5.3 Signature modes and formants

We can now return, yet again, to the three stringed instruments whose bridge admittances were plotted in Fig. 7 of section 5.1. Here is a repeat of that figure:

Figure 1. A repeat of Fig. 7 from section 5.1: scaled bridge admittance magnitude, for three different stringed instruments. Red curve: a violin; purple curve: a classical guitar; blue curve: a banjo. The frequency scale is expressed in semitones, starting from the lowest tuned note of each instrument. Successive octaves are indicated by ticks.

Before launching into a more detailed interpretation of these curves, we should have a look at the responses of more than one instrument of each type, to get an idea of which features are general, and which are special to a particular instrument. Figures 2 and 3 show results plotted in the same format as Fig. 1, for representative selections of violins and guitars, respectively. In both cases, some famous makers are included in the mix. Stradivari and Guarneri ‘del Gesu’ are represented among the violins. No maker in the guitar world is quite as famous (or expensive) as these two, but several leading makers are represented. Furthermore, several major styles of guitar construction are included. (For cognoscenti, these styles involve fan-braced, lattice-braced and X-braced instruments.) But the first impression from both plots is that ‘all violins’ are really rather similar, and ‘all guitars’ are also similar. It is something of a relief, though, to see that violins and guitars are significantly and consistently different from each other. Furthermore, the banjo response shown in Fig. 1 is significantly different from either.

Figure 2. Measured bridge admittance, in the same format as Fig. 1, for 7 violins: two by Stradivari, one by Guarneri ‘del Gesu’, and four by modern makers.
Figure 3. Measured bridge admittance, in the same format as Figs. 1 and 2, for eight acoustic guitars. The set includes classical, flamenco and steel-strung guitars; makers include Fleta, Smallman, Ramirez and Fylde.

Armed with this knowledge, we can try to describe and understand the features of the three types of instrument response, and explore what this tells us about how the instruments sound. We have already seen, in section 5.1, that any admittance usually shows well-separated modes at low frequency, but that the modal overlap factor progressively increases so that at higher frequencies modes lose their separate identities.

This distinction has an important consequence. Each type of instrument has a small number of low-frequency modes, sometimes called ‘signature modes’, that can usually be recognised in every individual instrument. To an extent, an instrument maker can control the shapes and natural frequencies of these modes by variations in material choice and constructional details. At somewhat higher frequencies it is still possible with careful measurements to determine the modes despite moderate overlap, but the shapes are likely to be much less recognisable from one instrument to another.

At higher frequencies still, things are even less simple. The modes overlap and blur together, and the variability of mode shapes between instruments is much greater. Although, mathematically speaking, the modes still determine the behaviour, the modal point of view becomes progressively less useful. A maker cannot hope to control everything at higher frequencies, but this does not mean that they cannot control anything. Some instruments show features called formants, that can offer a useful degree of control to the maker. We will return to that idea shortly, but first we will look at some signature modes for guitars and violins.

We will start with the guitar. Figure 4 shows a few examples, in the form of holographic images. In terms of the frequency scale of Fig. 3, these signature modes are found in the frequency range up to about semitone 30. In all four modes shown here, motion is largely confined to the top plate. The sides of the guitar are sufficiently rigid that there is little movement at the edges. Some modes do, however, involve motion in the back plate, not shown here: acoustic response in the air inside the soundbox provides some coupling to the back.

Response of the internal air also provides the explanation of the fact that the first two modes shown here look remarkably similar. Now, there is a very general mathematical property of mode shapes called ‘orthogonality’ (it was mentioned in section 2.2.5), which says that it is not possible for two modes to have shapes that are too similar. When a measurement shows apparently similar shapes like the images here, it always means that something else is participating significantly in the vibration, which has not been captured in the measurement.

In this case, we already know what this is from section ?. The guitar body is designed to have strong coupling between the lowest mode of the top plate and the Helmholtz resonance of the internal air, ‘breathing’ through the soundhole. The result of this coupling is a pair of modes, each involving significant amounts of both plate motion and air motion. But one of them has these components in the same phase, while the other has them in opposite phase: this difference allows them to satisfy the requirement of orthogonality. In actual fact, there are three coupled modes in the guitar body, because the lowest mode of the back plate also couples to the other two, but we haven’t shown the third one in Fig. 4.

