A. Background on the soundpost
After that lengthy discussion of the physics behind bridge adjustment on a violin, we now turn to the other major resource of a “tone adjuster”: the soundpost. This section will draw on work done by Myles Nadarajah as part of his PhD project [1]. It has also benefited greatly from discussions with Joseph Curtin, Luca Jost and Evan Davis.
First, we need to be clear about what the soundpost is, and where it can be found. A typical soundpost is a cylinder of spruce, similar to the wood used to make the top plate of the body. It is about 6 mm in diameter, small enough to be inserted through one f-hole using a special tool. The violin maker cuts the length and the angles on the ends of the post very carefully, so that it sits snugly against the two plates at the desired position.
Figures 1 and 2 give an impression of the normal position, roughly in line with the middle of the treble bridge foot but a few millimetres towards the tailpiece. The post is not fixed into position, but held in place by friction. This allows for small adjustments in position and tightness, and makers set great store by such adjustments when tweaking the sound and playing properties of the instrument.
We have already said a little about the role of the soundpost, back in Chapter 5. Figure 3 is a repeat of Fig. 3 from section 5.1, and it shows the main acoustical effect of the soundpost in schematic form. The oscillatory force from a vibrating string acts on the body at the string notch in the bridge. The main component of this oscillatory force is in the plane of bowing, so (at least for the two middle strings) it is a lateral force, as indicated in the figure. If the violin body was as symmetrical as it looks from the outside, this lateral force would produce antisymmetric motion of the violin top, indicated schematically by the red line in the left-hand plot. This antisymmetric motion would not create any change in volume of the body, and so it would not generate any monopole sound radiation. The result is that, at low frequency when the wavelength of sound is long compared to the violin body, the sound radiation would not be very strong.
But the violin is not symmetrical when we look at the inside: as the right-hand plot reminds us, both the bassbar and the soundpost disturb the symmetry. At low frequency, we might guess that the soundpost creates a “hard point” on the top plate, so that the vibration pattern might look more like the new red curve. The motion is no longer antisymmetric, so it now involves a change in net volume through the cycle of vibration, and the low-frequency sound radiation will be stronger. In summary, we would expect the addition of a soundpost to improve the bass response of the violin.
We can verify this prediction with some measured data. However, it would be risky to try measuring a fully-strung violin without its soundpost — as the right-hand diagram in Fig. 3 reminds us, the soundpost also plays a role as a supporting pillar, helping to carry the static load from the string tension. Instead, we can do a similar measurement to the ones shown in the previous section: with the bridge and strings removed, we can tap the body at the two bridge-foot positions, and measure the resulting radiated sound. To reduce the bias that would come from using any single microphone position, we will show results that are an RMS average of the sound measured at a ring of 12 microphones, equally spaced around the “equator” of the violin in the plane of the bridge.
Tapping at a single bridge-foot position does not, of course, give a good representation of the antisymmetric pattern of force we have just described, resulting from rocking of the bridge in response to a lateral force at the top. But we can remedy this by a simple trick. For each microphone position we measure the sound made by a tap at the bass foot, and also by a tap at the treble foot. If we subtract one from the other, we simulate an antisymmetric forcing pattern: downwards at one foot but upwards at the other foot. The RMS average of this antisymmetric combination around the circle of microphones should then give a reasonable representation of what we want to know, and we can safely repeat the measurement with and without the soundpost in place.
The results are plotted in the red and blue curves of Fig. 4: the red curve is with the soundpost, the blue curve is without it. The third curve, in green, shows the measured sound in response to a lateral tap at the bridge corner — but only for the case with the soundpost in place, so this curve should be compared with the red one. That comparison can be made in quantitative detail, because the plot has been scaled by a suitable factor deduced from the geometry of the bridge, so that the net moment applied to the violin top is the same as the one resulting from the difference of the two bridge-foot measurements.
This figure shows several interesting features. First, it very clearly demonstrates the predicted effect of the soundpost on low-frequency sound radiation. Around the lowest strong resonance, the “air resonance” A0 at about 300 Hz, the red curve lies substantially above the blue one. (The resonance frequency is also different with and without the soundpost: the stiffening effect of the post raises the frequency.) There is still a difference in peak heights at the next strong resonance, B1- around 400—450 Hz, but it is smaller. By 500 Hz, the peak heights are similar. Above 500 Hz the red and blue curves have broadly similar levels, except in the range 1500—2000 Hz where the blue curve wins by several dB. At the very highest frequencies, the two curves are more or less identical: it seems that the soundpost has very little effect above 2.5 kHz or so.
