Coupled piano strings

The fact that a piano typically has not one, but two or three strings tuned to a given pitch, and that these sets of strings cross the bridge at (almost) the same point, leads to significant dynamical coupling among their vibrations. Since the dominant dissipation mechanism is the non‐rigidity of the bridge, the rate of energy loss by one string is radically affected by the way that its partners are vibrating; for example, an “antisymmetric” vibration of a pair of strings is much longer‐lived than a “symmetric” one. This fundamental phenomenon is complicated by a number of factors, including (a) slight differences in the natural frequencies of the individual strings; (b) a bridge admittance which has a reactive as well as a resistive part; (c) two possible polarizations of the string vibration; and (d) hammer irregularities which cause nonidentical initial excitations of the strings. In this paper we develop theoretical predictions of the way that the rate of energy transmission from strings to bridge, as a function of time, depends on the various parameters mentioned; we then compare these predictions with experimental measurements of beats and “aftersound” under various conditions. Our data shows the time‐dependence of each polarization of string vibration amplitude, as well as the resulting sound pressure level, covering a dynamic range of 70 dB from the moment of hammer impact until the signal is lost in the noise. The agreement with theory is excellent. On the basis of this understanding we also explain the function of the una corda pedal in controlling the aftersound, point out the stylistic possibilities of a split damper, and explore the way in which an excellent tuner can use the fine tuning of the unisons to make the aftersound more uniform from note to note.