Musical instruments differ in the extent to which the sound can be fine-tuned after the instrument maker has finished their job. In a guitar with a glued-in bridge, there is relatively little that can be done beyond choosing strings. On the other hand, we saw (and heard) back in section 5.5 that some features of a banjo can be modified by the player, to tailor the sound to what they prefer — for example the head tension can be changed, and different designs of bridge can be used. Indeed, any stringed instrument with a tailpiece and a “floating” bridge offers the possibility of bridge modification. In a piano, the bridge, soundboard structure and string choice are all fixed unless a major refurbishment is done — but there is scope for fine adjustments to the action and to the stiffness of the felt that lines the hammers. A skilled piano technician can make significant changes to the sound and feel of an instrument, as shown in graphic detail in the film “Pianomania”. We will look at some of the subtleties of piano hammers in section 12.2.

In this section, we will look at the violin family. There are several features of a violin or cello that can be adjusted, to the extent that “tone adjustment” is part of the job description of many violin makers, with some of them specialising in it to the extent that players may bring an instrument across the world to be tweaked. The main ingredients of this tonal adjustment involve the bridge and the soundpost: small changes to both can have surprisingly significant effects, and it is this sensitivity that we will explore in this section.

We start with the bridge. We have already said a bit about the dynamics of a violin bridge back in section 5.3, but now we need to go into more detail. Figure 1 shows a typical violin bridge. When the strings are bowed, they apply forces at the corresponding string notches at the top of the bridge. Those forces are predominantly in the plane of the bridge and in the direction of bowing, as indicated by the red arrows. The body of the violin is driven by the forces generated at the bridge feet, indicated by the green arrows. The feet are small, so we can think of these forces as being applied to the body at two points.

To understand the role of the bridge, we need to know two things. For the influence on the *sound* of the violin, we need to know how a force in some specified direction on the top of the bridge is converted into the two forces through the feet. These foot forces will drive body vibration and hence sound radiation, in a way that will not change if the bridge is altered. To understand the influence back on the bowed string itself, to determine “playability”, we need to know the motion induced at the string notch, described by a frequency response function we have met earlier, the *bridge admittance*.

As we have already seen, there are two types of bridge deformation we need to consider for a violin. These are most easily visualised by thinking of the bridge clamped by its feet in a heavy vice. The bridge would then have two in-plane vibration modes within the frequency range of interest: a rocking mode usually around 3 kHz and a bouncing mode usually around 6 kHz. We can idealise these as simple mass-spring oscillators. Figure 2 shows the idealisation of the rocking mode. A rigid base carries the two feet, and the rocking motion is visualised as a weighted bar, pivoted to the base and controlled by a torsional spring (like an old-fashioned clock spring). The rocking frequency is determined by the ratio of this spring stiffness to the *moment of inertia* of the weighted bar.

The corresponding idealised version of the bouncing mode is shown in Fig. 3. This time, a mass is supported on a linear spring, connected to a rigid base similar to the one in Fig. 2. The bouncing frequency would be governed by the ratio of the spring stiffness to the mass. If we ignore the slight asymmetry of the bridge shape, a transverse force at the centre of the bridge would excite only the rocking mode, while a vertical force would excite only the bouncing mode. These forces are indicated by blue arrows in the figure.

Both these idealised models are straightforward to analyse: the details are given in the next link. The link also gives details of a more complicated model which combines the two into a single analysis. This combined model removes some approximations made in the simplified models, and also allows us to use excitation at the various string positions and angles shown in Fig. 1, rather than just the ideal cases of excitation at the bridge centre point as in Figs. 2 and 3.

We are not, of course, really interested in the case with the bridge feet clamped in a vice: we want the feet to be resting on the body of a violin. We can capture the relevant behaviour of that violin body by a set of frequency response functions, which can be measured on a violin with its bridge removed as illustrated in Fig. 4. Two small accelerometers have been fixed to the positions where the bridge feet will sit, then a small impulse hammer is used to tap on one or the other of these accelerometers. Additional detail relating to this measurement is given in the next link: there are some important subtleties.