The violin also has ‘signature modes’, but they are more complicated. Some examples are shown in Fig. 5. In terms of the frequency scale of Fig. 2, these modes are found at frequencies below about semitone 20. Compared to the guitar modes in Fig. 4, the violin shows a lot of movement around the edges: the ‘rib garland’ of a violin is made of thin strips of wood (usually maple), and it is very flexible. For all four modes shown here, the node lines (visible as white stripes) run off the edges of the top plate, and reappear in more or less the same places on the back plate. The motion is truly three-dimensional.

There has been a long history of researchers trying to visualise and understand these modes, and this has resulted in a rather non-intuitive set of labels that are commonly used to describe the modes. The first mode, top left in Fig. 5, is usually called ‘A0’. It is essentially the Helmholtz resonance of the internal air, modified by the flexibility of the plates. The body of the violin is ‘breathing’ in and out. You might immediately think that this must make the mode an efficient radiator of sound, because the volume is changing so the body can behave as a monopole sound source (recall the discussion in section ?).

Well, that conclusion is correct, but not for the obvious reason. As was explained in section ?, the result of coupling a Helmholtz resonance to flexible plates is that this lowest resulting mode has volume variations of the box structure which are in the opposite phase to the volume variations resulting from air flow in and out of the f-holes. The air flow contribution is the bigger of the two. So the mode is indeed a good radiator of sound, but the structural motion seen in the plot serves to reduce the net radiation a little. However, it is very important that there is some structural motion associated with the mode, because otherwise it would not show up in the bridge admittance, and the vibrating strings would not be able to excite it.

The second mode shown in Fig. 5, top right, is usually called ‘CBR’ for historical reasons. For this mode, the top and back plates are moving in very similar ways: the entire body of the violin is vibrating rather like a thick bending plate. The result is that there is very little change in volume, and this mode is not a good radiator of sound.

The two modes on the bottom row of Fig. 6 are ‘twins’: the top plate motion in the first of them is very much like the back plate motion in the second, and vice versa. An evocative name for them both is the ‘baseball modes’: each has a single sinuous nodal line that snakes around the box in a pattern like the seam of a baseball. However, they are more commonly labelled ‘B1-‘ and ‘B1+’. Both modes have significant volume variation, and both are excellent radiators of sound. Both are strongly influenced by the presence of the soundpost inside the violin body, producing asymmetry and coupling the top and back plates together: see the discussion in section ?.

Until recently, no very persuasive description had been given of why the low modes of a violin body should take these particular forms, or how an instrument maker might be able to influence the details (and in particular the sound radiation from them). However, an extensive series of explorations by Gough using finite element (FE) analysis has given some answers to these questions [1]. He explored a series of models starting from very simple assumptions, then gradually adding in the complications of violin design one at a time so that the progressive evolution and emergence of the signature modes can be charted, and the relative influence of various contributory factors assessed. The full picture revealed by Gough’s work is too complicated to include here: an abbreviated summary showing some key aspects can be found in [2].

We will move away from signature modes now, and look higher up the frequency range. The banjo hasn’t been forgotten, but a discussion of that instrument is best left until we have introduced an important new idea. The concept of a formant has come from the study of the human voice, whether speaking or singing.

It is simplest to think about singing. If you sing a steady note, there are two choices you can make: the pitch of the note, and which vowel sound you want the listener to hear. You set the pitch by the frequency of oscillation of your vocal chords, but how is one vowel distinguished from another? You know what you have to do in practice: you have to position your tongue and lips in a particular configuration, different for each vowel. The sound is made by your vocal chords, but it has to pass through your vocal tract before it can emerge from your mouth as external sound waves. The vocal tract has resonances, like any other acoustical duct (see section ?): these resonances are called formants. The effect of changing the positions of your tongue and lips is to shift these resonances around. Because the walls of the vocal tract are made of soft flesh, the resonances have quite high damping, so their half-power bandwidth is quite large.

The result is a frequency spectrum of the vocal tract filter which will look a bit like the schematic version shown as a dashed line in Fig. 6. The animation in this figure shows what happens when the singer performs a chromatic scale. The individual harmonics change with every note, but the heights of the peaks always mark out (to a greater or lesser extent) the positions of the two formants included in this plot. Your brain makes use of that information to recognise that the same vowel is being sung at all these different pitches. The full story of vowel sounds, and speech perception in general, is too complicated to go into detail here; to learn more, look at the Wikipedia article.

Figure 6. Schematic example of formants. The red lines mark the harmonics produced by a singer, performing a one-octave chromatic scale starting at $G_3$ (196 Hz). The dashed line shows a schematic version of the frequency response of the vocal tract, in a configuration corresponding to a particular vowel. As the pitch of the note changes, the filter shape is still ‘dotted out’ by the heights of the harmonic peaks, so that your brain is able to perceive which vowel is being sung.

We now want to apply this idea of formants to musical instruments. The interpretation is slightly different in that context. The example in Fig. 6 shows a series of peaks, which sketch out the shape of a larger-scale structure associated with the formant filter. In the application to stringed instruments, a somewhat similar structure is sometimes seen, but now it involves many peaks from the resonant behaviour of the body, whose heights are systematically modulated to mark out a larger-scale structure.

The example that has been most thoroughly studied relates to the violin. It produces a feature usually called the bridge hill [3,4]. We have already seen the effect of this hill, without really being aware of it. It is directly responsible for the difference between violins and guitars seen in Figs. 2 and 3, in the frequency range around semitones 24–48. The feature we are interested in becomes much more obvious if we plot the phase as well as the magnitude of the bridge admittance. An example, for a typical violin, is shown in Fig. 7. The upper plot shows a magnitude plot that is by now quite familiar. The ‘hill’ feature appears in the frequency range indicated by the horizontal dashed line. The lower plot shows the phase response, and this does something very dramatic in the same frequency range: the phase drops systematically towards $-90^\circ$.

Figure 7. The bridge admittance of a typical violin, this time showing the phase (lower plot) as well as the magnitude (upper plot).

To understand what is going on, it helps to look at a simpler system. Instead of the complicated violin body, we can imagine a system which has regularly spaced resonance frequencies, all with the same damping factor (or Q-factor), and all having the same modal amplitude at the point where we choose to drive the system. We can calculate the admittance of this simple system using the formula in eq. (12) of section 2.2.7: for a particular choice of parameter values the result is shown in the left-hand plot in Fig. 8. This super-simple system has no eye-catching and distracting features, so that the effect of the next stage will be very clear.

We now choose to drive this simple system in a different way. Instead of applying a force directly, we drive it through a mass-spring resonator system as sketched in Fig. 9. As explained in the next link, the modified admittance can be calculated in terms of that of the original system, plus the assumed values of the stiffness $k$ and the mass $m$. An example of the result is shown in the right-hand plots of Fig. 8. It should be immediately clear that this shows a strong resemblance to the violin response in Fig. 7: the amplitude shows a ‘hill’ where the level has been lifted by some 20 dB, while the phase plot shows the dramatic drop towards $90^\circ$ that we noted earlier. The formant-like nature of the amplitude plot should also be obvious.

Figure 9. Schematic sketch of an instrument body with admittance $Y_v(\omega)$, being driven through a simple mass-spring oscillator.

SEE MORE DETAIL

It is useful to find a way to see the shape of the underlying formant ‘hill’ without the distracting details of the individual body modes. This can be done in a very simple way by using an approach known as Skudrzyk’s mean value method [5]: the details are described in the next link. The essence of the argument is to distinguish the direct field of vibration radiating out from where the force is applied, and the reverberant field made up of all the reflected waves returning from the boundaries, and then bouncing around until they die away from the effects of damping. Skudrzyk showed that the mean trend of a decibel plot like the ones we have been looking at corresponds to the direct field only: all the detail of modal peaks is the result of adding in the reverberant field. The dashed lines in Fig. 8 were calculated this way, from the direct field. They follow the mean trend of the decibel plots of amplitude very well, and they also catch the phase behaviour convincingly.

SEE MORE DETAIL

The argument behind Skudrzyk’s method can also be used with measured data. The phase of the reverberant response will vary in an irregular way with frequency, as the interference effects between different wave paths change, and it may therefore be reasonable to expect that the reverberant field will average to zero over a frequency band of sufficient width to contain several modes of the system [6]. Computing a suitable moving average of a measured admittance should therefore remove the reverberant field and leave the direct field, which is what we want to give the Skudrzyk estimate of the trend.

The result, applied to the admittance from Fig. 7, is shown as the blue dashed line in Fig. 10. The agreement is excellent. Now, you might think this is not surprising: we have used an averaging process, and got a result which predicts an average trend. But we have computed a linear average of the complex admittance, which will include a lot of phase cancellation, whereas the trend we have successfully matched is the logarithmic mean of the magnitude (because it is a decibel plot). Really, this agreement is quite surprising, even if it was predicted by the theory!

