i.e., into the volume vacated by the inward motion of the plate. This flow of room air into the dent takes place at the same time that other air is expelled into the room through the (Rules from within the cavity.

From the point of view of the room, then', there is no net flow of air into or out of the region immediately surrounding the violin as a whole (and so no production of sound), as long as these two flows compensate each other exactly. This equality of flow is what one observes at low frequencies of excitation, so that at low frequencies a fiddle body provided with f-holes is almost totally unable to radiate sound into the air! As the bridge excitation frequency rises toward the aircavity resonance frequency, the oscillatory flow in and out of the f-holes becomes progressively more vigorous and so overcomes the cancellation produced by the oppositely moving body walls. Above resonance, the motion of the enclosed air reverses in its relation to the driving force exerted by the walls (see sec. 10.1, statements 4 and 7), so that now the maximum outward flow coincides in time with the outward motion of the walls ant the two contributors to the radiation act in concert.

The dotted curve marked A in figure 24.3 shows the influence of the first air =22' resonance of a violin body on the perceived loudness of the sinusoid one would hear if a constant - amplitude sinusoidal driving force were applied to the bridge 460 Hz (we are assuming that nothing else is going on). At the bottom of the figure a set of lines is drawn which are labeled with the note names of a whole-tone scale beginning at the bottom note of the violin's playing range (G3). Each line has marked on it dots at the frequencies of the various harmonic components of the corresponding note, so that you can understand how the loudnesses of these components are affected by the resonance peak.

C. The Main Wood Resonance and Its Connection with the Air Resonance.

The next item of information one can extract from a study of the Saunders loudness curves is evidence for the existence of a strong sound output peak for string excitations taking place in the neighborhood of 440 Hz. This peak, which is usually referred to as the main wood resonance. has been traced to a vibrational mode of the wooden body itself. The upper part of figure 24.4 shows the part of this vibration which is observable on the top plate of a violin. The back plate hits a similar but somewhat more symmetrical and much weaker motion. Notice that this mode is particularly easy to excite by means of the bridge and bass bar since these act in the region of maximum topplate excursion. This type of oscillation is sometimes called a "breathing mode," since the body as a whole expands and contracts its total volume. Such a mode (acting by itself) can function as a very effective source of excitation for sound in the room. The dotted curve marked W in figure 24.3 shows how the loudness perceived by a listener in a room would vary if this main wood-resonance mode were to act in the absence of any other property of the violin body. You will recognize that the air resonance whose radiation consequences are illustrated by curve A in figure 24.3 is excited by the same oscillatory breathing action of the cavity walls that gives rise to curve W, except that we earlier imagined the walls to be driven inexorably, with constant amplitude, by some mechanical device.

Fig. 24.4. Upper. coupling between first air mode and main wood resonance of a violin; lower, schematic diagrams of the air pressure distributions of the next four air modes within the violin body.

The solid curve in figure 24.3 shows how the air and wood resonances combine their influences in controlling the sound of a real violin. It is based on a calculation reported in 1962 by John Schelleng and confirmed by various experimental studies.; This overall curve has an interpretation that is very similar to that for the vocal-tract curves of chapter 19 (see figs. 19.5 and 19.6). A listener does not of course perceive enormous changes in the loudness of the complete cone when the strength of a single partial is altered. However, he will have no trouble in hearing a clearly marked change in tone color as a note with changing pitch slides some partial through the resonance peak (see sec. 19.5). We should notice in passing that for violin notes between G3 and A4 the fundamental component and/or its second harmonic always has its loudness considerably enhanced by the joint effect of the main air and wood resonances. Similar remarks can be made about the lower notes of the ocher members of the bowed string family of instruments.

D. The Influence of Other Air Resonances. In part B of this section we learned that because the walls of the violin are yielding, the first air-mode resonance is lowered quite significantly. We can recognize that this yielding of the walls is simply the response of the main wood-resonance mode to the pressure variations of the enclosed air, the excitation taking place well below the natural frequency of the walls. In a series of experiments carried on since 1972, Erik Jansson in Stockholm has found chat this coupling behavior of the air and wood modes works both ways: he and Harry Sundin have shown that on a violin the second mode of air vibration can have a significant effect on the frequency of what we have been calling the main wood resonance.' Let us see how this comes about and at the same time make the acquaintance of some of the other air-cavity modes.

The lower half of figure 24.4 shows diagrammatically the acoustic pressure distributions and nodal lines for air modes 2 through 5. The dashed lines indicate nodes and the regions marked 0 are places where very little oscillatory pres

sure variation is detectable. Mode 2, whose natural frequency lies in the neighborhood of 460 Hz, is a simple sloshing of air back and forth between the ends of the cavity; this mode closely resembles the first air mode of a pipe chat is closed at both ends in having a pressure maximum at each end and a node ac or near the middle. Comparison of the top-plate vibration pattern shown in the upper part of figure 24.4 with the pressure pattern for air mode 2 shows that the large excursion of the lower half of the plate (on the tailpiece side of the f holes) strongly drives the lower half-hump of the air mode standing wave- an internal excitation chat is not canceled by the weaker vibrations of the upper half of the plate which act on the oppositely varying air pressure in this region.

Jansson has shown chat the mutual influence of air mode 2 and the main wood resonance is so strong that the peak marked W in figure 24.3 is generally split into two peaks that can have quite a deep notch between them. The exact behavior of the sound output in the neighborhood of what we have been calling the main wood resonance thus turns out to be a complicated version of the behavior we first noticed in the kettledrum; it is not correct to consider air and mechanical properties independently- the two peaks have frequencies that are determined jointly by the air and by the walls, and one should not in general assume that the predominant motion is to be found in either of the two subsystems. The fact that air mode 2 has a nodal line running across the waist of the instrument cells us chat very little air will be driven in and out of the f-holes by this type of air motion. The radiated sound associated with both parts of the split W-curve peak is thus produced almost entirely by the wall vibrations acting directly on the outside air.

The higher-frequency air modes will be excited to a greater or lesser extent by the various higher modes of the violin body, although their influence on these higher wood resonances is not expected to be very large. However, we can look for contributions to the radiated sound at the frequencies of chose air modes having pressure maxima near the positions of the f-holes.

24-2. High-Frequency Radiation Properties of Bowed String Instruments

We have just completed a close examination of two prominent peaks which are found at the low-frequency end of every violin-family instrument's range. At higher frequencies we still find many peaks and dips, but these do not in general show very much similarity as we go from one violin to another, for example, or from one cello to another. The overall trend of the transmission behavior is very similar for all stringed instruments, however, and we can gain a fairly good understanding of the reasons for this trend.

Before we begin to list the various acoustical properties of the body which help to control this trend, we should remind ourselves that, to a reasonably good approximation, the magnitude of the driving force Fn dir exerted on the bridge by each component of the played tone is roughly constant. For instance, we learned in section 23.4 that in the simplest form of the theories of Helmholtz and Raman all of the direct-excitation Fn's have exactly equal amplitude. Furthermore, in section 24.1 we learned that the indirect excitation arising from oscillatory variations in the string downbearing has a set of driving-force components Fn ind chat decrease only gradually as we shift our attention to the higher-numbered modes. Since the two forms of bridge excitation give us roughly equal driving forces at all frequencies, in our attempts to understand the sound output of an instrument we need consider only the varying ability of the body to convert a driving force into sound in the room.

