The main study of the banjo was all conducted with a single instrument and a single bridge. One conclusion was that the dynamic behaviour of the bridge, together with its mass and height, is responsible for creating formants. As a postscript to that study, a number of alternative banjo bridges were fitted to the standard instrument, and the bridge admittances at the 1st and 3rd string positions measured. These measurements could then be used to synthesise sound examples. The combination of measurements and synthesised sounds can give a first impression of the range of variation possible, and of how different these can sound. Some of the bridges are commercially available, others were made or modified specifically for this project. All the bridges have the same height, and all except the last one have the same mass (2.2 g). The bridge makers involved are Greg Deering, Donald New, Jeffrey Weitzl and Bart Veerman. The selection and adjustment was made by my co-conspirator on the banjo project, David Politzer.
Figure 1 shows the original Deering bridge, together with the most extreme variant: a solid block of poplar, shaped to give the correct mass and height. The white patches visible in this figure, and in others to follow, are pieces of reflective tape added to allow the laser vibrometer measurements. Figure 2 shows the measured admittances with this “bridge”, contrasted with the corresponding results with the Deering bridge. The same format will be used in all subsequent admittance plots. Results for the featured bridge, the poplar block in the case, are plotted in solid lines while the Deering results are plotted in dotted lines, the same in all subsequent plots. The black and blue curves show the admittances at the first string notch, the red and magenta curves show the admittances at the 3rd string notch, at the centre of the bridge.
For the poplar block bridge, lacking a hard ebony cap like the other bridges, it was not possible to obtain a hammer impact of a very high bandwidth: the poplar is simply too soft. This is the reason that the measured admittances become very noisy at high frequency. However, the comparison of the plots reveals exacatly what would have been predicted based on the earlier study. Because the mass and height were the same as the original bridge, the low-frequency formant is very similar with both bridges. Up to about 2 kHz, the pairs of curves agree quite closely. Above that, the Deering bridge shows formants (discussed in detail in section 5.5). But the poplar block shows very little evidence of such features, except for a small hump around4 kHz.
The final step is to produce synthesised sound examples. The same strategy has been used in all cases. The two measured admittances have been used to compute estimates of admittances at the 5 string notches. Strings 1 and 3 were directly measured. String 5 is assumed to have the same admittance as string 1 because all the bridges are symmetrical. For strings 2 and 4, a simple average of the results for strings 1 and 3 was used. The resulting sound clips for the original bridge and the poplar block are given in Sounds 1 and 2. To my ears, the two sounds are similar but by no means identical. I should emphasise that there is no claim that these synthesised sound will capture all nuances of the actual bridges. For example, the softer wood of the poplar block may well have a local influence on the string where it enters the bridge notch: such effects are not included in the sounds here.
The next set of bridges, Shown in Fig. 3, are variations on a theme. The starting point was three nominally identical bridges with a violin-like shape. The first bridge is the original. For the second, the middle leg has been trimmed back so that it does not touch the banjo head when fitted. The third bridge has a similar shortening of the leg, and then has a cut in the lower part to reduce the bending stiffness.
The corresponding admittances are shown in Figs. 4, 5 and 6. Again, in all three cases the low-frequency formant shows a close correspondence with the original Deering bridge. But the higher formants are significantly different, as would be expected with these different designs. The clearest difference is in the formant appearing strongly in the bridge-centre measurements. With the Deering bridge (magenta dots) this formant appeared around 5 kHz. In the first two of the present trio, it is seen around 8.5 kHz, then when the cut is added for the third bridge in this set it falls to around 6 kHz. This formant was shown to be associated with a first bending resonance of the bridge, modified by contact with the banjo head. These frequency variations all follow a pattern that makes sense in terms of that interpretation.
The variations seen when comparing the black and blue curves are less easy to guess. With the Deering bridge, a formant-like feature was seen around 3.5 kHz. The finite-element modelling described in Section 5.5 suggested that this was associated with detailed motion of the feet of the bridge. Without carrying out similar detailed modelling on the other bridge shapes to be shown here, it is very hard to guess what we expect to happen to this feature — and indeed a feature looking somewhat similar was seen in Fig. 2 with the poplar block, which of course does not have feet at all. What is clear in Figs. 4, 5 and 6 is that something changes when the middle leg is shortened: a clear feature is seen in Figs. 5 and 6 around 3 kHz, whereas Fig. 4 showed a broader and less prominent feature, somewhat higher in frequency.
Synthesised sounds based on these three bridges are given in Sounds 3, 4 and 5. It is hard to put into words one’s impressions of such sounds, but I think I hear a progressive change from “more muffled” to “more clear” through the sequence of sounds. But exactly which features of the response are giving me that impression is very hard to say. It would require some careful investigation and systematic listening tests to resolve that question: remember the discussion of the traps and pitfalls of psychoacoustical testing from chapter 4.
The next two bridges, shown in Fig. 7, are commercially-available bridge with designs that are radically different from the 3-foot design of the Deering bridge. Admittances for the two are shown in Figs. 8 and 9, and corresponding synthesised sounds in Sounds 6 and 7. The first bridge shows admittances that are similar in general form to the Deering original, but different in details. The second bridge is more different, as is not surprising from this design which has obvious higher bending stiffness.
The final set of bridges is shown in Fig. 10. Admittances are shown in Figs. 11, 12 and 13, and synthesised sounds are given in Sounds 8, 9 and 10. The first pair of bridges have a design similar to the Deering bridge, but made of a different wood. The difference between the two is that in the first one, the feet are shaped with a curved contour while the second one is flat. The third bridge is the only one in the set tested here which has a different mass: roughly half the mass of all the other bridges. Both measurements and sound examples suggest very little difference between the first pair. But the third bridge behaves quite differently, as we would expect. The lower mass makes the low-frequency formant extend to higher frequency, as is clear in Fig. 13. The corresponding sound (Sound 10) is strikingly different from all the others.