Figure 10. The violin admittance from Fig. 7, with blue dashed lines added to show the result of a linear average with bandwidth 1000 Hz and a Hanning window, to estimate the direct field contribution.

So the ‘bridge hill’ in the violin seems to be the result of driving the body through some kind of resonant system. But when you look at a violin you see no obvious sign of the mass and spring from Fig. 9. There is a kind of resonator, but it is not immediately apparent: as you might guess from the name, the violin bridge plays a central role. Figure 11 shows the ‘island’ area of a violin: the bridge, with its curious-shaped cut-outs, sits in the rather flexible region of the top plate between the f-holes. The strings sit on the top curve of the bridge, and when a string is bowed the force applied to the bridge is predominantly in the side-to-side direction.

Figure 11. The ‘island’ area of a violin. Out of sight inside the body, the bassbar passes underneath the left-hand bridge foot, and the sound post is close to the right-hand foot.

The bridge tends to rock in response to this force. Now, if the bridge was removed from the violin and its feet clamped in a vice, a typical violin bridge turns out to have a resonance somewhere around 3.5 kHz, in which it bends at the waist in a rocking motion, as sketched in the left-hand plot of Fig. 12. The right-hand plot shows a simple mass-spring model of a bridge, representing this rocking motion in a form that makes it look a little more like Fig. 9. This bridge resonance was first described by Reinicke, back in the 1970s [7].

As Reinicke was already aware, when the bridge is on the violin the corresponding rocking resonance is usually a lot lower in frequency, because the top plate of the instrument is more flexible than the bridge itself. This is the resonance that is responsible for the bridge hill, and it is the reason that the bridge admittance of a violin is some 20 dB higher than the corresponding admittance of a guitar in the low kHz range. It is governed by the mass of the bridge, particularly the top part above the waist, and by a combination of two stiffnesses. One is associated with the flexibility of the bridge itself, the other with the rotational flexibility of the ‘island’, when driven by the pair of forces through the bridge feet. These two stiffnesses are connected ‘in series’, so that whichever is the lower of the two has the dominant effect: usually, this is the stiffness of the ‘island’.

Violin makers are very well aware that bridge adjustment is an important resource for tonal tweaking of a violin. They will think carefully about choosing a bridge blank with the right material and foot spacing, and then about the final thickness and the details of the cut-outs. What they are doing, mainly, is adjusting the bridge hill. The model outlined here can be used to understand the influence on the hill of the different parameters — see reference [4] for more detail.

One particular ‘bridge adjustment’ is very familiar to all violinists: when a player fits a mute, they are simply increasing the mass $m$ in Fig. 12 and thus lowering the frequency of the bridge hill. This results in what a scientist would call a low-pass filtering effect: the roll-off of response above the hill, seen in Figs. 2 and 7, starts sooner so that the high-frequency content of the violin sound is reduced. The sound might then be described as more mellow, less strident and penetrating.

So now, finally, we have enough information to return to the three-way comparison of the violin, guitar and banjo. The reason for choosing these three instruments will now become clear: they exhibit three contrasting balances between the effects of signature modes and formants. Figure 13 shows a final version of the comparison of the scaled admittances: this time, the three averaged curves have been added.

Figure 13. A final comparison of the admittances of the three instruments from Fig. 1, this time including the result of averaging as in Fig. 10. Blue: banjo; red: violin; black: guitar.

The violin

The violin (red curves) has just been discussed. At low frequency it has the set of signature modes shown in Fig. 5. The air resonance A0 shows as a small peak around semitone 5 (270 Hz), and the three modes CBR, B1- and B1+ show as a cluster of high peaks in the semitone range 12–20 (corresponding to a frequency range around 400–700 Hz). After that, modal overlap increases steadily, and the only other clear thing visible in these curves is the bridge hill peaking around semitone 40 (2.3 kHz), as just discussed. This is not quite the end of the story, though. Although it is not visible in the particular plots shown here, recent work has shown that there is often a second ‘hill’ in a typical violin, in a higher frequency range. This one is associated with ‘bouncing’ motion of the bridge, rather than rocking. The cello, with its very different bridge design, has been studied far less than the violin, but it is likely to have one or more similar formants of its own.

Whether or not they think in these terms, violin makers manipulate both the signature modes and the bridge hill formant(s). Both play important roles in aspects of the sound quality and ‘playability’ of an instrument.

