of Microprocessors

DIGITAL TONE GENERATION TECHNIQUES 433
fixed-point rather than floating-point arithmetic. Only the sin function poses
any computational difficulty and that can bypassed through use of a sine
table with a length equal to the waveform table being filled. Chapter 18 will
describe the use of integer and fractional arithmetic to maximize the speed of
digital synthesis computation.
Dynamic Timbre Variation
In a voltage-controlled synthesizer, the tone generators usually put out
an unchanging waveform that is dynamically altered by processing modules.
Although the same can be done in direct synthesis with digital filters, the
table method of tone generation lends itself well to types of timbre variation
not easily accomplished with filters.
One technique that is quite practical involves two waveform tables and
interpolation between them. The idea is to start with the waveform in Table
A and then gradually shift to a different waveform in Table B. The arithmetic
is actually quite simple. First, a mixture variable ranging between 0 and 1.0
is defined, which will be called M. The contribution of waveform B to the
resultant waveform is equal to M and the contribution of A is 1.0 - M.
The actual resultant samples are computed by evaluating
Sr=(l-M)Sa+MSb, where Sr is the result sample, Sa is a sample from the
A table, Sb is the B-table sample, and M is as before. Thus, as M changes
from 0 to 1.0, the mixtute changes in direct proportion.
In actual use, M should be updated frequently enough so that it
changes very little between updates. Also the same table pointer should be
used on both tables to insure that they are in phase. If these rules are
followed, the transition is glitch- and noise-free even for very fast transitions.
The technique is not limited to two tables either. One could go through a
whole sequence of tables in order to precisely control a very complex tonal
evolution.
Speaking of evolution, it would be desirable to know exactly what the
spectrum of the tone does during the transition. If each harmonic has the
same phase in the two tables, then the amplitude evolution of that harmonic
will make a smooth, monotonic transition from its amplitude in tone A to its
new amplitude in tone B.
Things get quite interesting, however, if they are not in phase. Depending
on the phase and amplitude differences between the two tables, any
number of things can happen as shown in Fig. 13-8. The graph shows
amplitude and phase variations of an arbitrary harmonic during a linear
transition from wave A, where this harmonic has an amplitude of 0.4 units,
to wave B, where its amplitude is 0.9 units. Its phase in wave A is taken as a
zero reference and its phase in wave B is the parameter for the curves. When
the phase of this harmonic in wave B is also zero, the amplitude transition is
linear. It becomes progressively nonlinear and even dips momentarily along
its rise as the phase difference approaches 180 0 •