ï~~increase in the reflection and absorption at the point of
impact until the hammer has attained its maximum push
against the string. From this point on there is a gradual
decrease in the reflection until the hammer leaves the string
entirely. When this occurs, the string becomes one continuous bi-directional waveguide from the capo d'astro to
the bridge.
Since the real string is not an ideally flexible medium, it
is necessary to take into account the inharmonicity of its
partials due to the effects of stiffness of the steel wire. This
is well known physically ([12]) but would be too expensive to
compute in the classical manner. Instead, it is assumed that
all of the effects of stiffness at each point along the string
can be "lumped" together and accounted for solely at the
junction of the string and the bridge. This has proved to be
a reasonable assumption in the case of musical instrument
strings that require substantial amplification (in this case
by the soundboard) in order to be heard. Accordingly, a
specially designed allpass filter is added at this junction as
described in [20].
To simulate the effects of multiple strings on a single
pitch, the present model uses multiple bi-directional waveguides closely coupled together.
3. Bridge
As with the real piano, the model of the bridge must
maintain a delicate balance between transmission and reflection. If too much string energy is transmitted the vibrations in the string would die away too quickly, and if too
much energy is reflected back into the string the soundboard
will have nothing to amplify and the piano would be inaudible. These reflection and transmission characteristics
are dependent on the relative characteristic impedances of
the string and the bridge. The string impedance can be calculated or estimated for a given piano from measurements
of mass, tension and type of material. In the present model,
I use a 1-multiply junction (see Figure 2) to connect each
string with the bridge and soundboard. The transmission
coefficient is on the order of.001, that is, 99.9 percent of the
string energy is reflected and the final tenth of a percent is
allowed to pass on through the bridge to the soundboard.
The bridge is also responsible for a considerable amount
of energy being lost due to friction and heating. This
has been effectively modeled in the past by a simple moving average lowpass filter at the bridge junction ([7]). Its
difference equation can be stated as:
$n) = udx(n) + gax(n - 1))
Here, gi specifies the overall gain or damping characteristics, and g2 determines the cutoff frequency. See [15] for an
introduction to difference equations and filter theory. This
filter is inserted into the string waveguide right after the
junction connecting the string to the bridge.
4. Soundboard
The model of the soundboard is the least fully developed
of all the components. It is simple to match the frequency
characteristics noted by Suzuki ([22]), but to capture the
effects of the time-domain response is not at all simple.
The most accurate way would probably be to model it as a
three dimensional waveguide. Unfortunately, this rapidly
gets away from a reasonable compute time. Instead, I
have concentrated on developing what is essentially a two
dimensional model, albeit a rather limited two dimensions.
This extremely oversimplified view nonetheless has yielded
surprisingly good results. I have limited it to a structure
of six waveguides each of which is connected directly to
the bridge at a single location. I use a MIMO junction
to handle this interconnection. In this way, some of the
energy from a sounding string can be passed through the
soundboard and can be returned to that string or to any
other string who's damper has been removed. This provides
a simple means to obtain some pedal effects as well. Each
of these interconnected waveguides also contains a lowpass
filter with a rather large damping factor. This soundboard
network is tapped in several places and its output is sent
to the DACs. By tapping it at widely separated points and
sending each separate tap to a different output channel rich,
decorrelated, multi-channel outputs can be had for almost
no additional cost.
V. Conclusions
The basic elements of a modern acoustic piano have been
considered and a method for cheaply and accurately modeling them digitally has been discussed. Work is underway
at Stanford's Center for Computer Research in Music and
Acoustics to develop a performance model based on this
research. The basic model has been realized on a realtime digital synthesizer and exploratory work has been conducted on Symbolics Lisp Machines with FPS array processor hardware. The goal of future research is to expand the
model, making it more physically accurate and meaningful,
and to develop in the process a set of physical modeling
tools that can be used in the design and study of other types
of instruments. Thus, it should soon be possible for computer musicians, computists, to use the reed mechanism of
a clarinet to drive a violin string which in turn is connected
to a piano soundboard. Then the distinct advantages of
physical modeling will be manifest.
94
1987 ICMC Proceedings
