The coupled motion of piano strings

Gabriel Weinreich

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As everyone knows, the word "piano" for the instrument with whose acoustics this seminar is concerned derives from the name given to it by its inventor, Bartolommeo Cristofori, who shortly after its invention in c. 1709 had his creation described as "gravicembalo (~clavicembalo) col piano e forte" because, unlike the ordinary clavicembalo (that is, harpsichord), it was capable of varying its dynamic level. The description is, in some ways, even more apt than its originator intended. In the preceding lecture, Donald Hall has described how the radical nonlinearity of the hammer produces, along with the dynamic range, a correspondingly large range of different tone colours, giving the phrase piano e forte added significance. My lecture will be concerned with an even more peculiar fact, namely that, in a certain sense, the gravicembalo piano e forte can be said to be playing piano and forte at the same time!

Loud and sustained

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To understand how this phenomenon originates, let us consider a hypothetical engineer assigned to develop a musical instrument in which a string which is initially excited by a hammer blow (or, for that matter, by plucking) is to be coupled to a soundboard, and so produce a musical sound. Obviously, this engineer would have to make a decision with regard to the degree of coupling between the string and the soundboard, i.e., how efficiently the string vibrations are transferred to the soundboard. Here he would immediately encounter a difficulty, for the following reason. A strong coupling will lead to a sound which is initially loud but decays very quickly in time, since the same strong coupling must lead to a high rate of energy transfer from string to soundboard. If, on the other hand, the engineer wishes to have a long sustained sound, so as to give the instrument more of what one might call a "singing" quality, he must keep the coupling to the soundboard weak - and hence be constrained to a sound which is quiet and lacks carrying power.

It is true, of course, that even without any soundboard coupling the string vibration would not last forever, since other mechanisms of energy loss - such as internal friction of the string, viscous dissipation in the air, and direct sound radiation from the string motion - do exist, but in practice this is irrelevant. To the best of my knowledge, in all musically useful string applications the dominant mechanism of string damping is motion of the end supports, that is, coupling of the bridge to the soundboard.

To a practical engineer, the competition between loudness and sustaining power would point to the need for some sort of compromise or "trade-off;" but a musical instrument is not, by its nature, an "engineering compromise." Most objects capable of producing sounds are, after all, not used as musical instruments. The ones which are selected for such a purpose always represent some type of "miracle," that is, a situation in which a seeming engineering limitation is overcome in an unexpected way.

In connection with the piano, it turns out that our ear - or, more correctly, our ear plus brain - has a way of judging both loudness and sustaining power in a way that might not have been predicted. Specifically, a sound is perceived as loud if it starts out loud, even if it then decays quickly; and it is perceived as sustained if some part of it is sustained, even if that part is rather weak. Thus a sound which starts out with a high but quickly decaying amplitude, and which then, having reached a rather low level, switches to a much smaller rate of decay - so that there is a sustained but subdued "tail" or "aftersound" - is perceived as being both loud and sustained. And that is precisely the miracle of the piano tone.

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This lecture is one of Five lectures on the Acoustics of the piano

1990 Royal Swedish Academy of Music