We learned in sections 11.2 and 12.4C chat the radiating power of a loudspeaker or other sound source in a room rises steadily as we go to higher frequencies until the dimensions of the source become comparable with the hump dimensions (half wavelengths) of the room modes. At higher frequencies the excitation becomes progressively less effective, for reasons that we first met in connection with the excitation of strings by a broad plectra and hammers (see secs. 8.1 and 8.2). For a violin-sized object we would expect this dimensional limitation on its ability to radiate to begin advertising itself with a gradual leveling-off of the sound output above the 1000 Hz.

As the excitation frequency applied to the body by the strings rises, it excites the plates into increasingly complicated vibration modes, each one having more nodal lines than the one before.[6] This is a way of saying that the vibrating surface divides itself up ever more finely into vibrating segments each of which acts oppositely on the room from its neighbors. A glance at the plate and drumhead vibrational shapes illustrated in chapter 9 will confirm this. A violin driven at the bridge in the frequency region between 1500 and 2000 Hz shows vibration patterns having two or three dozen humps distributed over the entire body surface. An engineer who forgets that the violin is not a loudspeaker might criticize it for tieing an extremely inefficient radiator of sound at these frequencies, since these vibrational humps (which may be only 2 or 3 cm across) have a span that is very much shorter than the 8-to-12-cm widths of the room-mode humps in this range of frequencies. The presence of many small humps gives us a second reason to expect a failing-off in the high-frequency sound output of a violin, this time with a limitation that becomes significant above about 2000 Hz.

Studies of the energy lost within the wood itself show that the damping produced by both cross grain and along-thegrain frictional losses rises sharply at frequencies above about 3500 Hz.[7] Above this frequency, then, an ever-increasing share of the string excitation is diverted away from its tortuous path to the room, spending its effort instead on frictional hearing within the structure of the instrument. This gives us yet another reason to expect a reduction in the strengths of the high-frequency partials of a violin tone.

When all three of the high-frequency limitations described above are taken into account, we would expect the partials of a violin tone that lie above about 2000 Hz to be very much attenuated. Even when we take into account the increasing sensitivity of the ear for high-frequency sounds, we should expect an extension of the curve shown in figure 24.3 to fall to very small values indeed above about 3000 Hz. Let us see what actually happens.

Figure 24.5 shows the loudnesses of the various partials as a function of frequency (I have calculated these loudnesses on the basis of measurements made by many different experimenters). Below 500 Hz (about C5) the curve is simply a re plotting of the information contained in figure 24.3. It is at higher frequencies that we notice something surprising: while this high-frequency region contains many sharp peaks and dips (whose positions vary from instrument to instrument), the output averaged over the peaks always shows a rising trend that extends past our expected 2000-Hz limitation and continues up to about 3000 Hz before the strengths of the higher partials begin to be strongly attenuated! We are forced to recognize that something in the complete vibratory system is able to do much more than merely counteract the attenuating effects listed earlier.

We do not have to go far to discover the explanation for this modified behavior. In section 24.1, part A, I pointed out that it is proper to treat the bridge as a simple lever only at frequencies well below the 3000-Hz first-mode resonance of the bridge itself. It is not difficult to show mathematically that as the frequency of the string driving force on the bridge rises toward the bridge's own resonance frequency (as measured with the feet clamped), the effective lever ratio of the bridge grows so as to magnify the force available at the bridge feet to drive the top plate. Walter Reinicke has measured not only the resonance frequencies of violin and cello bridges, but also the actual transformation ratio between the string and foot forces." Reinicke's figures for the resonantly peaked driving efficacy of the bridge account for the increased strengths of the string partials shown around 3000 Hz in figure 24.5. Reinicke was also able to use data on the properties of the bridge to explain the variations he observed in the damping of the A-string modes that we made use of in section 23.3. The measured dampings of the string modes correlate with the ability of the bridge to steal the vibratory energy of the string by passing it along to the violin body and thence to the air.

24.3. Characteristic Features of the Violin, Viola, and Cello; A Recent Development:

The New Family of Large and Small True Violins

In the preceding two sections of this chapter we have learned of three stable features of the acoustical behavior of the body of atypical bowed string instrument which underlie its predominant tonal characteristics. Two of these features are resonance peaks: (1) the strong resonance associated with the lowest mode of oscillation of the air enclosed within the body of the instrument and (2) the equally strong resonance associated with the simplest of the vibrations of the body's wooden parts. These resonances exert their influence on the lower notes of the instrument by altering the strengths of the first and second partials of the tone. The third stable feature is a broadly rising amplitude of the higher partials up to a frequency that can be predicted from a knowledge of the first-mode resonance of the bridge itself (as measured with its feet clamped). In the following paragraphs we will look at how these features are related to the tunings and sizes of the violin, viola, and cello.

The violin has its four strings tuned in fifths to the notes G3, D4, A4, and Es, and on a good instrument the air resonance lies near 290 Hz, within a semitone of the fundamental frequency of the D-string. Similarly, the so-called main wood resonance (which is in fact the joint consequence of a body resonance and mode 2 of the air within it)

is located around 440 Hz, within a semitone of the A-string tuning. On a violin the strengths of the low A3 and its two neighbors are enhanced greatly by the fact that the second partials of these tones sic more or less on top of the main wood resonance. All these things taken together explain why a Saunders loudness curve typically shows maxima for the notes near A3, D4, and A4. One also frequently gets strong notes in the general regions of C5 and C6. In the neighborhood of 3000 Hz the peaks and dips follow a trend having a broad maximum that is controlled by the resonant force-transformation properties of the bridge.

Because violas are built in more widely varying dimensions, we find less uniformity among different instruments. However, the following figures are reasonably representative. The strings are tuned a fifth below those of a violin, ac C3, G3, D4, and A4. The first-mode air resonance is often around 230 Hz (near B3b), being somewhat lower on large instruments and higher on small ones. Already we can see why the lowest notes on a viola tend to be somewhat weak and dull: the air peak lies about ten semitones above the bottom C3, so the fundamental components of the lowest few notes are very weakly radiated. The viola's main wood resonance is likely to be around 350 Hz (near F4), so that the two resonances are related by approximately a musical fifth, as they were in the case of the violin. (Having made this remark, I must hasten to warn my readers not to make too much of its direct musical significance. The tolerances of the locations of these resonances are easily sufficient to permit this interval to range on good instruments from as little as a fourth to as much as a sixth-the particular relationship is not important in itself.) The musical characteristics of the lower viola notes from E3 on up are reminiscent of the notes of a violin going up from G3. The resemblance can be traced to the similar placement of the resonances relative to these notes on the two instruments. Because of the differences in proportion between violins and violas, the mode-2 air resonance of a viola is somewhat higher in relation to the wood resonance than is is for a violin. As a result, in the Saunders loudness curves of a viola one can see evidence for the separate identities of these resonances. Data are unfortunately not available on the resonance frequencies of the viola bridge, but there is evidence to suggest that the spectrum has its high-frequency maximum in the general region of 2000 Hz. In brief, the string tunings and playing range of a viola are transposed a fifth below those of the violin, and the high-frequency behavior seems also to be transposed downward by this amount. However, the crucially important lower two resonances are not transposed down a fifth, and this change in the overall relationships gives the viola a musical character distinctly different from that of the violin. It is not a closely related larger brother of the violin in the way that a B b tenor saxophone is the lower-pitched brother of the Eb alto.

