Since the violin is a linear system, the same Fourier components or "partials" appear in the output of the violin. The amplitude of each partial in the radiated sound is determined by the response of the instrument at that particular frequency. This is largely determined by the mechanical resonances of the bridge and by the body of the instrument. These resonances are illustrated schematically in figure 3, where typical responses have been mathematically modelled to simulate their influence on the sound produced.
At low frequencies the bridge simply acts as a mechanical lever, since the response is independent of frequency. However, between 2.5 and 3 kHz the bowing action excites a strong resonance of the bridge, with the top rocking about its narrowed waist section. This boosts the intensity of any partials in this frequency range, where the ear is most sensitive, and gives greater brightness and carrying power to the sound. Another resonance occurs at about 4.5 kHz in which the bridge bounces up and down on its two feet. Between these two resonances there is a strong dip in the transfer of force to the body. Thankfully this dip decreases the amplitude of the partials at these frequencies, which the ear associates with an unpleasant shrillness in musical quality.
The sinusoidal force exerted by the bridge on the top plate produces an acoustic output that can be modelled mathematically. The output increases dramatically whenever the exciting frequency coincides with one of the many vibrational modes of the instrument. Indeed, the violin is rather like a loudspeaker with a highly non-uniform frequency response that peaks every time a resonance is excited. The modelled response is very similar to many recorded examples made on real instruments.
In practice, quite small changes in the arching, thickness and mass of the individual plates can result in big changes in the resonant frequencies of the violin, which is why no two instruments ever sound exactly alike. The multi-resonant response leads to dramatic variations in the amplitudes of individual partials for any note played on the violin.
Such factors must have unconsciously guided the radical redesign of the bridge in the 19th century. Violinists often place an additional mass (the "mute") on the top of the bridge, effectively lowering the frequency of the bridge resonances. This results in a much quieter and "warmer" sound that players often use as a special effect. It is therefore surprising that so few players - or even violin makers - recognize the major importance of the bridge in determining the overall tone quality of an instrument.
One of the reasons for the excellent tone of the very best violins is the attention that top players give to the violin set-up - rather like the way in which a car engine is tuned to get the best performance. Violinists will, for example, carefully adjust the bridge to suit a particular instrument - or even select a different bridge altogether. The sound quality of many modern violins could undoubtedly be improved by taking just as much care in selecting and adjusting the bridge.
The transfer of energy from the vibrating string to the acoustically radiating structural modes is clearly essential for the instrument to produce any sound. However, this coupling must not be too strong, otherwise the instrument becomes difficult to play and the violinist has to work hard to maintain the Helmholtz wave. Indeed, a complete breakdown can occur when a string resonance coincides with a particularly strongly coupled and lightly damped structural resonance.
When this happens the sound suddenly changes from a smooth tone to a quasi-periodic, uncontrollable, grunting sound - the "wolf-note". Players minimize this problem by wedging a duster against the top plate to dampen the vibrational modes, or by placing a resonating mass, the "wolf-note adjuster", on one of the strings on the far side of the bridge. However, this only moves the wolf-note to a note that is not played as often, rather than eliminating it entirely.
