of Microprocessors

DIRECT COMPUTER SYNTHESIS METHODS 119
even a simple sample computation. Unfortunately, implementation of the
technique is complicated somewhat by the fact that, for most ftequencies,
there is a nonintegral number of samples per cycle of the waveform. The
problem is solved by tabulating the waveform at a frequency for which there
is an integral number of cycles per sample. When table lookup is tequited at
a diffetent frequency, the table lookup routine, for example, might be
requested to "find the 8. 763rd entry" in the table. Interpolation between the
8th and 9th sample may be performed or, if the table is large enough, the
nearest tabulated entry, which would be the 9th, would be used. There is
thus a tradeoff between table size, lookup time, and sound quality, since
errors result in background noise being added to the waveform. Since the
costs of memory are declining faster than the speed of memory or computers
is increasing, table lookup without interpolation is a frequently used
technique. In fact, completely table-driven synthesis programs are possible.
Note that the table method can also be applied to envelopes and any other
curve that is likely to be used more than once. Table Lookup methods wilL be
studied in detail in Chapter 13 and again in Chapter 20.
Hardware Aids
Another way to increase computation speed is to connect some
specialized hardware to the computer that is designed to be especially efficient
in the calculations typically needed for music. For the microcomputer
user, a hardware multiplier wouLd be of great benefit if his microprocessor
chip does not have multiply and divide built in. The difference in processing
speed can easily be 10: lover muLtiplication in software and might resuLt in
a three- to fivefold increase in overall music program speed. The multiplier
could be expanded into a muLtiplier-summer, which would speed up sumof-products
calculations such as a Fourier series. A full array processor normally
performs high-speed operations on matrices and vectors such as multiplication,
addition, and even inversion. In sound synthesis, it could be used
to compute severaL samples of several sounds at once and thus greatly speed
things up. A device known as a "fast Fourier transform processor" is specifically
optimized for computing Fourier transforms. Although normally used
to rapidly compute the spectrum of a waveform for sound and vibration
analysis, most can also compute a waveform given the spectrum. Although
there are limitations in using a Fourier transform processor for generating
changing waveforms, its availability can be a great asset.
Digital Sound Modification
Nearly all of the sound modification techniques described in Chapter 2
can be easily implemented in a direct computer synthesis system. Spectrum