The guitar

The guitar admittance shows two large peaks corresponding to the first two signature modes shown in Fig. 4. The first of these is the equivalent of mode A0 in the violin, but air motion is coupled more strongly to plate motion in the guitar, which is why the peak is so much more prominent in the bridge admittance. Relative to the tuning of the instrument, this mode is placed lower than in the violin: around semitone 3 (i.e. around the note $G_2$ near 100 Hz). The next mode is roughly an octave higher, at a very similar frequency (relative to the tuning) to the mode CBR in the violin. Indeed, the guitar and the violin both show a cluster of three strong modes in very much the same range on this semitone scale.

After that, the guitar diverges strongly from the violin. The average curve is almost featureless: it simply trends gently downwards. This more-or-less flat trend line is what would be expected for bending vibration of any flat plate, based on Skudrzyk’s method. Recall that the Skudrzyk prediction of the mean trend is the admittance of an infinitely-extended system: in this case, an infinite plate. That is a system with a simple mathematical solution: the response of an infinite plate to a point force is exactly the same as an ideal mechanical resistance (or dashpot): the admittance is not complex but is a real number, and it is independent of frequency. So the Skudrzyk mean prediction would have a constant amplitude, and a constant phase of $0^\circ$: like the left-hand plot of Fig. 8, in fact.

The averaged line for the guitar in Fig. 13 may just possibly show a weak hill-like feature peaking around semitone 52 (1.7 kHz), but it would need a careful and systematic study to find out whether this is a consistent feature, and whether it has audible consequences.

But in general terms the description of the guitar bridge admittance we have just given is in accord with the received wisdom among guitar researchers. Most research has been devoted to understanding individual modes at low frequency, and their associated sound radiation characteristics. There is no equivalent in the guitar world of the research literature devoted to the bridge hill in the violin.

The banjo

Like the other instruments, the banjo has well-defined modes at low frequency, of course. We already know what they look like: they are essentially the same as the mode shapes of the drum, shown back in section 2.2 and examined in detail in section 3.6.1. Adding the bridge and strings to the banjo membrane changes the frequencies, and the asymmetric position of the bridge means that the degenerate pairs of modes in the drum are separated to significantly different frequencies. But the mode shapes remain very similar to those of the drum: measured versions of a few modes of a fully-strung banjo are shown in Figs. 14–16. Note that these laser vibrometer measurements were made with the resonator back of the banjo removed in order to access both sides of the membrane. This changes the frequencies a little, as discussed earlier.

Figure 14. A mode (or, strictly, an operating deflection shape) of the banjo, at 297 Hz
Figure 15. A mode (or, strictly, an operating deflection shape) of the banjo, at 500 Hz
Figure 16. A mode (or, strictly, an operating deflection shape) of the banjo, at 716 Hz

But Fig. 13 reveals something else: the averaged curve for the banjo shows a large formant-like feature, peaking around semitone 33 (1 kHz). It also suggests another similar (although smaller) feature around semitone 55 (3.5 kHz): we will return to this in a moment. But first, we need to understand the low-frequency formant, which dominates the bridge admittance of the banjo: even the ‘signature modes’ fall within the range of influence of this formant.

The formant is determined by an effective mass and stiffness acting at the bridge, just as in the violin bridge hill, and the idealised case in Fig. 9. The mass is clear enough: it is the mass of the bridge. The bridge of this particular banjo weighs only 2.2 g, but the Mylar membrane weighs only about 18 g so the added mass of the bridge is quite a significant fraction of that. But it is not at all obvious where the stiffness comes from.

It turns out that there are two sources of stiffness that are relevant, both of them requiring rather careful investigation to pin down (see [8,9] if you really want to know). One is associated with the fact that a membrane has a lot more modes than a plate: we won’t go into details here, but the argument from section 3.2.4 can be applied to this case, with the result that the modal density of a membrane, on average, increases proportional to frequency. For a plate, we showed in section 3.2.4 that the modal density was on average constant. The growing modal density of a membrane means that the combined effect of all the high-frequency modes adds up to something that has a significant effect on stiffness at low frequency.

The second source of stiffness comes from the stiffness of the strings along their lengths. When the bridge moves in response to string vibration, the geometric configuration of the strings and tailpiece, with a break angle of about $13^\circ$ over the bridge, means that the strings must be stretched a little. The banjo has metal strings, and the stiffness involved in this stretching is enough to matter. It acts just like a spring attached to the bridge. There is a similar stiffness contribution from the in-plane stiffness of the membrane itself: the downbearing force of the strings means that there is an effective ‘break angle’ where the membrane deforms around the bridge feet.