The cello has its strings tuned an octave below those of a viola (a twelfth below chose of the violin) at C2, G2, D3, and A3. The main air resonance is found to lie in the neighborhood of 125 Hz (between B2 and C3). This is even higher in relation to the bottom-string tuning than is the case for the viola. While the actual sharpness and tallness of the air-resonance peak of a cello are roughly the same as on the smaller instruments and while the peak's presence is clear audible, its visibility on a Saunders loudness curve is considerably less, for reasons that we will consider shortly. The main wood resonance of a cello lies near 175 Hz (about F3), which places it therefore somewhat more than halfway in pitch between the upper two strings of the instrument. Notice that so far the properties of the cello and viola appear to be quite consistent with one another, since the corresponding resonances, as well as the string tunings, are an octave apart on the two instruments. In fact their behavior is quite different, one reason being connected with the peculiar behavior of the cello's air response. The other distinction comes about because the tall bridge of a cello leads to an extremely strong response of the body to string excitations having a frequency near the main wood resonance. This response can sometimes detune the string's own mode-1 frequency sufficiently to disrupt the formation of a normal regime of oscillation; in its stead, various more complicated types of vibration may take place that are collectively known to musicians as wolf notes.

Because a cello bridge has legs proportionately much longer than those of a violin bridge, its first-mode vibrational shape has a rather different appearance. Nevertheless, Reinicke finds, as before, a large increase in the ability of the string to drive the body at the bridge's third mode (near 2000 Hz), and there is a deep notch in the transmission ability at an intermediate frequency a little above 1500 Hz. Both the notch and the second maximum lie within the musically important range of a cello spectrum, whereas the analogous features of a violin bridge transmission curve lie at about 5000 and 6000 Hz, too high to be of much significance.

Let us turn now to an examination of the cello's behavior when is is played near the main air resonance. As expected, the air resonance has clearly audible effects. To pick it out, one does not listen for loudness changes (since loudness is a property of all the harmonic partials taken together); instead one listens for changes in tone color and for the special smoothness of tone that is associated with sounds whose components are placed on transmission resonances. The main air resonance, which is easy enough to hear that with a little practice one can notice it under the rapidly changing conditions of musical performance, shows up on a Saunders loudness curve as a peak of surprisingly modes[ dimensions. This points out the danger of coo much reliance on readings from a sound level meter, which can register only the combined soundpressure contributions from all the harmonics of the played cone. This means chat is may overlook a change in the amplitude of some partial of particular interest, such as the main air resonance, and allow it to be partially masked by the welter of other components. An example of how the sound level meter can shortchange the strength of a resonance occurs when some higher partial of the tone falls into a dip in the radiation curve at the same time that the fundamental component is placed on a peak. The two effects manage to offset each other in the meter reading even though they give rise to an easily recognized auditory sensation. Let us look at an example of such behavior, since there is reason to suspect that a typical cello shows a weakening of the radiated second harmonic of the tone whose fundamental is reinforced by the first air resonance.

The dimensions of a cello body are such as to give its second air mode a frequency that is very nearly an octave above the frequency of its first air mode (rather than a wide fifth above, as on a violin). As a result both of these modes will be strongly excited when a note is played at the main air-resonance frequency, since they match the first and second vibrational components of that note. A glance at figure 24.4 will remind us, though, that the second air mode will not radiate much even though it may be strongly excited, because the f-holes lie in the nodal region of the second mode of vibration. This means that we should not expect this resonance to enhance the second harmonic component of the sound. However, two acoustical consequences can be expected from the excitation of air mode 2. First, we find that the cello's top plate is made to "feel" more than normally rigid to the bridge feet when the air mode is strongly excited, thus reducing the transmission of vibratory energy from the string to the body. Second, the frictional losses and other losses of energy incurred by the non radiative sloshing of the second-mode air oscillation will absorb some of the excitation from the string, once again. reducing the sound output from the instrument. Both of these phenomena will show a broadly tuned effect: air mode 2 need not lie exactly an octave above the mode-I frequency for the reduced second partial of the string tone to offset the resonant increase in the strength of the fundamental component significantly, thus producing only a small peak in the sound level meter reading for this note.

Bowed instruments of the violin family were perfected during the seventeenth and eighteenth centuries, giving us the violin, the viola, and the cello. The lowest member of the bowed string tribe today, the bass viol, is a descendant of the acoustically different family of viols, which otherwise exists today only in antiquarian surroundings. Contrary to the almost universal practice of wind-instrument makers since the Renaissance and of the early makers of the viols, the early violin makers were not successful in developing a complete set of instruments having overlapping playing ranges spaced apart in fifths or fourths (e.g., soprano, alto, tenor, and bass). The violin and viola have this relationship, but there is a member of the family missing between viola and cello, and another between cello and bass viol. From time to time over the centuries efforts have been made to fill these gaps, but until recently the resulting instruments proved to have shortcomings that prevented their acceptance for serious musical purposes.

In 1958, during a series of intensive experiments carried on by Carleen Hutchins and Frederick Saunders on the effects of moving violin and viola resonances up and down in frequency, the composer Henry Brant and the cellist Sterling Hunkins proposed the development of eight instruments in a series of tunings and sizes to cover the entire musical range, all of these to have their main air and wood resonances placed close to the frequencies of the two middle strings, as they are on the conventional violin. This suggestion was timely both from scientific and musical points of view, because an attack on the design problems connected with such a project promised to reveal a great many things about the acoustics of conventional instruments.

In the years since 1958, Hutchins has herself worked indefatigably and has enlisted the cooperation and aid of many others to bring this "new family of fiddles" into existence. The musical and scientific rewards of these efforts have proven to be at least as great as was originally hoped." The family has two instruments that are above the violin in pitch: the treble, with strings tuned an octave above the violin, and the soprano, with tunings a fifth higher than the violin. The alto, which is the viola member of the new family, has a length of about 82 cm (in place of the 70 cm typical of a viola). This added length is required because an ordinary viola is physically too small to have its resonances placed in the desired manner. Some people play the alto vertically on a peg, cello-fashion, while others place it under the chin as is done with a conventional viola. Next comes the tenor, which is somewhat smaller than an ordinary cello (107 cm rather than 124 cm in body length), with its strings tuned a fifth above the cello. This instrument fills the tuning gap that is normally left between viola and cello. Below the tenor comes the baritone, which has the same string tunings as a cello but a larger body. Finally there are small and a large bass (these now being true violins), with their strings tuned in fourths, at A1, D2, G2, C3 and E1, A1, D2, G2.

