/ Modeling Piano Sound using Waveguide Digital Filtering Techniques
ï~~A typical piano has one string for each pitch in the lowest octave, two strings per pitch above that until about C3, then, for the rest of its range, it has three strings per pitch The effects of this multiple stringing are complex. See especially, [24) for a thorough treatment. For my purpose it is sufficient to note two effects: the multiple strings are always slightly detuned and thus contribute a mild, slow beating to the timbre; second, multiple strings contribute tc the effect known as a double decay-the amplitude envelope of the piano dies away very rapidly at first and then, after the first few tenths of a second, much more slowly. This double decay shape would seem to be partly, and perhaps more significantly, caused by the presence of at least two spatially polarized modes of vibration. There is a vertical mode that dies away rather quickly and a horizontal mode that takes a much longer time to decay. Again, see [24] for a more detailed discussion of this. 3. Bridge and soundboard The bridge serves the purpose of transfering the vibrational energy of the strings to the soundboard and to each other. In most pianos the bridge is actually two seperate bridges, one for the lowest, cross-strung, strings and another for the rest. A significant effort has been made by piano designers and manufacturers to balance the impedance characteristics of the bridge with the impedance of individual strings so that each string will resonate for the longest time possible. The way this is done is to ensure that the bridge has a much higher impedance than the string, thus tending to reflect the string's vibration back into the string, but not so much that insufficient energy is transmitted to cause the soundboard to vibrate. Too much bridge impedance means the soundboard will get too little energy and we won't hear a thing. Too little bridge impedance means the string will die away too quickly. The job of the soundboard is simply to transmit the vibration of the strings to the air with large enough amplitude for it to eventually reach our listening ears. Usually, a large piece of laminated pine is used for this purpose. A study by Suzuki [22] shows quite clearly the first few vibration modes of a Steinway soundboard. He reported measuring six low-frequency peaks in the spectrum: 49.7, 76.5, 85.3, 116.1, 135.6 and 181.1 hertz. These apparently correspond to the fundamental vibration modes of the particular soundboard studied. It must be noted, however, that the soundboard studied by Suzuki was without the cast-iron plate and strings and thus may only partially correspond to the resonances of soundboards in the complete piano. Preliminary studies by the present author and Julius Smith have shown similar frequency characteristics in fully functional pianos (see Figure 6); there are some shallow, Figure 8. The measured frequency response of the soundboard in a Yamaha Conservatory model grand piano low frequency resonances followed by approximately 14 dB drop per octave thereafter. This frequency response, by itself, would be relatively simple to model with low-order filters. However, the time domain characteristics of the soundboard are not so simple, as can be seen from Figure 7 and Figure 8. Figure 7 is an impulsive signal that was fed into a soundboard. The resulting response of the soundboard is shown in Figure 8. Figure 7. An impulsive signal used to excite a soundboard. Figure 8. The response of the soundboard in a Yamaha Conservatory model grand piano to the signal in Figure 7. 92 1987 ICMC Proceedings
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