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On the “Bridge Hill” of the Violin

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On the “Bridge Hill” of the Violin ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 91 (2005) 155 – 165 On the “Bridge Hill” of the Violin J. Woodhouse Cambridge University Engineering Department, Trumpington St, Cambridge CB2 1PZ, U.K. [email protected] Summary Many excellent violins show a broad peak of response in the vicinity of 2.5 kHz, a feature which has been called the “bridge hill”. It is demonstrated using simplified theoretical models that this feature arises from a combination of an in-plane resonance of the bridge and an averaged version of the response of the violin body at the bridge- foot positions. Using a technique from statistical vibration analysis, it is possible to extract the “skeleton” of the bridge hill in a very clear form. Parameter studies are thenpresentedwhichreveal how the bridge hill is affected, in some cases with great sensitivity, by the properties of the bridge and body. The results seem to account for behaviour seen in earlier experimental studies, and they have direct relevance to violin makers for guiding the adjustment of bridges to achieve desired tonal quality. PACS no. 43.75.De 1. Introduction The characteristic high bridge of the violin and cello, and indeed of most bowed-string instruments, presumably de- veloped initially for ergonomic reasons. In order to pro- vide the range of angles needed to bow each string indi- vidually, the strings must be raised clear of the instrument body. Also, the high bridge plays an essential role in “ro- tating” the transverseforcefrom the vibrating string into normal forces applied to the instrument body through the Figure 1. A violin bridge with an indication of the motion in the bridge feet, which can then excite bending vibration of the lowest in-plane bridge resonance. body [1]. Compared to the low, robust bridges of the pi- ano or guitar, the violin bridge may seem to be a necessary evil: it is fragile and requires regular attention to keep it significant variation with frequency. The lowest bridge res- straight and properly fitted. However, research of recent onance is usually found around 3 kHz when the bridge feet years shows that this type of bridge has provided, perhaps are held rigidly (for example in a vice), and the motion somewhat fortuitously, a crucial means for adjustment of consists of side-to-side rocking of the top portion of the thestring-to-body impedance characteristics which has al- bridge as sketched in Figure 1. lowed the violin family to acquire its familiar loudness and Since the work of Reinicke and Cremer, several exper- tonal colouration. imental studies have been carried out which relate to the The oscillating force provided by the vibrating string influence of this lowest bridge resonance. First, measure- can only excite vibration of the instrument body by first ments have been made by D¨unnwald [3] and Jansson [4, 5] passing through the bridge. The bridge thus acts as a fil- of the frequency response of a wide variety of violins. Both ter, and it is nosurprise that the material properties and authors found that violins of high market value showed geometric configuration of the bridge can have a signif- astrong tendency to exhibit a broad peak of response in icant influence on the sound of an instrument. The first the vicinity of 2–3 kHz, in a feature originally named the systematic study of the transmission properties of the vio- “bridge hill” by Jansson. The name was given because he, lin bridge was made by Reinicke and Cremer [1, 2]. They and indeed Cremer [1], attributed this feature to the filter- showed that a normal violin bridge has internal resonances ing effect of the lowest bridge resonance just described. within the frequency range of interest for the sound of the Atypical example is shown in Figure 2: the plot shows instrument, so that the filtering effect of the bridge has very the input admittance (velocity response to unit force am- plitude) in the direction of bowing, measured at the string position on a violin bridge using a method similar to that Received 5 May 2004, described by Jansson [4]. The characteristics of the bridge accepted 7 June 2004. hill are seen clearly in Figure 2: a rise in amplitude around c S. Hirzel Verlag EAA 155 ACTA ACUSTICA UNITED WITH ACUSTICA Woodhouse: On the “bridge hill” of the violin Vol. 91 (2005) -10 was already stressed by Reinicke and Cremer. The results of Jansson have fleshed this idea out with empirical data. -20 The task of this paper is to explain the pattern of be- -30 haviour and show how the frequency, height and shape of the bridge hill are influenced by the constructional pa- -40 Admittance (dB) rameters of the bridge and