Now look at the comparison with the green curve. In the range up to around 500 Hz, the green curve shows similar peak heights to the red curve, vindicating the approach we have taken in these measurements. The A0 peak occurs at essentially the same frequency in both, but the next two strong resonances are shifted to slightly lower frequencies in the green curve. This shows the combined effect of the bridge and strings on the body vibration. As frequency rises, the green curve tends to be above the red one until 2 kHz or so, then it falls systematically below the red curve above 3 kHz. This high frequency roll-off is a consequence of the “bridge hill”, as discussed in detail in the previous section.
We can do another thing with these measurements. If we take the sum rather than the difference of the sounds resulting from tapping at the two foot positions, the result is a symmetrical pattern of force, which is directly comparable to the effect of a vertical tap at the bridge centre. Figure 5 shows a comparison for this case, in a similar format to Fig. 4. Again, the blue curve is for the unstrung body without soundpost, the red curve is the same measurement with the post in place, and the green curve is the recorded sound from a vertical tap at the bridge centre, with the post.
At low frequency, the pattern is the opposite of what we saw in Fig. 4. For the first few strong resonances the peaks are far higher without the post: a vertical force can excite the low resonances very effectively, and installing the soundpost is detrimental because it stiffens the top. This is the reason that a guitar would not benefit from a soundpost — a guitarist can pluck the strings in a way that generates a vertical component of force at the bridge. But a violinist doesn’t have this option: they can only bow a given string at a rather small range of angles, predominantly lateral. In the low-frequency range the green curve in Fig. 5 tracks quite close to the red one, apart from the shift in resonance frequencies that we have already commented on. Above about 1 kHz, all three curves are fairly close together. Again, it appears that the soundpost makes rather little difference to the radiated sound above 2.5 kHz or so.
B. Tightness, position, fit
These results have shown the drastic effect of installing a soundpost, compared to not having one. Now we can move on to the far more subtle question of soundpost adjustment: what is the physics behind the experience of violin makers, that rather small tweaks to the soundpost can make a significant difference to the sound? Some experienced tone adjusters tell us that they consider three aspects: tightness, position and fit, in that order of priority. Not everyone agrees, though: some say that a good fit is compulsory, then tightness and position are considered. On the face of it, this priority sequence is quite surprising, and we need to explore what physical influence these three factors might exert on the vibration behaviour of a violin body. This will tell us what factors we need to include in a model, which will be developed later in this section.
Of the three factors, position is the most clear-cut: moving the ends of the soundpost will surely have some influence on the vibration modes of the body, and hence on the sound and playability of the instrument. The role of tightness and fit is far less obvious, and to an extent these two factors are linked in terms of their physical representation in a model. We need to focus on what happens at the ends of the post, where it meets the arched plates of the body shell. Two important aspects of this contact region are shown in schematic form in Fig. 6.
On the left is a sketch of a well-fitted post versus a poorly-fitted one. The distinction is obvious: the well-fitted post makes contact with the plate over the whole cross-section, whereas the poorly-fitted one only touches near one edge. But “contact” isn’t quite so simple. On the right is a schematic close-up of a region of the contact area for the well-fitted post. Both the plate and the post end are not in reality perfectly smooth: as a result of wood grain and cell structure, and also tool marks, both surfaces will be slightly rough (shown in the plot with an exaggerated vertical scale). As a result, actual contact between the surfaces is restricted to relatively small regions around the high spots, known as asperities. If the two surfaces are pressed together with more force, the asperities are progressively squashed flat and the regions of real contact grow.
In the context of a model to describe the vibration of the body with the soundpost in place, both these effects are relevant to the mechanics of coupling between the post and the plate. The two can interact via a force and via a moment (or torque), as illustrated in Fig. 7. From the point of view of the post, the force applies an axial stress and will excite axial vibration, while the moment will cause bending of the post and excite bending vibration. From the point of view of the plate, the force will interact with the motion in the contact region, while the moment will interact with rotation around that region (associated with bending motion of the plate).
The balance between the force and moment is influenced by the tightness and fit of the post end against the plate. In the extreme case of a poorly-fitted post like the left-hand sketch in Fig. 6, the contact is confined to a very small area near an edge of the post. A force can be transmitted through this small area, but a moment cannot. With a well-fitted post, on the other hand, the contact area is much bigger and now a moment can be transmitted. In the former case it is as if the post were joined to the plate through a hinge, while in the latter case it is as if the post were rigidly glued in place.