John Schelleng worked out the scaling rules that determine the proportions of the new family. We can summarize here some of the main requirements that his scaling design had to meet to ensure the musical usefulness of the instruments.

1. String lengths had to be scaled to fit human proportions: a half-length string tot the treble would be too small for the playing of a chromatic scale, and a bass string length of 3.6 meters (twice the height of a man would clearly be beyond the abilities of the most athletic bassist.

2. Once string and body lengths are chosen to fit the needs of the player, one has only the thickness (and to some extent the arching of the plates available for adjustment to gc the wood resonances in the desired position! It turns out that the plates of the smallest instruments must be an astonishing 5 mm thick. On the large bass the astonishment has an opposite cause the plates are so thin that one feels he could punch holes in them by vigorous tap with a pencil.

3. The frequency of the main air resonant (i.e., air mode 1) depends chiefly on the volume of the body cavity and on the area of the f-holes. Since the plate sizes and also the f-hole dimensions are chosen to satisfy the requirements listed earlier, the chief recourse here is to adjust the depths of the ribs. Eve this does not suffice in the treble violin sin, an over-shallow body not only looks peculiar it also lacks sufficient strength to withstand twisting forces. For this reason, the ribs a fairly deep, but they have extra vent holes bring the air-resonance frequency up to the desired value near 2 X 290 = 580 Hz. The problem is also difficult at the bass end of the scale: one cannot build too deep a body or the player will not be able to put his bow at around it. However, the yielding of the thin walls of the body makes it possible to get the resonance down to the desired frequency. Another possible problem is that if the violins of the new family were all to be built with rigid walls, the large instruments would have exceedingly narrow air resonance peaks of unacceptable tallness. Fortunately, the motion of the progressively thinner walls provides enough extra damping to keep the peaks within limits of tallness and breadth that give good acoustical results.

4. Once the body proportions of each member of the family are set, corresponding string sizes must he assigned. As we learned in our study of pianos and harpsichords, it is important to get a proper relationship between the wave impedances of the strings and of the body (as mediated by the bridge). This means that the thicknesses of the strings on each instrument must be chosen along with their tensions to meet simultaneously the needs for correct vibrational frequency and for a suitable string-to-body wave impedance ratio.

Two sets of the new violin family of instruments have been built. They have excited a tremendous amount of interest and enthusiasm wherever they have been demonstrated. Their tonal homogeneity poses a challenge to composers who are used to the distinctly different sounds of the violin, viola, and cello; for instance, care must be taken in part-writing to prevent the various musical voices from running together into a full but somewhat bland overall sound. The new instruments cannot normally be used as replacements for the conventional ones, because of their different tone and power, but for certain purposes they have begun to make their way into standard usage. For example, the fullness and power of the alto violin will tear up a string quarter if it is substituted for the viola, but the alto can serve beautifully on occasion as the solo voice in a viola concerto where it must compete with the entire orchestra. The superior power and tonal fullness of the bass members of the family as compared with the conventional bass viol have also aroused considerable enthusiasm on the part of players and conductors.

The success of Carleen Hutchins and her co-workers in building a consort of true violins in accordance with John Schelleng's scaling procedures is impressive. Their instruments' musical usefulness is a tribute to the combination of scientific understanding and craftsmanship of a high order that went into the making of them. Once the first set of new instruments was in existence, it was natural to want to find a way to cross-check the acoustical relationships against their perceptual analogs. In the spring of 1964 it seemed to me worthwhile to compare the tone of various members of the Hutchins family of instruments with the tone of a good conventional violin that had been tape-recorded and played back at altered speed in order to transpose its sounds to the pitch ranges of the various new instruments. The violinist Edith Roberts and I made a preliminary tape of this sort which was promising enough to warrant our carrying out a more careful experiment in 1968.[11]

In such an experiment there are several musical and technical implications to the required alternation of recordings and playbacks made at two-thirds and onehalf speed. The tempo is drastically altered along with the pitch change, as is the rate of vibrato. For instance, to make an acceptable imitation of the tenor instrument (which plays an octave down), it is necessary to play at a very fast tempo (approximately double) so that the music will come out at a reasonable pace on playback. Recording and playing back at differing speeds brings about alterations in the frequency response and internal noise properties of the equipment, and these must be carefully compensated.

The recording was done in the livingroom/music-room area of my home, a region that is large enough to guarantee that hundreds of room resonances will be excited by any one of the violin partials. Use was also made of the fact that the statistical fluctuations in transmission of sound in the room are considerably reduced if the microphone is placed very close to a corner formed by two walls and either floor or ceiling. The player stood in the middle of the room, moving about freely to ensure a good averaging of the room modes as she played. Formal tests of the uniformity of transmission as well as more ordinary tests of listening to the recorded music confirmed the correctness of our approach. Moving the microphone even a hand's breadth away from the corner produced an unpleasant roughness in the sound, and more conventional microphone placements in the room produced nothing better than the usual amateurish sound of a home recording.

The final tape put together from our recordings has a very pleasant sound but, far more interesting, it is easy to recognize that the tonal characteristics of the various new instruments are present in the transposed sound of the ordinary violin. A particular example of this is the presence of an almost unpleasant squawkiness in the tones of both the treble violin and its transposed counterpart. Hutchins and I verified that increasing the damping of the air resonance of the treble violin by stuffing a certain amount of cotton into its f-holes would eliminate the difficulty.

This shows that an air resonance who tallness and sharpness contribute to what we like very much in the tone of a viol is not suitable for "best" sound when high-pitched instrument is built.

24.4. The Adjustment of Violin Plat and the Required Properties of The Material

The making of instruments of the viol family has always been among the me demanding of arts. There are so many variables involved and so much time elapses between the carving of a plate a its assembly into an instrument ready I testing that the maker can hardly lea from experience unless he is possessed o perfect memory, remarkable intuition, fine ear, and endless patience. Many craftsmen can make a respectable instrument, but it is given to very few any generation to create a superb one, a these special individuals are not always able to pass on their knowledge.

Because she is a skillful instrument maker in the conventional sense as well an expert in musical acoustics, Carle Hutchins has been able to add greatly our fund of reachable knowledge on he to adjust the various parts of an instrument in the course of construction. Her success in this activity and that her collaborators have encouraged increasing numbers of instrument makers learn and to make use of acoustical telling as a guide in their work." We ca not detail here many of the ways in which acoustical science provides information the maker, but it is worthwhile to outline some of the complexities of the problem as well as some of the ways in which the complexities can be exploited or circumvented .

