Coupled piano strings

Abstract

The admittance of the piano bridge has a crucial effect on piano tone by coupling together the strings belonging to one note into a single dynamical system. In this paper, we first develop theoretical expressions that show how the rate of energy transmission to the bridge as a function of time (including the phenomena of beats and "aftersound") depends on bridge admittance, hammer irregularities, and the exact state in which the piano is tuned. We then present experimental data showing the effects of mutual string coupling on beats and aftersound, as well as the great importance of the two polarizations of the string motion. The function of the una corda pedal in controlling the aftersound is explained, and the stylistic possibilities of a split damper are pointed out. The way in which an excellent tuner can use fine tuning of the unisons to make the aftersound more uniform is discussed.PACS numbers: 43.75.Mn, 43.40. Cw INTRODUCTIONWith very few exceptions, and starting with the earliest instruments known, pianos have been built with more than one string per note; yet relatively little attention has been paid to this feature by acousticians. It was noted first by Martin and Ward, t and later in more detail by Kirk, •' that the strings comprising a single "triplet"--that is, the group struck by a single hammer-were not tuned in perfect unison by excellent tuners; and that, indeed, listeners often preferred the sound produced when there was a small discrepancy among the individual frequencies.Setting aside the subjective "preferences," whose cause and nature--and, hence, significance--is difficult to assess, the practice of artist tuners must be taken with extreme seriousness. We note that in this context "discrepancies from unison" refers to frequency differences observed when the strings are sounded individually, with the other members of the same triplet damped. These differences are not equal to the beat frequencies heard when the strings sound together, due to the coupled nature of the string vibrations.A number of years earlier, Martin s had reported on the characteristic "double decay" of piano tones, whereby the sound amplitude dies away with two distinct rates, breaking from an original "fast" decay to a later "slow" decay, or "aftersound." This feature is important in producing the characteristic piano tone. Martin did not at that time suggest any mechanism for the double decay, but in later work found that the phenomenon is connected with the presence of more than one string per note, and l{hat the amount of aftersound is affected by the exact manner in which the unisons are tuned. 4The suggestion that the phase relations among the ß strings play an important role appears to have been made first by Hundley, Martin, and Benioff, s and in more explicit form by Benade. 6 Benade points out that when three strings vibrate in phase, the motion of the bridge is three times what it would be if one string were vibrating alone; hence, the rate of energy loss of each string is tripled. He suggests that, after some ...

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