The effect of surface roughness, illustrated in the right-hand sketch of Fig. 6, brings another phenomenon into play. The effect of the asperities can be represented using springs: an axial spring to transmit the force, and a torsion spring to transmit the moment. These springs represent the combined “squashiness” of the asperity contacts, and so the spring stiffnesses will depend on the extent of the real area of contact, and hence on the tightness of the post: a post barely held in place, with the minimum of force, will have a very low “contact stiffness”; a tightly-fitted post will have far higher stiffness. So here is one way that tightness might influence vibration: the contact with a very loose post, or a poorly-fitted one, will behave like a hinge, whereas a well-fitted post installed tightly will approach the behaviour of a glued joint. The well-fitted but loose post will also have a very soft contact spring to transmit the axial force, which will have an effect on the vibration as we will see later in this section.
There is a more subtle aspect of surface roughness and contact stiffness. A post may be sufficiently well fitted that there is some contact across the whole cross-sectional area of the post end, but this does not guarantee that the contact stiffness is uniformly distributed. There may be more pressure between post and plate at one edge than the other, and this would probably mean that the effective contact stiffness is higher on the side that is pressing harder, as shown schematically in Fig. 8.
This would have two consequences. In terms of the axial stiffness, the net stiffness will be an average over the cross-section, and this value will be affected by the slight tilt, even though this is invisible to the naked eye. Probably more significantly, the effective contact point with the plate will be biased in the direction of the side which is pressing harder: the right-hand side in Fig. 8. In other words, there is a linkage between all three factors: tightness, position and fit. A tiny tilt, too small to produce a visible gap, could shift the effective post position over a range approaching the diameter of the post, about 6 mm. That would regarded as a very large positional shift if it was done by moving the post bodily sideways.
These are not the only possible effects of post tightness and fit. So far, we have been using the usual approach of treating small-amplitude vibration using linear theory: that is where the description in terms of (linear) contact springs comes from. But it may often happen that the end of the post almost fits, but leaves a very thin gap of the kind sketched in Fig. 6 (right-hand half of the left-hand plot). When the plate vibrates, this will involve some rotation near the post end. Even if the vibration has very low amplitude, it might be comparable with the size of the thin gap, so that significant changes occur during the vibration cycle.
Under those circumstances, a linear spring ceases to be a good approximation. If the surfaces were rigid, they would come in and out of contact, leading to buzzing. With the effects of roughness things will be more gradual, but still the effective contact stiffness for the rotational motion will vary through the cycle, leading to nonlinear effects. Such nonlinear effects might well have audible consequences for the sound of the instrument, although it is hard to guess just how those would be described by a player or listener.
Another effect, which we have ignored up to now, is that the wood in the contact region around the post ends will deform slowly, under the continued loading. The post ends will gradually “bed in” more snugly. Indeed, the whole body shell of a new instrument will deform a little in response to the string tension as well as the soundpost pressure: this will be part of the “settling in” process of a new instrument. It is very commonly reported that a new instrument will need to have a slightly longer soundpost fitted after it has settled for a while, as a result of this deformation.
Over a longer time-scale, many famous old violins are observed to have distorted arching, bulging around the soundpost area. This is the result of successive interventions to keep the post tightness high, by fitting longer and longer soundposts. In extreme cases, restorers may have to reshape the arching of such a deformed instrument using careful application of heat and pressure. More commonly, a soundpost patch will be fitted to the underneath of the top plate around the soundpost, to repair damage inflicted by the post digging in to the soft spruce. You can see one of these patches in Fig. 1. If you look carefully at that figure, you can also see distortion in the arching of both the top and the back plates, making the internal cavity a bit deeper on the soundpost side.
C. The cigar-box violin
To gain some traction on the subtle questions surrounding soundpost adjustment we can’t easily use an approach based directly on measurements of a real violin, as we did when looking at the effect of bridge adjustment. Instead, to get at least a qualitative impression of how soundpost adjustments might affect the sound of a violin we will use a severely simplified model: a “cigar-box violin”. This model has a lot in common with the “square banjo” model we used back in section 5.5.