The vibrational properties of any part of a violin, viola, or cello depend not only on the easily measured size, thickness, and arching of the wood, but also on the elasticity, density, and internal damping-properties which change from sample to sample, and even from day to day as tile temperature and humidity change." From the earliest days instrument makers have intuitively recognized that the less-tangible properties of the wood affect the vibrational properties of the isolated plate as well as those of the finished instrument. Because of this, an extensive lore has grown up on how to listen for certain sounds called tap toner that can be heard when the plate is held in certain ways and tapped at particular spots. Such tests are of course informal explorations of the characteristic modes of the plate-not merely their natural frequencies of oscillation but also the nature of their vibrational shapes. Hutchins and others have systematized the exploration of tap tones with the aid of laboratory apparatus that can extricate one sinusoidal component at a time from the complete collection that we perceive as the tap tone. It is much easier to tell someone what spectral components are to be sought in making a viola plate than it is to teach him by repeated example exactly what sort of woody, ringing sound he is supposed to listen for. This in turn makes it easier to explain where to scrape and carve in order to arrange the various sound components into a desired relationship.

Another approach that has proved immensely fruitful is to mount the plate on a well-standardized system of supports (clamps or rubber bands) and then to excite it at a carefully chosen point by a magnetic drive coil. The resulting vibrations are detected either by a pickup located somewhere on the plate or by means of a microphone placed a short distance away. Response curves plotted in this way contain a great deal of useful information about the vibrational properties of the plate, especially when the peaks and dips observed in one experimental arrangement are correlated with those in another (see the tin-tray experiments in sec. 10.7). Once a craftsman has taken the time to become familiar with two or three major features of the response curve of properly carved plates, he can then carefully work over each new plate until its vibration signature, as evidenced by these characteristic features, is of the proper sort. It is of course very helpful for any instrument maker working in this way to have a fairly good idea of the vibrational shapes of the various plate modes, so that perturbation techniques of the sort outlined in section 9.4 can guide his efforts.

The positions of plate resonances are not at all easy to deduce on the basis of response curves made with a microphone. When the microphone is placed only a short distance away from the plate (1 to 50 cm), it responds in a very complicated way to the sound output of all parts of the plate and displays certain consequences of the local flow of air across the nodal lines and also around the edges of the plate. A vibrational mode of the plate may manifest its presence in the response curve by a paired dip and peak, by a simple peak or an unsymmetrical peak, or even by a dip. One also finds extra peaks and dips in the microphone response curve that have no counterparts in the modal frequencies of the plate. Despite these complexities, measurements using carefully placed microphones have proved immensely useful in practice. Techniques based on sound-pressure averages made in a reverberant room are also available. These tend to give plate resonance data in much more direct form than similar averages gathered from microphones placed at large distances from the plate in an anechoic chamber.

It is not difficult in principle to discover the characteristic shapes of the various plate modes. One has merely to drive the plate at the proper frequency and then map out the vibrating surface, either with an optical or magnetic probe, or by means of a small microphone held so that the distance between the vibrating surface and the microphone diaphragm is much less than the microphone diameter (the distance must be very short, otherwise the microphone signal consists of a surprisingly equal mixture of disturbances coming from all parts of the plate). A quicker but much more elaborate way to obtain the vibration pattern of a plate mode is to use photographically recorded laser holograms, following a technique first applied by Karl Stetson.

In practice, working out the characteristic shapes of the free-plate modes is almost as difficult (and treacherous) as is the determination of the characteristic frequencies themselves. Much of the difficulty, in fact, lies in finding these platemode frequencies (as discussed in an earlier paragraph). If one does not drive at a plate resonance, there will be significant excitation of at least two adjacent modes, so that the plate is moving in a complicated way having a peculiar set of nodal lines and hump regions that are in reality the result of superposing two characteristic shapes. Such shapes are easily misinterpreted. In certain cases, two or more of the natural frequencies may lie so close together that it is impossible to separate their contributions. Under these conditions considerable ingenuity is required to extricate the true patterns of the individual modes. Here the non-holographic methods often show an advantage.

The vibrational properties of a wood plate arc very much dependent on the fact that the cross-grain stiffness is only about ten percent of the stiffness along the grain (see the digression on wood plates in sec. 9.2). If one wanted to simulate the vibrational behavior of a wooden violin plate in metal or plastic, it would be necessary to cut it so that the ratio of length to width would be close to unity, instead of the customary ratio of about three-to-one. Let us understand the implications of this remark with the help of figure 24.6, which shows in schematic form the first five mode shapes of the top or back plate of a violin. Mode 1 is the twisting mode that we first met in figure 9.3. Two versions of mode 2 appear. Though rather dissimilar in appearance, they have very nearly the same frequency-about 50 percent higher than that of mode 1. The theory of vibrating plates agrees with experiment in predicting both versions of the mode-shape; theory also predicts its frequency relative to mode 1, but only for plates that are functionally square. Modes 3 and 4 have vibrational shapes that are very reminiscent of the motion of a free disc vibrating with three nodal lines crossing its diameter. In one of these two modes we find a nodal line running parallel to the grain, while in the other the corresponding nodal line is at right angles to the grain. Different pieces of wood may reverse the order of these two vibrational patterns, as indicated in the figure, or further work on a given plate, such as thinning, may result in reversal. Theory and experiment agree in placing the frequencies of these two modes at somewhat more than double the mode-1 frequency. The frequency spacing between modes 3 and 4 can vary from 10 percent to nearly 50 percent. The reversibility of the order of these two modes and the near equality of their frequencies constitute a direct proof that the plate is very nearly "square" in its acoustical properties, since on a perfectly square or perfectly circular plate they would have identical frequencies. Modes 3 and 4 are particularly difficult to separate in holographic experiments- one or the other may be overlooked or obscured.

String-instrument makers find chat it is especially important to obtain the correct vibrational shape for mode 5, which is often called the ring mode. The inner parts of the plate move in one direction while the outer parts move in the other. It has a frequency about 4 times chat of mode 1. When the vibrational shape is that sketched for mode 5, the arching of the violin plate adds great stiffness to the vibrating system, and this stiffness raises the frequency. If the plate were fiat, the mode having this shape would be recognizable by mathematicians as forming a pair with the upper version of mode 2. (The stiffness due to arching has relatively little influence on the lower modes we have sketched since these are primarily of a twisting character.)

Every string player knows of instruments that play well in dry weather, and others chat perform best when the humidity is high. The reason is simple to find once we realize that the two stiffnesses of wood (measured along and across the grain) change differently with changes in humidity. This means chat an instrument can only be in its optimum vibratory condition with a single sort of weather. The maker is left with the choice of tuning the two plates of a string instrument relative to each other under identical conditions or finishing each on a different day in the hope of building an instrument that performs at least acceptably under all conditions.