These effects, the bridge mass and the various sources of stiffness, combine to produce something like the schematic system shown in Fig. 9. The result is the formant we saw in Fig. 13, behaving much like the violin bridge hill but acting at significantly lower frequency. In section 5.5 we will be able to listen to a collection of synthesised sound examples based on a computer model of a banjo. This will allow us to hear the effect of changing various parameters, like the bridge mass or the head tension. Those sounds give a strong suggestion that any parameter change which affects the formant has a clearly audible consequence, whereas a change which shifts the individual resonances but leaves the formant in the same place gives only a rather subtle change of sound.

We won’t be sure until some serious psychoacoustical studies have been done (see chapter ? for some discussion of what that would involve), but if this speculation turns out to be right, the banjo gives a rather extreme example of the interplay between signature modes and formants. Individual body modes seem to have rather little perceptual significance, and the sound of the instrument is dominated by the effects of at least one formant. This is the opposite extreme to the guitar, where there don’t seem to be any interesting formants, and signature modes are the only game in town.

I say “at least one formant” above, because we already noted the appearance of a second, admittedly smaller, formant-like peak in the banjo admittance. There is also a third, at even higher frequency. These additional formants show more prominently if admittance measurements are made at different positions on the banjo bridge. Figure 17 shows a comparison of three different measurements on the same banjo, plotted over a wider frequency range with one additional octave. The red curves show the admittance near the top string, as in the case we have been looking at so far. The black curves show the result of measuring at the bridge centre, near the 3rd string. The blue curves show the admittance measured in the perpendicular direction: tapping and measuring horizontally on the top corner of the bridge, to give the admittance that would be relevant to string vibration in the plane parallel to the banjo head.

Figure 17. Scaled admittance measured at three different positions on the banjo bridge, with averaged curves shown in dashes. Red: near the top string as in previous plots; black: near the middle string; blue: horizontally on the side of the bridge.

The three measurements are very different from each other. The second peak we noted in the previous measurement appears far more strongly in the blue curves. The third formant shows up most prominently in the black curve. To track down what has caused these new formants required a detailed computer model of the banjo and its bridge: for details, see [9]. The bridge of this particular banjo has a typical three-legged shape, shown in Fig. 18. Both high-frequency formants turned out to be associated with resonances of this bridge, modified by the behaviour of the banjo membrane that it is sitting on. The lower of the two, around semitone 52 in Fig. 17, involves bending of the three legs so that the feet rotate; the higher one, around semitone 62, involves a bending resonance of the bridge rather like the first mode of a free-free beam as seen in Fig. 1 of section 3.2.

Figure 18. A close-up of the banjo bridge. The white patches are reflective tape, used for the laser vibrometer measurements.

In section 5.5, among the various sound examples will be some that illustrate the audible effect of these high-frequency formants. But before we are ready for that, we need to know how to add strings to the instrument body so that we can synthesise sounds based on an accurate representation of the physics of the instrument.

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[1] C. E. Gough. A violin shell model: vibrational modes and acoustics. Journal of the Acoustical Society of America 137, 1210–1225 (2015).

[2] J. Woodhouse. The acoustics of the violin: a review.  Reports on Progress in Physics 77, 115901.  DOI 10.1088/0034-4885/77/11/115901 (2014).

[3] E. V. Jansson and B. K. Niewczyk: On the acoustics of the violin: bridge or body hill? Journal of the Catgut Acoustical Society, Series 2, 3 23–27 (1999).

[4] J. Woodhouse. On the “bridge hill” of the violin.  Acta Acustica united with Acustica 91, 155–165 (2005).

[5] E. Skudrzyk: The mean-value method of predicting the dynamic response of complex vibrators. Journal of the Acoustical Society of America 67, 1105–1135 (1980).

[6] J. Woodhouse and R. S. Langley. Interpreting the input admittance of violins and guitars.  Acta Acustica united with Acustica 98, 611-628.  DOI 10.3813/AAA.918542 (2012).

[7] W. Reinicke. Die Uebertragungseigenschaften des Streichinstrumentenstegs. Doctoral dissertation, Technical University of Berlin (1973).

[8] banjo I

[9] banjo II