The complicated baroque shape of a normal violin body will be replaced by a rectangular box, which makes some computations a great deal easier. Within the limits of this crude model, though, the geometry and material properties are chosen to be similar to a real violin. The top plate has roughly the right area, aspect ratio and thickness, and is assigned the material properties of typical soundboard spruce. (Later, in section 10.3, we will learn more about wood properties and how to measure them.) The back plate is given a suitable thickness and assigned the material properties of maple. The sides are assumed to form a rigid rectangular frame, with a mass typical of the rib garland of a normal violin. The top and back plates are assumed to be hinged to this frame on all edges.
Inside this box, we place a fairly realistic model of a soundpost: a rod with length, diameter and material properties typical of a violin soundpost. This rod can vibrate axially, and can also execute bending vibration in any transverse direction. It is coupled to the two plates via springs, representing the contact stiffnesses discussed above. There are a pair of identical linear springs to couple axial motion to displacement of the plates, and also a set of torsion springs to couple soundpost bending to plate bending. We can drive the top plate of the cigar-box violin with the detailed bridge model developed in section 7.5. Figure 9 shows a sketch of the whole model, including the footprints of the two bridge feet. The details of the model, including the chosen parameter values, are described in the next link. Figure 10 shows, separately, the linear and torsional contact springs at the ends of the post.
It is hard to have any intuitive sense of how stiff the contact springs should be, so Figs. 11 and 12 give a guide. They show the natural frequencies for axial motion (Fig. 11) and bending motion (Fig. 12) of the model soundpost when it is fitted between rigid walls at both ends, for a very wide range of contact stiffnesses. In both cases, increasing the stiffness makes all the natural frequencies go up, as would be expected. Figure 12 shows each bending mode changing from a hinged mode (at low stiffness) to a clamped mode (at high stiffness). Figure 11 shows a similar transition for the axial modes, from having free ends (with low stiffness) to having fixed ends (with high stiffness). The next link explains how these figures were computed, and gives a more detailed discussion of the interpretation. These plots tell us that if we want to model a post that is fitted so tightly that it is effectively glued into place, we need an axial stiffness of at least $10^9$ N/m, and a torsional stiffness of at least $10^4$ Nm/radian. For reasons explained in the next link, our chosen datum values of these stiffnesses are lower: they are indicated in Figs. 11 and 12 by dotted lines.
The cigar-box violin is not, of course, a very accurate representation of a normal violin. It has no arching, no f-holes, no “island area”, no bass bar. But it has the big advantage of being simple, intuitive, and fast to compute. The hope is that the model may give useful qualitative insights about the effects of changes, for example to the position of the soundpost relative to the bridge, or to the stiffness of the various contact springs. As an aside, we should note that there is a real instrument that shares some features with our model. Figure 13 shows a trapezoid violin, built to a design by the 19th century French scientist Félix Savart. This has flat plates, and an outline that is somewhat similar to our rectangular box. Players report that a well-made trapezoid model can sound better than a lot of student violins.
It is time to explore how the cigar-box model behaves. We will start with the case that will be our datum, using the best estimates for realistic contact stiffnesses (see the previous link for details). Figure 14 shows a prediction of bridge admittance, compared to a normal violin. Specifically, this is the violin (made by the author in 1990) that was used extensively in section 7.5. The bridge parameters used in the cigar-box model are derived from this violin, via the fitting process leading to Fig. 8 from section 7.5. The cigar box is perfectly symmetric, having no equivalent of the bass bar, so in order to disturb the symmetry slightly, the bridge has been displaced by 5 mm from the centre-line of the box. The soundpost is positioned in a conventional place, with its centre 6.5 mm behind and 1 mm towards the edge from the centre of the treble bridge foot.
The comparison in Fig. 14 reveals that below 1 kHz or so, the cigar-box model has about the right number of clear resonance peaks but that the detailed pattern of frequencies and heights is quite different from that of the normal violin. This is no surprise: the rectangular box cannot be expected to have modes that closely resemble the “signature modes” of a violin. The first few mode shapes are shown in Fig. 15. For each one, the top plate and back plate are shown separately. The displacement indicated by the colour shading is in the same direction for both plates, so the same colour would indicate outward motion of the top, and motion into the box for the back plate. The positions of the sound post and of the bridge feet are indicated.
The first mode, at 222 Hz, looks very similar on both plates. The top and back are moving up and down in synchrony, carrying the soundpost along with them. The mass of the ribs is moving in the opposite direction, as evidenced by the faintly blue colour all around the edge. The two bridge-foot positions show similar colours, so there is very little rocking of the bridge — this is why the mode is barely visible in the bridge admittance (red curve of Fig. 14). The second mode, at 299 Hz, is far more prominent in Fig. 14, and the reason is revealed by the very different colours at the two foot positions. There is strong motion in the top plate, much bigger at one bridge foot than the other. The back plate is hardly moving in this mode, and the soundpost is near a node line for both plates (indicated by a greenish colour). A cross-section near the soundpost and bridge would reveal motion very similar to the right-hand sketch in Fig. 3.