The measurement gives four frequency response functions, but two of them should be exactly the same by virtue of a general property of vibrating systems called *reciprocity*. For *any* linear vibrating system, if you tap at one position and measure at a second position, you should get exactly the same response if you swap the tapping and measuring positions. A typical set of measurements is shown in Fig. 5. The red and blue curves show the driving-point admittances at the two foot positions, while the black and green curves show the two cross terms. These are indeed very similar, as reciprocity requires. The red and blue curves illustrate a pattern which is common to all normal violins. At low frequencies, the red curve for the bass bridge foot is higher than the blue curve for the treble foot. But at high frequencies this is reversed, with the blue curve lying higher than the red one.

Armed with these results for the body response, we can calculate several interesting things from the two idealised bridge models shown in Figs. 2 and 3 (and from the combined and extended model described in the previous link). We can predict the bridge admittance when the bridge feet are resting on the violin, and also the response at the bridge feet to excitation by an impulse hammer or by string forces at the top of the bridge. We can then go further: we can try to find the parameter values of the models (mass, spring stiffness, moment of inertia) by best-fitting the predicted results to the actual measured bridge admittance. We can then do “virtual bridge adjustment”: the model can be used to explore what would happen to the bridge admittance or the bridge-foot forces if those masses or stiffnesses were varied.

The first step is to find out if the models work well enough to be useful in this way: can they give a convincing prediction of measured bridge admittance, with plausible values for the model parameters? The rocking model sketched in Fig. 2 has three parameters: the stiffness of the torsional spring, the moment of inertia of the weighted bar (representing the top part of the bridge), and the distance from the pivot point to the bridge top. To use the combined model, we also need to include values for the two parameters of the bouncing model sketched in Fig. 3: spring stiffness and mass. Measurements of the rocking admittance and the vertical admittance at the bridge centre can be used to determine these parameters. We will concentrate mainly on the rocking case, which is of most direct interest to normal violin playing, but the same procedures have also been applied to the vertical admittance to complete the set of estimated parameter values: we will see an example later.

Choosing particular values (of three parameters for the simplified model or five parameters for the combined model), the rocking admittance can be calculated. This can be compared with the measured version, and we can invent a metric to put a number on how different they are. For the results to be shown here, that metric is inspired by the plot we normally use to assess by eye how good the match might be. The two frequency responses are plotted on a logarithmic frequency scale, over a range from 180 Hz (just below the lowest note of a violin) to 8 kHz (high enough to capture the two formant features, “hills”, that we are interested in). Amplitude is plotted on a decibel scale. Our metric is then the root-mean-square (“RMS”) difference between these two decibel plots, a commonly-used measure. Note that because the decibel difference is first squared, it doesn’t matter which curve is higher and which is lower. All that matters is the vertical distance between the two curves.

We can now run the calculation with a grid of values of the three parameters, and see how the metric varies. The combination that gives the lowest value is our best match, at least according to this particular metric. An example is shown in Figs. 6 and 7. Figure 6 shows a contour map of the metric, for a wide range of values for the moment of inertia and the rocking resonance frequency (which then determines the torsional spring stiffness). The contours are spaced 0.5 dB apart, starting from the lowest value, which in this case was an RMS difference of 3.4 dB. That minimum occurs in the oval region in the upper left of the plot, corresponding to a rocking frequency of 2.9 kHz and a moment of inertia value of $7.5 \times 10^{-7}\ \mathrm{kg~m}^2$. To see what this RMS difference means, Fig. 7 shows the measured and predicted admittances. It can be seen that the two curves are gratifyingly close.

Next, we need to check that if the same bridge is installed on a different violin, the fitted bridge parameters remain essentially the same even though the admittance will be different. This test is difficult to do with normal violin bridges, because the feet are carefully carved to fit the arching profile of each individual instrument, and would not be expected to give a proper fit on a different one. However, there is a type of commercially-available violin bridge (aimed presumably at the amateur market) in which the feet can swivel so that there is a much better chance of fitting to two different instruments. These bridges are ideal for the test we want to perform.

Figure 8 shows three of these bridges, modified to give very different total masses so that we can also get an impression of how much difference the bridge might make to the admittance. The middle one of the three bridges is fairly normal, with a mass of 2.4 g. The right-hand one has had some “extraneous” wood removed, and it is also a lot thinner. The result is a mass of only 1.5 g. The left-hand bridge has been left thick, and in an effort to raise the frequencies of the rocking and bouncing modes the cutouts have been filled with epoxy. The result is a total mass of 3.3 g.

