of Microprocessors

330 MUSICAL ApPLICATIONS OF MICROPROCESSORS
Figure 10-7 shows rhe effect of sampling a low-frequency square wave
with such a degraded SAH. The output is a series of rising and falling
arpeggios, but the pitch intervals start large and then decrease after each
direction reversal. Altering the series resistor changes the step size and rate of
interval decrease considerably. The effect on sampled white noise is also
interesting. As the resistor is increased, the random output changes from
total disorder toward a more correlated result. Unfortunately, the standard
deviation also decreases, which would have to be counteracted with a VCA in
a practical application. Obviously, voltage control of slew limiting would be
a useful SAH feature.
Controlling Randomness
As was mentioned, a slew-limited SAH module is capable of imparting
a degree of order in random sequences. Actually, it is possible to get exactly
the same results by passing the signal to be sampled through a single-pole
R-C low-pass filter first. If white noise is to be sampled, then this amounts to
filtering the noise. The slew-limited SAH module is actually a discrete-time
low-pass filter, which is the first step toward a digital filter!
A sequence of random numbers is actually sampled white noise. Thus,
one can easily write a program to simulate analog sampling of white noise by
using the RND (random number) function available in the BASIC programming
language. If a synthesizer is interfaced to the computer, then
random numbers can.be fed to an oscillator to produce the same kinds of note
sequences available with analog setups. One point to be aware of is that most
random number generators have a uniform probability distribution, generally
between 0.0 and 1.0. A good approximation to a gaussian distribution may
be had by adding up 12 random numbers (distributed uniformly between 0.0
and 1.0) and subtracting 6.0 from the sum. The mean of the result will be 0
and the standard deviation will be 1.0.
The name "stochastic music" refers to music (melodies) that orginates
from sequences of random numbers. It should be apparent that raw random
numbers, regardless of the probability distribution function, would create
rather uninteresting music. Each note is an independent entity, with no
relation to what came before and no influence on what follows.
A very simple algorithm can be applied to a random sequence, however,
to produce a highly correlated sequence that might be more interesting.
The basic idea is to use random numbers to determine the direction and
magnitude of pitch movement rather than the pitches themselves. As a simple
example, let's say that the pitches are to be notes on the chromatic equaltempered
scale and that the maximum allowable interval between successive
notes is an octave. Thus, a sequence of random integers falling between -12
and +12 inclusive is needed. The BASIC expression INT(25*RND(I»
-12 will produce such a sequence. To produce a note sequence, numbers