There is an important thing to note about these modes, and the admittance comparison in Fig. 14. The frequency of the first strong peak in the bridge admittance is significantly lower in the cigar box than in the normal violin. The main reason for this is the stiffening effect of arching at low frequencies. The flat plates in the cigar box do not have this stiffening effect, and the result is a tendency to lower frequencies. But the effects of arching diminish as frequency rises, so the typical spacing of modes in the violin will catch up with the spacings for the cigar box as frequency rises. Sure enough, at higher frequencies in Fig. 14, the two curves are much more similar than they are at low frequency. The normal violin has a marked “bridge hill” around 2—3 kHz. The cigar-box model does not show such a clear “hill”, probably because it lacks a flexible “island area”. However, both curves show a similar drop-off above 3 kHz or so, associated with the filtering action of the bridge.
The next step is that we would like to learn something about radiated sound from the cigar-box model. However, within the constraints of this crude model we can only expect a rather rough approximation. First, we only attempt to estimate the far-field sound, a long way away from the box compared to its own size, and also compared to the wavelength of sound. At the lowest frequencies, this will be dominated by a monopole field driven by the net volume change as the box vibrates. At higher frequencies, we can use a different approximation based on the Rayleigh integral (described in section 4.3.2). Specifically, we treat two cases of a microphone positioned almost directly above the centre of the top plate, and we ignore sound radiated by the back plate (on the argument that it will be in an acoustic shadow, round the back of the box). We will look at two cases: one in which the microphone is exactly above the top plate, and one in which it is displaced by $10^\circ$ from that perpendicular direction. For each microphone position, we can combine the two very different approximations into a single estimate: the details are explained in the first of the side links above.
The red line in Fig. 16 shows the prediction for the second microphone position, for excitation in the bowing direction at the G-string bridge notch. It is compared with the corresponding measured sound radiation in more or less the same direction by the violin used for the admittance measurement in Fig. 14. We need to exercise caution in comparing these two curves, though. We have just explained that the theoretical estimate is crude, but there are also two significant issues with the measurement. First, the microphone distance was by no means large enough to place it in the far field. Second, the measurement was made in a slightly reverberant domestic room, so that features of the room acoustics contribute to the plotted curve.
Given all these caveats, the comparison in Fig. 16 is in fact quite encouraging. We already saw in Fig. 14 that the low-frequency behaviour of the cigar box was different in detail from the normal violin, and of course this difference carries over to the sound radiation. At high frequency, Fig. 14 showed that the trends of the two admittance curves were quite similar, and this is also reflected in Fig. 16. It seems that the simple model of radiated sound may be doing a reasonably good job across this wide frequency range.
Figure 16 shows that the cigar box has a strong peak arising from the resonance at 299 Hz. Figure 15 shows that this is indeed the first mode involving a large volume change, so that it has strong low-frequency radiation. The normal violin also has a strong peak in the sound radiation curve near this frequency, but that comparison is rather misleading: the peak comes from the “air resonance” A0, and the cigar-box model has no equivalent air resonance because the box is completely closed, with no soundholes.
D. Virtual soundpost adjustment
We can now see, and hear, some examples of virtual soundpost adjustment using the cigar-box model. We begin with an extreme case: removing the soundpost entirely. Figure 17 shows a comparison of the predicted bridge admittance with and without a post, and Fig. 18 shows the corresponding comparison of the predicted sound radiation. In both cases the red curve is our datum case, the same as the red curve in Figs. 14 and 16 respectively.
The blue curves in both figures show the effect of reducing all the contact stiffnesses to zero. Figure 17 shows surprisingly little effect on the bridge admittance, but Fig. 18 shows a drastic reduction of the sound radiation at the lowest frequencies. This is exactly as we expected, and is in broad agreement with the measured responses shown in Fig. 4: the difference between the red and blue curves in that figure can be compared with Fig. 18 at low frequency. However, remember that the curves in Fig. 18 include the effect of the bridge, whereas the red and blue curves in Fig. 4 do not include the roll-off at high frequency caused by the bridge.
