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  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  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.
1987 ICMC Proceedings