Recent developments in the science and engineering of artificial materials have encouraged serious work on the possibility of making musical instrument bodies out of suitably designed composite materials. Carleen Hutchins, Donald Thompson of the C. F. Martin Company, and Daniel Haines of the University of South Carolina have recently demonstrated guitars (1974) and violins (1975) whose cop plates are made in the form of a sandwich." The inner core is a kind of paper over which are laid long strands of carefully aligned carbon fibers held together by epoxy cement. The desired ratio between the stiffnesses along and across the grain is achieved through the enormous tensile strength of these fibers combined with the flexibility of the epoxy. The relative thicknesses of the paper and the outer coverings are adjusted to provide the desired density and also the variation of internal damping with frequency of the sort chat is needed for a successful imitation of wood. Holograms of the vibrations of the carbon-epoxy violin plate show essentially no difference from those for a wooden violin plate of good quality. The assembled violin plays very well and has excited the serious consideration of a manufacturer interested in dependable production on a commercial basis.

24.5. Musical Properties of Bowed String Instruments

Certain special properties of the sound from bowed string instruments set these instruments apart from other members of the orchestra. Each member of this film has a pair of strong air and wood resonances that influence the radiated sound of its lower notes. Moreover, one finds the total radiated sound of each member of the family a large number of high frequency peaks and dips fluctuating about a broadly humped maximum whose frequency is determined largely by c resonance properties of the bridge (<. fig. 24.5). If we stop our consideracions here, we are led to think of the string sound as being determined in a manner almost strictly analogous to the transmision of the human voice: a more or less autonomous source has its oscillations transmitted to the room by way of a filter that has a number of transmission peaks. In other words, the air and main wo resonance peaks appear to be simple at logs of the first two voice formant peaks (see figs. 19.5 and 19.6).

However, when radiation behavior considered, we recognize that the analogy sketched above is a gross oversimplification. Voice sounds are emitted by a small aperture that functions as a simple source to radiate almost equally in all directions. By contrast, the complicated vibration shapes of the violin body cause it to send into the room an exceedingly complex pattern which, for example, is different for every direction in which the sound can go in an anechoic chamber. [15]

On an average basis, the violin radiates its low-frequency partials equally in directions; its higher components are radiated in a progressively righter beam a direction perpendicular to the plate (this behavior is reminiscent of the progressively increasing directionality sound components emitted by a trumpet or a woodwind; see secs. 20.8 and 22. However, superposed on this average behavior are the elaborate directional patterns of the separate partials mentioned above. It is this complicated radiation pattern for each partial of a violin tone (a pattern that changes drastically for any change in frequency) that distinguishes the violin family from other instruments.

Because of the integrative abilities of our hearing mechanism, we are able to collect all of these radiative complexities as they come to us via multiple reflections in the room. The vibrato (taking place at the rate of about half a dozen cycles per second) plays a particularly interesting role among the bowed string instruments. It supplies a sort of timing cue for the relationships among all the partials, whose strengths fluctuate more or less randomly in amplitude but concurrently (at least at the source) in time. There are many implications to be drawn from the fact that the 30-to-50 millisecond "collecting time" of the hearing mechanism associated with the precedence effect (see secs. 12.2 and 12.4) is comparable with the 80-millisecond time it takes for the vibrato to sweep the component frequencies from maximum to minimum or back. One's thinking can also be stimulated by the fact that each of the first half-dozen harmonic partials of a tone lies within its own critical bandwidth for the ear (see sec. 13.5) and so has its fluctuations processed for loudness, etc., more or less as an individual, whereas the higher partials are spaced closely enough relative to the critical bandwidth (which is approximately onethird of an octave) for overlapping collections of them to be processed together. This aggregate processing on the one hand tends to average out the radiation and room fluctuations; on the other hand, it can lead to a harshness of tone if these higher partials are too strong relative to the lower half-dozen.

The difference between the ways in which we aurally process the loudnesses of low- and high-frequency phenomena helps to explain why we had to pay such close attention to the details of the lowfrequency end of the curve in figure 24.5, whereas we looked only at the general trend of the high-frequency part of the curve. We also gain some insight into the reasons why a violin (or any other instrument) must be provided with a means for ensuring a reasonably small acoustic output at high frequencies.

The fact that our hearing mechanisms can winnow out the common elements provided by the body resonances of a violin or cello while at the same time permitting us to enjoy the fluctuating variety of the unprocessed sound provides us with a unity in the midst of diversity that is extremely difficult to imitate.[16] We can readily understand the limited success of attempts at electronic synthesis of bowed string sounds, even when the vibrations of an actual bowed string are picked up electrically and run through a fixed set of filters on their way to a highfidelity loudspeaker.[17] No matter how elaborately the peaks and dips of the filter transmission curve are matched to the radiation of a violin in a given direction, our ears have no difficulty recognizing the artificiality of the sound. The successive versions of the sound that reach us from different parts of the room all share the same common origin-the filter and loudspeaker. One would require at least several filter sets separately radiating into the room to simulate the diversity of the sound reaching us from a normal instrument." This gives us a hint why even such simple sounds as those produced by tapping a board with a stick or snapping a rubber band stretched across a cigar box are so difficult to synthesize by conventional means. It is not so much the particular frequency components or the damping of the modes that gives us such a clear impression of the woodiness or the twang in these sounds, but rather the fact that they are radiated in a way that is characteristic of vibrating plates.

There is one more feature of string tone that has a very large influence on its musical behavior. There is an inherent unsteadiness to the bowed string tone that has been noticed from the earliest days. On a bad instrument the unsteadiness of the oscillatory regime becomes a splutter or scrape (whose dynamical implications were pointed out by Helmholtz in his first paper), whereas on the best instruments we find this unsteadiness becoming a sort of warmth and richness.

While I was still an undergraduate I noticed that one does not hear clear-cut beats between mistuned violin tones of the sort that painfully advertise slight errors between two wind instrument sounds. 1t was not difficult for me to recognize at that time that the weakness of the beats implied unsteadinesses in the sticking and slipping of the rosined bow on the string. In 1963 and later, correspondence with John Schelleng raised the question again. Examination of published photographs showing string motion at the bowing point confirmed that there are small fluctuations in the oscillation. Rough measurements of the separated fundamental and second-harmonic components of a violin tone showed the variations to be essentially random and spread over a frequency range of somewhat less than one percent. Lothar Cremer and others have more recently made careful measurements of the periodicity of the overall sound (rather than of the individual components), getting a spread somewhat greater than one percent (about 20 cents), as would be expected from the combined influence of all the partials.

The fact that each partial of a string tone is spread over a bandwidth of about 20 cents means that there is a diffuseness to the string tone which has enormous implications for the musician. On the one hand it allows larger tuning errors to be made in ensemble playing before the discrepancies become unacceptable, and on the other it permits the composer to write a wide variety of chords having many degrees of consonance and dissonance. We have here an elaboration of phenomena we met in connection with the multiple stringing of pianos (see sec. 17.3). The diffuse string sound explains to a large degree the greater versatility of the string quartet as compared with a wind ensemble. The skilled wind group can produce on demand chords of oily smoothness or dissonances of astonishing harshness, neither of which are attainable to the same degree by stringed instruments. But the tendency of the wind ensemble sound to push consonance anti dissonance toward their extremes means that the subtleties of the middle ground must inevitably be neglected, and this is just the region where the string ensemble is unsurpassed.