Sounds 1 and 2 allow you to hear the effect. The bridge force from a recorded fragment of violin playing has been convolved with the two frequency responses in Fig. 18, in a similar way to the examples of virtual bridge adjustment presented in section 7.5. The drastic change in sound without the post is very clear: the reduction of low frequency makes the sound very thin and nasal. It does indeed sound very much like the actual effect of knocking the soundpost down in a violin — but don’t try this experiment yourself unless you know what you are doing! The way to do it in safety is to slacken off the top three strings of the violin, leaving only the G string under tension. This reduces the downbearing force from the string tension, and the soundpost can be knocked out while someone plays on the G string. But you need to know how to put the post back in afterwards.
Now for something far more subtle. Figure 19 shows the predicted radiated sound for three different values of the stiffness of the torsional contact springs, keeping everything else fixed. The black curve is a repeat of our datum case, with moderate stiffness. The dashed red curve is the case in which the torsional stiffness is reduced to zero, keeping the axial stiffness high. This is an idealised version of a tilted post, carrying the axial load but not coupling at all to bending of the post or plates. The curve is shown dashed so you can see what is happening: over much of the frequency range it is almost on top of the black curve.
The similarity of the black and red curves might suggest that the torsional stiffness has no effect. However, the blue curve shows the opposite extreme, with contact stiffness so high that the post is effectively glued in. This curve shows small but possibly significant differences compared to the other two. Fitting the post more tightly tends to increase the contact stiffness, so this difference might point towards one effect of post tightness. Sounds 3, 4 and 5 give you a chance to hear the effect. Sound 3 is a repeat of the datum case. Sound 4 corresponds to the red curve, the “tilted post”. Sound 5 corresponds to the blue curve, the “glued post”. To my ears, Sounds 3 and 4 are so similar that I do not think I could tell them apart. But Sound 5 is subtly but definitely different.
Next, we vary the stiffness of the axial contact springs. Figure 20 shows the predicted sound radiation for three cases, with the torsional stiffness set to the datum value 4.5 Nm/rad throughout. The black curve has the axial stiffness set to $2 \times 10^{6}$ N/m, the red curve has $10^5$ N/m and the blue curve has $10^{10}$ N/m. Sounds 6, 7 and 8 let you listen to these three cases. As Figure 20 suggests, Sounds 6 and 8 seem very similar while Sound 7 is significantly different. This case may be indicating a more significant effect of post tightness: with low contact stiffness, corresponding to a rather loose post, we see changes across the frequency range. The frequencies of the low modes are shifted, and there are differences of several dB in various higher frequency ranges.
It is worth showing a few mechanical response plots to give some insight into what the model is doing as the various contact spring stiffnesses are varied. We will compare three cases: the original datum case (plotted in black); the result when the axial stiffness is reduced to $10^5$ N/m (plotted in red); and the result when all contact stiffnesses are very high, as if the post were glued in (plotted in blue). Figure 21 shows the bridge admittances for all three cases. The black and blue curves are almost identical but the red curve is significantly different, so axial stiffness has made a difference to the admittance.
Figure 22 shows what happens when we look at the difference in plate velocities between the two ends of the post. The blue curve shows that there is some relative motion between the plates arising from the post’s own compressibility, even when the contact springs are effectively rigid. When the axial stiffness is progressively reduced, in the black and red curves, there is significantly more relative motion between the top plate and the back plate at the post position.
Figures 23, 24 and 25 show the velocity we would measure at the centre of the soundpost in the $x$, $y$ and $z$ directions respectively, in response to a force on the bridge. In Figs. 23 and 24, the red and black curves lie close together, but the blue curve is different. Reducing the stiffness of the torsional springs brings the bending resonance frequency of the post down, so that the response is reduced at the highest frequencies compared to the very stiff “glued in” post. Then in Fig. 25 we see the blue and black curves being very similar, while the red curve is different. Reducing the axial stiffness has brought the first axial resonance of the post down into the frequency range of the plot (look back at Fig. 11 to see this).
Next, we investigate movement of the soundpost. The contact stiffnesses are all set back to the datum values. Figure 26 then shows the predicted sound radiation when the post is shifted in a “North-South” sense. A relatively large shift of 5 mm has been chosen: this may seem large to a violin maker, who might not think they ever move a post that far. But remember the statement earlier that a subtle tilt of the post might shift the effective contact point across the width of the post. Since the post diameter is 6 mm, a 5 mm shift doesn’t sound so large in this context. The black curve is a repeat of the datum curve, as in Figs. 18 and 19. The blue curve corresponds to moving the post 5 mm further from the bridge foot (i.e. towards the tailpiece, if this had been a real violin). The red curve corresponds to a shift by 5 mm in the opposite direction, so that the post is lying underneath the bridge foot.
