of Microprocessors

418 MUSICAL ApPLICATIONS OF MICROPROCESSORS
One of the big problems with the generation of tones of arbitrary
frequency and waveform is the avoidance of alias distortion. As it turns out,
it is quite possible, sometimes unavoidable, to digitally generate a tone
having frequency components above one-half the sample rate. When this
occurs, the higher frequencies are reflected down as lower frequencies that
then sail right through the DAC's low-pass filt~r, no matter how sophisticated.
The audible effect is usually harsh distortion or excessive noise, and it
occurs even with single tones. The only way to positively avoid this source of
alias distortion is to avoid generating significant amounts of excessively high
frequencies.
Direct Waveform Computation
One of the easiest ways to generate tones digitally is to simulate the
operation of an analog voltage-controlled oscillator. In an actual synthesis
program, the "oscillator" would probably be a subroutine that accepts some
arguments (control "voltages") such as one or more frequency parameters and
returr~s one or more results representing samples on one or more output
waveforms. The subroutine would probably also require some carryover storage
from one sample to the next.
The analog VCO that was described in Chapter 6 consisted of three
major parts: control input acquisition, the sawtooth oscillator proper, and
waveshaping circuits. A digital equivalent will require the same setup. Concentrating
on the oscillator part, it was seen to consist of a current source, an
integrator, a comparator, and a discharge (reset) circuit. Fortunately, the
digital equivalents to all of these are exceptionally simple and, most important,
require very little computation time.
Digital Sawtooth Oscillator
An integrator is an accumulator, somewhat like a bucket that integrates
the flow rate of water entering it. A computer accumulator is also an
integrator;. it accumulates discrete-sized "pieces" represented by numbers
added to it via the "add" instruction. When a bucket overflows, it remains
full and only the excess water is lost. When an accumulator overflows, its
entire contents are dumped and integration starts anew. Thus, it should be
apparent that if one considers a number that is repeatedly added to a computer's
accumulator to be a "current," the accumulator contents to be the
integrator's output, and the overflow phenomenon of binary arithmetic to be
a combination comparator/discharge circuit, one has a sawtooth oscillator
just like the analog one.
This can be clarified by examining the program segment below, written
in 6502 assembly language: