/ Modeling Piano Sound using Waveguide Digital Filtering Techniques
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