violin. A first step has been -50 taken by Beldie[9],whohas shown that a reasonable fit 200 500 1000 2000 5000 Frequency (Hz) to Jansson’s results can be obtained if the behaviour of the body beneath the bridge feet is approximated by simple 90 springs. In particular, his idea explains why there can still 45 be a “bridge hill” with Jansson’s plate bridge without cut- 0 outs: the mode consists of side-to-side rocking motion of the entire bridge, with a restoring force provided by the -45 Phase (degrees) “springs” under the feet. However, Beldie gave no expla- -90 nation for what determines the stiffness of these “effective 200 500 1000 2000 5000 Frequency (Hz) springs”, and more importantly, no-one appears to have Figure 2. Input admittance of a violin, showing a typical bridge addressed the question of what determines the height and hill (in the frequency range indicated by the dashed line in the up- bandwidth of the bridge hill. per plot). The upper plot shows the magnitude in dB re 1 m/s/N. The approach here will be to explore the bridge hill us- ing simplified theoretical models which are “violin-like” butwhich remain simple enough to analyse without need- F eiwt ing elaborate finite-element calculations This simplifica- tion is convenient, but more importantly it can be com- m bined with recent developments in vibration theory to al- lowacalculation of the underlying “skeleton” of the fre- k quency response curve, not readily accessible from a de- tailed prediction such as a finite-element model. Once it has been shown that the skeleton calculation is reliable, Yv()w this approach will allow attention to be focused on the pa- rameters which influence the “hill” without the distract- ingdetails of the individual body resonance peaks. All Figure 3. Sketch of a generic system driven through a resonator, the general features reported by Jansson can be demon- as described in the text. strated with these simplified models. One particular con- clusion of this study will be that the bridge hill observed the peak frequency of the “hill”, followed by a steady drop by D¨unnwald [3] and Jansson [4, 5] in many valuable in- in amplitude at higher frequencies, and over the same fre- struments is probably not primarily a property of the in- struments as such, but a result of the fact that more care quency range a downward trend in phase towards . The second relevant set of experimental studies is the and attention has been devoted to the fitting of appropriate extensive series reported by Janssonandco-workers [6, 7, bridges to these instruments. This suggestion is in keep- 8] in which various structural modifications were made to ing with the experience of experts in violin set-up, who the bridge and/or the top plate of the violin, and the ef- attribute great importance to bridge adjustment. fect on the input admittance measured. These studies have shown that the appearance of a “bridge hill”, and its fre- 2. Modelling the bridge and body quency, are not determined by the bridge structure alone. In particular, they reveal that the feature can sometimes The amplitude and phase characteristics seen in Figure 2 still be seen when the standard bridge is replaced by a are precisely what would be expected when a multi-modal “plate bridge” with no cut- outs, so that the vibration mode system (the violin body) is driven through an intermedi- showninFigure 1 is no longer possible. Jansson’s mea- ate system exhibiting a resonance. The simplest example t surements alsoshowedasignificant influence arising from i e is shown schematically in Figure 3. A force F is ap- the spacing of the bridge feet [7] and the stiffness/mass plied to a mass m,which in turn drives a system with in- of the top plate in the vicinity of the bridge [6]. These Y k put admittance v through a spring of stiffness .(The results led Jansson to become less satisfied by the name subscript “v ”indicates “violin”.) It is straightforward to “bridge hill”, and he has advocated “bridge-body hill”, or show that the input admittance of the combined system is just “hill” [6]. However, it will be argued that the bridge is still the central defining element in the phenomenon, and kY i v Y b (1) the original name “bridge hill” will be retained here. k m ikY The fact that the frequency of a bridge resonance will v be significantly affected by the coupling to the violin body where the subscript “b”indicates “bridge”. 156 Woodhouse: On the “bridge hill” of the violin ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 91 (2005) Y 0 ing factor. Then Figure 4a shows a plot of v ,andFig- Y (a) ure 4b shows a typical plot of b .Parameter values are -20 giveninthe caption. The dashed lines will be discussed | (dB) in Section 3, and should be ignored for the moment.
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