24.6. Examples, Experiments, and Questions

A good preliminary to other violin family experiments is to find the first mode air resonance and main wood resonance of a violin. To locate the air resonance, bow the G-string and slide your finger up and down the fingerboard to produce a tone whose pitch varies above and below D4. As the fundamental component of the tone sweeps past the air resonance frequency there will be a distinct change in tone quality and a certain increase in loudness. A few traversals of the resonance will help you to pinpoint its position. Comparison of the tone played at the resonance with notes on a piano will allow you to estimate its pitch (estimations to about 25 cents are easily made). Verify that you have actually found the air resonance by making sure that your resonance falls in frequency when one of the f-holes is partly closed off by a finger, and that it weakens and disappears as more and more tufts of cotton are tucked into the f-hole apertures.

Continue now by bowing in the neighborhood of A4 on the D-string to find the main wood resonance lying near 440 Hz. Verify that tinkering with the f-holes does not make changes in the frequency and strength of this resonance. Having found the main wood resonance, you can try to excite it by playing an octave lower (near A3). The second harmonic component of this tone should make the main wood resonance ring out, helped somewhat by indirect excitation produced by the fundamental component.

2. Experiments with a metal rod similar to the one described in section 23.6 can usefully be carried out on the plate of a violin. (Note: it is advisable to protect the finish of the violin by covering the end of the rod with a disc of vinyl electrical tape or masking tape.) You may be able to detect the slight lowering of the main wood resonance frequency produced by pressing the end of the rod gently but firmly against the top plate next to the bridge foot under the G-string. Why would you nor expect much change from pressing next to the other foot of the bridge?

3. It is possible to get a very good idea of the influence of yielding walls on the frequency of an air resonance with the help of an empty plastic squeeze bottle of the sort commonly used to hold white household glue. When a 4-oz (near 100 cm3) bottle of this sort is held gently at its edges and blown across like a flute, a tone can be coaxed from it whose pitch lies close to F4 (near 350 Hz). When the tone is sounding, the flat sides of the plastic bottle will be vibrating vigorously. If one sticks more and more lumps of modeling clay onto the sides to increase their inertia, the walls become progressively less able to move in and out in response to the internal air-pressure variations, thus making the bottle act more and more like a hard-walled container. The sound produced by blowing on the bottle will rise progressively in pitch to the neighborhood of C5 (near 520 Hz). Exactly the same rise in pitch is produced by gluing stiffening braces onto the bottle, although the physics of the situation is somewhat altered (see sec. 22.7).

4. Many interesting acoustical games can be played with half- and threequarter-size violins, violas, and cellos. For example, the air resonance of a three quarter-size violin can often be brought down to near 290 Hz, typical of a full size instrument, by covering one f-hole almost completely by a strip of vinyl tape. This change will greatly improve the tone of the lower notes, even though the wood resonance is still too high.

You might find it interesting to use one of these undersized instruments to approximate one or another of the Hutchins family of violins. First move its air-resonance frequency downward once again, this time to a frequency two-thirds that of the main wood resonance (i.e., a fifth down from it in pitch) and then tune the strings in fifths (as usual), choosing the pitch of these fifths so that the middle two strings will lie violin-fashion near the two resonances.

5. If you have a cheap violin to experiment with, much can be learned about the acoustical properties of the bridge and body by attaching lumps of modeling clay at various places on the places or on the bridge, by slipping paper clips onto different parts of the bridge, or by carving away here and there on an expendable bridge. For example, the first-mode resonance of a violin bridge can be lowered by cutting away wood in the horizontal part where the hips run out to join the legs (see fig. 24.2). Thinning the legs themselves will have the same effect on a cello bridge. What would you expect to change in the overall spectrum of an instrument when such alterations are made to the bridge?

6. Notice from figure 24.2 that the rocking leverage exerted on the bridge of a violin by the side-to-side motion of the bowed strings is somewhat greater for the G-string than it is for the E-string. Does the difference in thickness (and hence in wave impedance) between these strings tend to offset or increase the resulting variation in the driving ability these strings possess (see sec. 17.1 and 17.5)? Because the indirect excitation process depends on an oscillatory variation in the downbearing force exerted by the strings on the bridge, you will be able to deduce from figure 24.2 that the E-string should produce relatively little indirect excitation of the violin body, while the lower strings become progressively more able to exploit this possibility. The fact that the indirect excitation process drains appreciable energy from the strings only when they have a large vibrational amplitude means that the string modes have a heavier damping (and so a greater bandwidth) when bowed vigorously, which is exactly the condition under which the large-amplitude inharmonicities discussed by Shankland and Coltman become most important. You should try to work out some of the musical implications of these remarks, and also to devise various playing experiments to display them.

7. Some years ago I was sent a pair of violas for resonance testing and for musical comparison. One of these was a good quality mass-produced instrument of conventional design. Its mate was similar in all respects except that its bass bar was cut away so as to reduce greatly the stiffness in its mid-portion, directly under the bridge foot. Examine figures 24.1, 24.3, and 24.4, and try to work out some of the acoustical consequences of this alteration. Your considerations should include some of the following matters: changes (if any) in the first air mode and main wood resonance frequencies, changes in the heights of the corresponding loudness peaks, changes in the damping of the string modes, and thence changes in the bowing feel. Can you figure out why sound level meter readings were very similar for the two instruments, although the modified one sounded rather "boomy"? What would you predict for the relative durability of the two instruments?

8. Harking back to the discussion of the individual flavors of different musical key signatures in section 16.4, what musical values would you expect for the keys of I), A, and G as played on a violin (even if uhcn, unstopped strings are not used), compared both with each other and with other keys such as Bb or E major?

Musical pitch has fluctuated up and down considerably in the past three centuries. What does this imply about the key-flavor ideas that a violin-playing musician might hold in any given era? Since Bach's day, for example, the pitch has risen nearly a semitone. The Vienna Concentus Musicus, which specializes in baroque music, plays close to A-420. I have measured the main air and wood resonances of the unmodified Stainer violin used by Alice Harnoncourt as soloist with this group, and found that their frequencies are entirely similar to those measured on modern instruments and on most older ones. Most of these older violins were modified in the nineteenth century to give them more power and brilliance by stiffening the bass bar and using a taller bridge to produce more downbearing. Speculate on the acoustical implications of these modifications and on their musical correlates.

Notes

1. Walter Reinicke. "Ubertragungseigenschaften des Streichinstrumentenstegs ," Catgut Acoust. Soc, .Newslerter 19 (May 1973): 26-34. Notice in Reinicke's figures 7 and 16 that the vibrational shapes for modes 1 and 2 for cello bridges are quite different from those belonging to violin bridges.

2. F. A. Saunders, "The Mechanical Action of Violins, J. Acoust. Soc. Am. 9 (1937): 81-98. See also F. A. Saunders, "Recent Work on Violins," J. Acoust. Soc. Am. 25 (1953): 491-98.