Figures 9, 10 and 11 show the result of fitting these bridges in turn to the same violin used for the earlier tests. Each figure shows the contour map of the RMS difference metric, and also shows the best-fitted admittance compared to the corresponding measurement. The same vertical scale is used for all the admittance plots, and if you compare them carefully you can see clear differences. Figure 9 shows the lightest bridge, which produces the lowest fitted value of the moment of inertia, and gives the highest admittance levels around the “hill” in the 2—4 kHz range. Figure 10 shows the intermediate bridge. It has a higher moment of inertia, and the admittance in the hill region is lower.

Figure 11 shows the heaviest bridge. The admittance is lower still in the hill region, and we see something interesting in the contour plot. Instead of a clear-cut minimum marked by a small oval contour, the metric remains more or less constant throughout the upper right-hand part of the diagram. There is no clear prediction of a rocking resonance frequency — which is not surprising since this filled-in bridge is not really able to “bend at the waist” in the classic rocking resonance. The contour map is telling us that for a bridge like this, the admittance is very insensitive to the stiffness of the torsion spring, provided that stiffness is high enough. This is exactly what we should have expected: with a very stiff bridge, the admittance is dominated by the effective rocking stiffness of the “island” area of the violin top, where the bridge sits. If that island stiffness is much lower than the bridge stiffness, the bridge is simply carried along rigidly during the rocking motion, and the exact value of its stiffness makes little difference.

The same three bridges were then put in turn on a different violin. The corresponding results are shown in Figs. 12, 13 and 14. The detailed admittance plots are significantly different, most obviously at low frequency where the pattern of “signature mode” peaks is quite different. However, the trends across the three bridges are the same, and the contour maps are recognisably similar to the corresponding ones in Figs. 9, 10 and 11. In particular, Figs. 12 and 13 show the innermost oval contours in almost exactly the same places as Figs. 9 and 10. Figure 14 does not show a small oval like these, but it has a blank area in the upper right which looks very similar to Fig. 11. The conclusion is just as we would hope: the procedure of best-fitting the theoretical model gives robust predictions of the bridge properties, regardless of which violin is used.

It is useful to see examples of the corresponding fitting process applied to the vertical admittance, measured by tapping and observing in the vertical direction at the centre of the bridge’s top curve, between the D and A strings of the violin. Figure 15 shows the results, for the middle one of the three bridges on the original violin: the same combination as in Fig. 10. The admittance is plotted on the same vertical scale as the earlier ones, and it is immediately apparent that the values are significantly lower than in the rocking case. As we should have expected, the admittance in the right-hand plot of Fig. 15 shows a new “hill” around 5 kHz, based on the bouncing resonance of the bridge and the island area of the violin.

Figure 16 shows the corresponding results for the same bridge on the other violin, the same combination of instrument and bridge shown in Fig. 15. It shows similarly low values, and a slightly less obvious hill formant around 5 kHz. The comparison of the two contour maps is interesting. Both of them show a rather featureless blank area in the upper right, similar to Figs. 11 and 14. The interpretation is similar to the discussion of the rocking behaviour of the heaviest bridge: when it comes to bouncing, this less heavy bridge is sufficiently stiff compared to the island area of the violin that the admittance is insensitive to the exact spring stiffness. Under these circumstances we cannot expect to obtain accurate values of the bridge’s parameters by the fitting process — but that doesn’t matter very much because of the insensitivity.

We now have a reasonable level of confidence in the theoretical model, and we can use it to explore a question of great interest to violin makers. Starting from a given bridge on a given violin, how much difference could be made to the admittance or the sound of the instrument, by adjusting the bridge within the bounds of feasibility? Figure 17 gives an illustration of one way to address that question. We can take the violin/bridge combination studied in Figs. 5, 6 and 7, and vary the rocking resonance frequency and the moment of inertia by a certain factor upwards or downwards from the nominal fitted values. For this figure, the chosen factor is 30%: admittedly quite a large variation, but this makes it easier to see the trends.

