In the main text, sound examples are given for variations of parameters of immediate interest to players and makers of banjos. However, the synthesis models also require some assumptions about internal “housekeeping” variables. For completeness, these are discussed now, and some sound examples are provided to illustrate the effect of these variables. There are two main topics here: representing “sound”, and fine-tuning the treatment of damping.
The synthesis model does not attempt to calculate radiated sound, it simply calculates the motion of the body at the bridge following a plucked note. This calculation includes the physics relevant to all the parametric variations considered here, but in order to allow perceptual judgements, sound files must be created that give a plausible approximation to the sound that would reach a listener or a microphone.
The actual radiation of sound by the banjo, or indeed any other vibrating structure, is rather complicated. It will vary significantly with frequency, and also with position of the observing point. There is no simple, universal model for sound radiation, analogous to the general formula for structural response as a linear combination of modal contributions (as in section 2.2.5). It is possible in principle to compute radiated sound pressure: some results from a detailed Finite Element and Boundary Element (FE/BE) model were presented in refs. [1,2]. An example of synthesised sound based on this model will be presented shortly, but the method is too computationally intensive to be used for wide-ranging parametric explorations.
Measurements in which bridge response and radiated sound were simultaneously measured suggest that the trend with frequency of the sound broadly follows that of the bridge velocity. However, when synthesised sounds were made in which bridge velocity was the output variable, several different listeners concurred in thinking that the sound was not bright enough. A very simple filtering procedure was thus used to boost the high frequencies: the predicted body velocity signal is scaled by $(i \omega)^\beta$ where the power $\beta$ can be chosen for best effect. The value for the studies here is $\beta=0.4$. The effect of changing the value of $\beta$ can be heard in the sound examples below. Note that in this particular set of files, the loudness level is not preserved between examples. Changing the value of $\beta$ makes a big difference to peak levels, and so each sound file has been auto-scaled.
Next, we give some examples of sounds based on the FE/BE model. Sound X.7 uses the computed bridge admittance, plotted as the black line in Fig. 1, filtered with the standard power-law filter. Sound X.8 uses the same admittance filtered using the detailed FE/BE prediction. The red line in Fig. 1 shows the sound pressure at a typical point, and the dashed blue line shows the ratio, which gives the transfer function from bridge velocity to field pressure. This is the transfer function used to filter the sound of Sound X.8. Notice the broadly horizontal trend of this dashed curve, illustrating the comment above that sound very roughly seems to follow velocity. Sound X.9 is similar to Sound X.8, but it uses a magnitude-only version of the computed frequency response for the filtering process.
There is no doubt that the FE/BE model captures many aspects of the banjo response and sound radiation far more accurately than the super-simple square banjo model. However, the results illustrate a theme we have encountered before and will encounter many more times: it is never obvious which features of a model have the most perceptual significance. In the present state of this FE/BE model, the resulting sounds do not achieve a high degree of realism. The reason is perhaps associated with transient effects arising from narrow spectral features in the computed transfer functions. This links to the topic we will turn to next: the perceptual quality of synthesised sounds can be very sensitive to details of the damping model, and these are often not amenable to theoretical prediction.
Subsection 5.5(I) of the main text gives a general discussion of some problems associated with the treatment of damping within the synthesis models, and particularly in the square banjo model. In order to produce a datum case of that model which compared well in a visual sense with the bridge admittance of the real banjo, and which also produced a sound quality which was not marred by distracting “zinginess” artefacts, damping was incorporated in two different ways. Both involve a choice of parameter values, and the effect of those choices is illustrated in the remaining sound examples here.
The first concerns the modelling of the effect of the bridge. Once a good value of the mass and the added stiffness had been chosen in order to get the low-frequency formant in about the right place, the individual peaks within that formant were still rather too pronounced for realism. So additional damping was incorporated in the form of a mechanical resistance (or “dashpot”) at the bridge. A range of values of the dashpot coefficient are illustrated in the next set of sound examples. Sound X.10 shows the effect with no dashpot; Sound X.12 illustrates the choice of dashpot used in the datum model; Sound X.14 shows the effect of a much stronger dashpot.
Even with this dashpot included, synthesis with the square banjo model produced distracting “zinginess” which has been mentioned several times. The purpose of using the square banjo model for parametric investigations is to give a useful impression of the perceived effect on sound of physical changes, for example to membrane tension or bridge mass. For that purpose, it is essential that the datum model gives an acceptably banjo-like sound, so this problem of zinginess arising from low loss at high frequency needs to be addressed. The problem arises largely because of the concentrated mass used to represent the bridge in the model. At high frequency, the admittance is dominated by the effect of this mass, and so the real part becomes very small. This is not the case in the measured admittance, largely because of the effect of the bridge hill discussed in section 5.3. This hill feature was well captured by the FE/BE model, using a detailed model of the bridge, but the simple concentrated mass used in the square banjo model cannot reproduce it.
This issue has been addressed with an unashamed fudge. This takes its inspiration from the way measured admittance is processed. The laser vibrometer and data-logging system result in a small time delay between the two data channels, the hammer signal and the velocity response. This needs to be compensated in the computer in order to avoid a similar problem of the real part of the admittance becoming small or even negative at higher frequencies, leading to “zinginess”.
No such physical delay is present in the theoretical model, but it was found that if it is processed in the same way, shifting the phase to represent a very short delay of 20 $\mu$s, the mean value of the real part of the admittance is changed to have a magnitude comparable with the measured value. Being purely a phase shift, this change has no effect on the magnitude of the admittance. The result of this, admittedly non-physical, phase compensation is illustrated by the next set of sound examples. Sound X.16 in this set is based on the original prediction of the model, and illustrates the original problem. Sound X.15 makes things worse by using a 20 $\mu$s compensation with the wrong sign: the zinging sound is very much in evidence. Sound X.17 is the case used as the datum: the zinging is largely suppressed. Sounds X.18 and X.19 illustrate the effect of increasing the delay further.
 Jim Woodhouse, David Politzer and Hossein Mansour. “Acoustics of the banjo: measurements and sound synthesis”, Acta Acustica 5, 15 (2021). The article is available here: https://doi.org/10.1051/aacus/2021009
 Jim Woodhouse, David Politzer and Hossein Mansour. “Acoustics of the banjo: theoretical and numerical modelling”, Acta Acustica 5, 16 (2021). The article is available here: https://doi.org/10.1051/aacus/2021008