3. See, for example, Carleen M. Hutchins, "A Note on the Function of the Soundpost," Catgut Acoust. Soc. Newsletter 21 (May 197-1): 27-28. Hutchins is one of the founders of the Catgut Acoustical Society. She is a skilled and tireless experimenter and is a highly regarded maker of violins, violas, and cellos. She also has a most remarkable ability to stimulate others to think deeply and to experiment.

4. John C. Schelleng, "The Violin as a Circuit, J. Acoust. Soc. Am. 35 (1963): 326-38

5. E. Jansson, "Recent Studies of Wall and Air Resonances in the Violin," Royal Institute of Technology (KTH), Stockholm, Speech Transmission Laboratory, Quarterly Pmgrru and Static Report. 15 January 1973, pp. 34-39; E. V. Jansson, "An Investigation of Acoustical Properties of the Air Cavity of the Violin," Royal Institute of Technology (KTH), Stockholm, Speech Transmission Laboratory, Quarterly Progrm and Status Report. 15 April 1973, pp. 1-12; and Erik Jansson and Harry Sundin, "A Pilot Study on Coupling between Top Plate and Air Volume Vibrations," Catgut Acoust. Soc. Newsletter 21 (May 1974): I1-15.

6. There are many beautiful pictures of vibrational shapes and a very thorough discussion of them in E. Jansson, N.-E. Molin, and H. Sundin, "Resonances of a Violin Body Studied by Hologram Interferometry and Acoustical Methods," Physica Scripta 2 (1970): 243-56. See also W. Reinicke and L. Cremer, "Application of Holographic Interlerometry to Vibrations of the Bodies of String Instruments,"J. Acoust. Soc. Am. 48 (1970): 988-92.

7. A fair amount of information about the behavior of wood can be found in the papers listed in reference 14 below.

8. Reinicke, "Ubertragungseigenschaften des Streichinstrumentenstegs."

9. C. M. Hutchins, "The Physics of Violins," Scientific American. November 1962, pp. 78-93.

10. Carleen Maley Hutchins, "Founding a Family of Fiddles," Physics Today. February 1967, pp. 23-37.

11. Arthur Benade and Edith Roberts, "Tape Recorder Transposition of Standard Violin to Simulate the Highest Seven Members of C. M. Hutchins' Family of True Violins," Catgut Acoart. Soc. Newsletter 1.3 (May 1970): 8-10.

12. There is a large literature on this subject which may be discovered via the references in the following three reports: Carleen M. Hutchins and Francis L. Fielding, "Acoustical Measurement of Violins," Physics Today. July 1968, pp. 34-40; Carleen M. Hutchins, "Instrumentation and Methods for Violin Testing,"J. Audio Eng. Soc. 21 (1973): 563-70; and Carleen M. Hutchins, "Progress Report on a Method of Checking Eigenmodes of Free Violin Plates during Instrument Construction," Catgut Acoust. Soc. Newsletter 19 (May 1973): 17-20.

13. An enormous amount of scientific study has been devoted to the acoustical properties of wood during the past half century. Despite this, it is only recently that a coherent picture is beginning to emerge. Inadequate recognition of the influence of vibrational mode shapes on the measured elasticity and damping has often compounded the problem of reconciling partial measurements from various experiments. See H. Meinel, "Regarding the Sound Quality of Violins and a Scientific Basis for Violin Construction," J. Acontt. Soc. Ant. 29 (1957): 817-22; Schelleng, "The Violin as a Circuit," part Vill; John C. Schelleng, "Acoustical Effects of Violin Vamish," J. Acoust. Soc. Am. 44 (1968): 1175-83; Yoshimasa Sakurai and Eugen J. Skudrzyk, "Acoustic Properties of Wood" (abstract), J. Acoust. Soc. Am. 46 (1969): 124; and N. Ghelmeziu and 1. P. Beldie, "On the Characteristics of Resonance Spruce Wood," Catgut Acoutt. Soc. Newsletter 17 (May 1972): 10-16. See also the papers listed in note 14.

14. Daniel W. Haines, Nagyoung Chang, and Donald A. Thompson, "Can spruce be replaced?a guitar with a graphite-epoxy top plate" (abstract),/. Acoust. Soc. Am. 55 supplement (1974): S49, and Daniel W. Haines and Nagyoung Chang, "Violin with a graphite-epoxy top plate" (abstract), J. Acoust. Soc. Am. 57 supplement 1 (1975): S21. The following short paper gives an excellent summary of the acoustical properties of any material that can serve usefully as a violin top plate: Daniel W. Haines and Nagyoung Chang, "Application of Graphite Composites in Musical Instruments," Catgut Acoust. Soc. Newsletter 23 (May 1975): 13-15.

15. Jurgen Meyer, "Directivity of the Bowed String Instruments and Its Effect on Orchestral Sound in Concert Halls," J. Acoust. Soc. Am. 51 (1972): 1994-2009. The excitation mechanism used in these experiments has x high mechanical impedance, which causes the instrument to vibrate with mode shapes having very. nearly a node at the driving point. For this reason, details of Meyer's results do not quite correspond to the bowed-string case, even though the overall trends at high frequencies are directly applicable. See also Ion Paul Beldie, "Vibration and Sound Radiation of the Violin at Low Frequencies," Catgut Acoust. Soc. Newsletter 2' (November 1974): 13-14. The presence of string resonances makes the interpretation of these data complicated.

16. Paul C. Boomsliter and Warren Creel, "Research Potentials in Auditory Characteristics of Violin Tone," J. Acoust. Soc. Am. 51 (1972): 1984-93.

17. M. V. Mathews and J. Kohut, "Electronic simulation of violin resonances," J. Acoust. Soc. Am. 53 (1973): 1620-26. This is an interesting and useful paper, despite the fact that it was apparently impossible to match the spectrum envelope of the electronic instrument to that of the real violin used as a reference.

18. W. S. Gorrill, "Viola with Electronically Simulated Body Resonances" (abstract), J. Atourt. Soc. Am. 52 (1972): 147, and W. S. Gorrill, "Viola Tone Quality Study Using an Instrument with Synthesized Normal Modes" (abstract), J. Acoust. Soc. Am. 54 (1973): 311. Gorrill's instrument is a very successful variant of the electronic instrument devised by Mathews and his coworkers; Gorrill uses a small loudspeaker mounted in the body of a viola. Filtered signals to this speaker are further modified by the resonances and radiation properties of the body itself and of the loudspeaker, thus restoring the multi-source nature of the audible sound.

19. L. Cremer, "Des Einfluss des 'Bogendrucks auf die selbsterregten Schwingungen der gestrichenen Saite," Acustica 30 (1974): 119-36, fig. 23; and Lothar Cremer, "The Influence of 'Bow Pressure' on the Movement of a Bowed String: I(," Catgut Acouit. Soc. Newsletter 19 (bray 1973): 21-25, fig. 14. See also Irwin Pollack, "Detection and Relative Discrimination of Auditory 'Jitter.' " J. Acoust. Soc. Am. 43 (196R): 308-15. This is the first of a series of papers having considerable musical implication.