#### Computational Methods

Since finding the intersection of and requires an expensive iterative algorithm with variable convergence times, it is not well suited for real-time operation. In this section, fast algorithms based on precomputed nonlinearities are described. Let denote half-pressure , i.e., and . Then (9.43) becomes (10.45)

Subtracting this equation from gives (10.46)

The last expression above can be used to precompute as a function of . Denoting this newly defined function as (10.47)

(9.45) becomes (10.48)

This is the form chosen for implementation in Fig. 9.39 . The control variable is mouth half-pressure , and is computed from the incoming bore pressure using only a single subtraction. The table is indexed by , and the result of the lookup is then multiplied by . Finally, the result of the multiplication is subtracted from . The cost of the reed simulation is only two subtractions, one multiplication, and one table lookup per sample. Because the table contains a coefficient rather than a signal value, it can be more heavily quantized both in address space and word length than a direct lookup of a signal value such as or the like. A direct signal lookup, though requiring much higher resolution, would eliminate the multiplication associated with the scattering coefficient. For example, if and are 16-bit signal samples, the table would contain on the order of 64K 16-bit samples. Clearly, some compression of this table would be desirable. Since is smoothly varying, significant compression is in fact possible. However, because the table is directly in the signal path, comparatively little compression can be done while maintaining full audio quality (such as 16-bit accuracy). In the field of computer music, it is customary to use simple piecewise linear functions for functions other than signals at the audio sampling rate, e.g., for amplitude envelopes, FM-index functions, and so on [382,380]. Along these lines, good initial results were obtained  using the simplified qualitatively chosen reed table (10.49)

depicted in Fig. 9.42 for . The corner point is the smallest pressure difference giving reed closure.10.19 Embouchure and reed stiffness correspond to the choice of offset and slope . Brighter tones are obtained by increasing the curvature of the function as the reed begins to open; for example, one can use for increasing . Another variation is to replace the table-lookup contents by a piecewise polynomial approximation. While less general, good results have been obtained in practice [89,91,92]. For example, one of the SynthBuilder  clarinet patches employs this technique using a cubic polynomial. An intermediate approach between table lookups and polynomial approximations is to use interpolated table lookups. Typically, linear interpolation is used, but higher order polynomial interpolation can also be considered (see Chapter 4).
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Scattering-Theoretic Formulation
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