The central plot shows the nominal case, the same admittance as the red curve in Fig. 6. Superimposed on this is a “skeleton curve”: we introduced this idea back in section 5.3 when we first met the bridge hill. By averaging the admittance over a bandwidth wide enough to contain several resonances, we can obtain a curve which tracks the mean level of the decibel plot. This makes the “hill” more clear by smoothing out the detailed peaks and dips caused by the individual body resonances of the violin. Surrounding this plot are 8 others. In the left-hand column, the clamped-foot bridge rocking resonance frequency has been reduced by 30%, while in the right-hand column it has been increased by 30%. In the top row the moment of inertia has been reduced by 30%, while in the bottom row it has been increased by 30%. All other parameters are kept fixed.

All the plots use the same vertical scale, but even so it is a little hard to focus on the comparison between any two particular cases. To make such comparisons easier, Fig. 18 collects the 9 skeleton curves together in a single plot. The line colours and types match those used in Fig. 17: the solid red line is the reference case, changes in colour indicate variations in rocking frequency, changes in line type indicate variations in moment of inertia. This combined plot makes it clear that if variations as large as 30% in these bridge parameters could be made, very significant changes in admittance should result: for example, the height of the “hill” peak varies over a range bigger than 10 dB. The implication is that even with more modest changes to the bridge parameters, variations of a few dB should be possible.

So far, we have only talked about the influence of bridge properties on admittance. But of course we are also interested in the influence on the sound of the violin when played. We argued at the start of this section that the influence of bridge variations on radiated sound should be captured by a pair of transfer functions, relating the force at the two bridge feet to the force applied at the top of the bridge (by the vibrating strings or by an impulse hammer used for testing).

We can calculate these two transfer functions from a fitted theoretical model: but now it is important to use the combined and extended model, rather than the simplified ones. As explained in detail in the first side link above, both simplified models predict that the forces at the two feet will be *equal* in magnitude (in phase for bouncing, in opposite phases for rocking), but the combined model reveals that it is not so simple. Figure 19 shows an example, using the fitted model which produced the admittance shown in the blue curve of Fig. 7. The force at the bass foot of the bridge is shown in red, while the force at the treble foot is shown in blue. In the middle of the plotted frequency range, the two are rather similar; but at high and low frequencies the red curve lies significantly above the blue curve. This particular plot uses an input force to match the way the admittance was measured: the hammer tap was applied to the bass corner of the bridge, oriented parallel to the bowing direction on the nearby G string.

We can use the extended model to investigate how the various transfer functions vary between the different string notches, each with a corresponding main bowing direction (tangential to the top curve of the bridge, as indicated back in Fig. 1). Figure 20 shows the answer for the bridge admittance. The four curves correspond to the four strings (G in red, D in blue, A in green, E in black). The four admittances are remarkably similar, although there are differences amounting to a few dB in some peak heights.

But when we look at the corresponding transfer function to forces at the feet, shown in Fig. 21, there are much more obvious differences. For the G string (top left plot), the red curve for the bass foot lies above the blue curve for the treble foot, at both high and low frequencies. But this balance changes as we move across the strings, and by the time we reach the E string (bottom right plot) the pattern is reversed. Since the sound radiation behaviour from forces at the two feet will be different, this may translate into systematic differences of sound across the four strings.

It is interesting to repeat the “virtual bridge adjustment” with these force transfer functions. Figure 22 shows an example, for forcing at the G string notch. The nine cases are laid out in the same way as in Fig. 17, with $\pm30\%$ variations of rocking resonance frequency across the columns and $\pm30\%$ variations in moment of inertia across the rows. In each small plot, the transfer function to the bass foot is plotted in red (along with its corresponding “skeleton curve”), while the corresponding plots for the treble foot are shown in blue.

Differences can be seen, but to make comparisons easier the skeleton curves for all the cases have been collected together: Fig. 23 shows the results for the bass foot force, while Fig. 24 shows the corresponding results for the treble foot force. For these two plots, the code of colours and line types from Fig. 18 has been adopted so that these figures are directly comparable. Colours indicate variations in rocking frequency, line types indicate variations in moment of inertia. The pattern of variation is different in the two cases, and both are different from the pattern shown by admittance in Fig. 18.