Here is a photo of my experimental electronic implementation of a "physical modeling" clarinet. The circuitry produces clarinet-like oscillations in response to a control voltage that is analogous to blowing pressure. It also can be configured to demonstrate the "period-doubling route to chaos", a characteristic of a large class of nonlinear mechanical systems. Circuit details are discussed in the "Schematics" section.

Background

What's all this physical modeling stuff, anyhow?

All the objects around us in the universe, and the manner in which they interact with each other, are described by the laws of physics. But a complete physical description of even a very simple problem -- the trajectory of a baseball thrown through the air, for example -- is impractical, because of the great complexity of the motion and interactions of all the little particles involved. So physicists use sets of simplified assumptions and equations, known as models, to describe practical reality.

In the case of the baseball trajectory, a simple model ignores the finite size and composition of the ball, ignores air friction and ignores the turning of the earth under the ball while it is in flight. The latter effects would have to be included in a more exact model, for example if you wanted to hit a distant target with an artillery shell. However, you would probably never need a model that included the quarks and gluons that make up the nucleons in the individual atoms of the projectile.

Musical Instrument Physical Modeling

The study of musical-instrument physics is about as old as physics itself. A good summary of the current state of the art can be found in the book of Fletcher and Rossing, Ref. 1, which describes modeling studies done on a wide range of problems in musical-instrument physics.

The term "physical modeling", as used in the musical world to denote to a type of musical synthesis, actually refers to just one specific kind of physical model. As with the simplified model of a baseball trajectory mentioned above, this kind of model ignores many of the details of real instruments in order to concentrate on the important essentials of the problem, in this case the evolution in time of the oscillatory process. This class of time-dependent physical models was intensively developed around the late 1970's. An excellent summary can be found in the review article of McIntyre, Schumaker and Woodhouse (MSW), Ref. 2.

The essential features of the MSW-type models are shown in the figure below, as discussed in Ref. 2. A nonlinear acoustic-impedance element -- for example, the mouthpiece/reed combination of a woodwind instrument -- is driven by a steady supply of energy (blowing force) plus a time-delayed version of its own output (acoustic energy reflected from instrument's bell). Because of the negative differential impedance of the nonlinear element, sustained relaxation oscillations are obtained.

Because the MSW-type models require the performance of a convolution integral and the solution of an implicit mathematical equation at each point in time, real-time computer solutions were not feasible in the early 1980's. Further simplification and reworking of the models, led by groups at Stanford University (Ref. 3) and the Yamaha Corporation (Ref. 4), resulted in "digital delay-line" versions that could run in real time on small, inexpensive computers. These versions are the basis for commercial "physical modeling" synthesizers.

My approach to implementing a model clarinet follows the original MSW-type approach quite closely. The convolution integrals and the nonlinear mathematical equations are solved directly by analog circuitry, circumventing some difficulties encountered in digital implementations.

References

1. N. H. Fletcher and T.D. Rossing,The Physics of Musical Instruments, Springer (1998).
   
   
2. M. E. McIntyre, R. T. Schumaker and J. Woodhouse, "On the oscillations of musical instruments," J. Acoust. Soc. Am., vol. 74, p. 1325 (1983).
   
   
3. J. O. Smith III, "Musical applications of digital waveguides," CCRMA Report No. STAN-M-39 (1987); "Physical modeling using digital waveguides," Computer Music J., vol. 16, p. 74 (1992); "Digital signal processing using waveguide networks," U. S. Patent No. 4,984,276 (1991).
   
   
4. There are many Yamaha patents. Examples are: T. Kunimoto, "Musical tone waveform signal generating apparatus simulating a wind instrument," U. S. Patent No. 5,117,729 (1992); "Musical tone waveform signal generating apparatus," U. S. Patent No. 5,477,004 (1995).

Please see the "Up Close"and "Schematics" sections
for more details.