The plot suggests that the black and red curves are generally quite close together, while the blue curve is significantly shifted in some frequency ranges. Sounds 9, 10 and 11 allow you to listen to the three cases, and they tell the same story: Sounds 9 and 10 are only very slightly different, while Sound 11 is more obviously different.
Finally, we investigate the corresponding effect of “East-West” movement of the post. Figure 27 shows the predicted sound radiation for 5 mm shifts in the two directions. The black curve is a repeat of the same datum plot, the blue curve corresponds to a 5 mm movement towards the centre-line of the plate, and the red curve corresponds to a 5 mm shift towards the edge of the plate. This time all three curves look rather different. Notice that this is the only one of the adjustments we have looked at which has a big effect in the “signature mode” range below 500 Hz. The cigar-box model is not, of course, realistic down in this range, but this qualitative observation might possibly carry over to normal violins. Sounds 12, 13 and 14 allow you to listen to the effect, and this time all three sound quite distinct, as Fig. 22 suggests they would.
These sounds are audibly different, but it is hard to put a finger on what exactly you are hearing. To help with that, we can generate a graphical representation by making use of the gammatone filters described in section 6.4, which give an approximation to the time-varying excitation pattern on the basilar membrane in your ear as you listen to the sounds. Processing the datum case in that way gives the plot shown in Fig. 28. This looks rather like a spectrogram, with frequency along the horizontal axis and time running vertically upwards, but the plot is more blurred than a spectrogram would be, because of the finite width of the auditory filters.
We can now take two cases and compare them. I have chosen Sounds 12 and 14, the datum case and a 5 mm move Eastwards of the soundpost. The decibel difference between the two plots corresponding to Fig. 28 is shown in Fig. 29. The axes are exactly the same as in Fig. 28. This plot reveals differences by up to about 10 dB in certain frequency bands at certain times. To reduce this to a simple plot in a crude but useful way, we can take the decibel average along each column of Fig. 29. This gives the curve plotted in Fig. 30.
Finally, we can assemble curves like Fig. 30 for different movements of the soundpost. Figure 31 shows the result for a ring of post positions, centred on the datum position. These correspond to shifts in the directions SW, W, NW, N, NE, E,SE and S, reading upwards from the bottom in the plot. The case shown in Fig. 30 was for a move E, so that curve is the third one from the top. All 8 plots show a different pattern, and they all have peak levels of several dB. It is not surprising, then, that these sounds are all audibly different. Four of these sounds were included above, as Sounds 10, 11, 13 and 14.
E. Directional sound radiation and power input
I said at the start of this subsection that two different microphone positions were used in the simulations, but so far all the examples have used just one of them. It is time to explain why. Listen to Sounds 15 and 16: Sound 15 is a repeat of the datum case we have heard several times now, while Sound 16 is the result of the same simulation, but calculating the sound radiation at the second microphone position. These sounds are strikingly different — far more different than all the rather subtle effects we have heard resulting from soundpost adjustments.
This contrast illustrates a key feature of sound radiation: any vibrating object, whether it is a real violin or our idealised cigar box, will have a pattern of directional sound radiation which grows more and more complicated as frequency rises. We saw some examples back in section 4.3, but this is the first time we have had a chance to listen to the consequences. Figure 32 illustrates the behaviour, for our cigar box violin. It shows the radiated sound response computed at a number of microphone positions: the black line corresponds to the direction perpendicular to the top plate of the box, while the other lines show a ring of 8 positions, equally spaced around that perpendicular direction but all angled at $10^\circ$ to it. The case highlighted in blue is the position chosen for all the earlier sound examples, including Sound 15, while the black line corresponds to Sound 16.
Based on Fig. 32, it is surely no surprise that the two cases sounded very different. The black and blue curves differ by more than 20 dB in some frequency ranges. The blue curve is fairly typical of the cloud of responses, while the black curve seems quite extreme. It is reassuring that Sound 15 sounds more violin-like than Sound 16. This is the main reason for choosing to use the first microphone position for all the earlier comparisons: it gives a reasonably realistic sound for our datum case, so that the effect of changes can be judged more readily.
