Introduction

The piano is an example of a nonlinearly struck string. It is simple in some ways and highly complicated in others. It is simple in that only the hammer velocity matters as a control variable when the string is struck--the finger that presses the key has no significant mechanical connection to the hammer after it is launched into flight toward the string. That means MIDI, for example, provides a sufficient representation for piano performance, and the dimensionality of control (aside from pedals) is confined to one degree of freedom per key, per time instant--the so-called ``velocity'' parameter.

Piano strings are simple in that they are uniform, tightly stretched, and nearly rigidly terminated. As a result, they are highly linear under normal playing conditions. The digital waveguide approach to string modeling therefore works very well for the individual piano strings. The non-negligible stiffness of piano strings poses an increase in the expense of the implementation, resulting in the need for an allpass filter in the ``string loop''. Allpass filters of order $6-20$ or more are required for good fidelity [47,346,384]. Another complicating factor is the non-negligible coupling between strings that are hit by the same hammer [554]. Less important, but audible, are the horizontal and longitudinal directions of vibration (the main vibration being in the vertical plane). There is also significant coupling among all the strings when the sustain pedal is down. To fully simulate the linear behavior of each string, it is necessary to couple at least three digital waveguides together corresponding to the main types of wave propagation in and along the string (two transverse and one longitudinal). Efficient coupling of waveguide string models is discussed in §4.12. In key ranges in which the hammer strikes three strings simultaneously, nine coupled waveguides are required per key for a complete simulation. This section will address the case in which only the vertical plane of vibration is simulated for each string, and only one string is implemented per key. In a high quality system, the correct number of strings should be implemented, since they are detuned and cause important beating and aftersound effects [554].

The soundboard and enclosure as a whole are simple in that they are largely linear, time-invariant components, but they are complex in that they are large. Large vibrating objects generally have many more resonant modes in the range of human hearing than do small objects. Also, waveguide propagation in the soundboard and enclosure is not confined to one dimension as it is in a string. That means a complete digital waveguide model of the piano would require two- and/or three-dimensional waveguide meshes [522] to model the resonating soundboard and piano enclosure. In sum, the sheer size of the piano and its soundboard lead to very expensive direct modeling techniques, even after accounting for the fundamental efficiency advantages of the digital waveguide approach. However, the commuted synthesis technique [471,524] bypasses this difficulty and allows simple ``sampling'' of the soundboard/enclosure impulse response into a read-only memory which is ``played'' into the string in a manner determined by the hammer-string collision.

While only the velocity is necessary to specify state of the hammer prior to hitting the string, the string collision is highly nonlinear [501]. The nonlinearity comes from the felt covering the hammer: As it compresses, it acts like a spring whose spring-constant is rapidly increasing. But for this nonlinearity, the entire instrument could be approximated by a linear model. The main difficulty with nonlinearity in this context is that it prevents use of the commuted synthesis technique at first sight. This is because commutativity of system elements is only possible in general for linear, time-invariant elements. However, §5.3 will describe how the piano hammer can be linearized at each striking velocity so that the commuted synthesis technique can be applied.