But the comparison of Sounds 15 and 16 raises an important issue. Why do we not hear such striking differences of sound when we move around while a violin is being played? There are two ingredients to the answer. First, we rarely listen to a violin being played in an anechoic chamber, or in the open air. Usually we are in a reverberant space of some kind. Reflections from the floor, walls, ceiling and furniture mean that your ears receive a mixture of sounds that were initially radiated from the violin in different directions. The more reverberant the space, the more complete is this “mixing” of directions.
The second ingredient is what happens inside your head, and is far more complicated. Think about how your visual system works. You have two “cameras” in your two eyes, and they dart around all over the place, scanning the world around you. But your brain puts that together into an internal perception of a fixed, three-dimensional world that stays largely the same as your gaze moves around it. Your auditory system does something similar. You have two “microphones” in your ears, which receive sounds that change as you walk around the room or turn your head, or as the violinist moves. Your brain assembles the information into a perception of a sound world that is relatively constant. Probably this is assisted by interaction with your visual system: you can see the same violin being played as you walk around the player, and that may help you to hear the sound quality as being rather constant and consistent — apart, of course, from the deliberate variations that the musician injects as part of their performance.
An obvious question arises from the contrast between Sounds 15 and 16: do they show similar variations when the soundpost is adjusted? If they don’t then perhaps it is quite misleading to use any single microphone position for the kind of study we have done here. On the other hand, if they do agree, at least to an extent, then we would like to know why. One possible answer to that question is that differences in sound may follow from differences in the flow of energy from the string at the bridge. If a modification such as moving the soundpost leads to a difference in the energy input to the violin in a certain frequency range, surely that should mean lower sound levels in that frequency range as measured by any microphone? But energy input is not the only factor: the soundpost adjustment will change the mode shapes, and this will change the directional sound radiation. The result might be that sound in a certain range goes up in one direction, but down if measured in a different direction.
Figure 33 gives at least a glimpse of the answer to these questions. It shows three curves based on the difference resulting from shifting the soundpost Eastwards by 5 mm: the case shown in the red and black curves in Fig. 27, and Sounds 12 and 14. The blue curve in Fig. 32 refers to exactly this: it shows the decibel difference between those two curves in Fig. 27. Positive values indicate louder sound after the shift, negative values indicate the converse. The red curve in Fig. 33 shows the same comparison but based on the second microphone position. The black curve shows the change in power input between the two cases, calculated from the real part of the (complex) bridge admittance.
It is immediately clear that the three curves in Fig. 33 are by no means identical, but that they share some features in common. All three curves show about the same magnitude of fluctuations, and at least in places they show striking similarities. Furthermore, the results for the two microphone positions (blue and red curves) are as different from each other as either is from the power input (black curve). If we were able to compute the total power radiated as sound in all directions, perhaps this would follow the black curve more closely.
This comparison is important. If we want to listen to the effect of a soundpost adjustment, then of course we need to use a comparison based on radiated sound. But if we want to quantify the differences in physical terms, for example to compare the sensitivity to different adjustments, then the measure based on power input is far simpler and more reliable. It is independent of the complicated issues surrounding sound radiation, and indeed it would be directly measurable on a real instrument.
F. Summary
It is useful to summarise what we have learned. The cigar-box model certainly does not capture all aspects of normal violin behaviour accurately, but perhaps it is close enough in some ways to give useful qualitative information about soundpost adjustment. Equally, the approach we have used to estimate sound radiation is approximate at best, but perhaps it is good enough to be useful.
Having repeated those important caveats, the frequency responses and sound examples shown here surely capture some aspects of the physics correctly, and the results seem to be in general agreement with the experience of makers and tone-adjusters. Small tweaks to a soundpost can indeed produce audible changes, including all aspects of the “tightness, position, fit” mantra. In our model, “tightness” and “fit” are represented by the values of various contact stiffnesses at the ends of the post. The results suggest that the axial contact stiffness has a bigger influence than the torsional contact stiffness.
“Position” is, slightly unexpectedly, more complicated. The position of the centre of the post is, of course, crucial. But even without moving the post centre, a very small tilt of the post might redistribute the contact stiffness, and thus shift the effective contact point around within the footprint of the post end. This suggests that a tilt too small to create a visible gap might have the effect of shifting the soundpost by 2 or 3 millimetres: a change that would be regarded as large by a violin-maker.
[1] Myles Cameron Nadarajah, “The mechanics of the soundpost in the violin”, PhD dissertation, University of Cambridge (2018).