Sound Modelling
101
frequency modulation
FM
carrier frequency
modulation frequency
modulation index
phase modulation
instantaneous frequency
For composers, this means a unification of compositional metaphors on different
scales and, as a consequence, the control over a time continuum ranging from
the milliseconds to the tens of seconds. There are psychoacoustic effects that can
be easily experimented by using this algorithm, for example crumbling effects
and waveform fusions, which have the corresponding counterpart in the effects
of separation and fusion of tones.
5.3
Nonlinear models
5.3.1
Frequency and phase modulation
The most popular non-linear synthesis technique is certainly frequency modula-
tion (FM). In electrical communications, FM has been used for decades, but its
use as a sound synthesis algorithm in the discrete-time domain is due to John
Chowning [23]. Essentially, Chowning was doing experiments on different ex-
tents of vibrato applied to simple oscillators, when he realized that fast vibrato
rates produce dramatic timbral changes. Therefore, modulating the frequency
of an oscillator was enough to obtain complex audio spectra.
Chowning's FM model is:
x(n) = A sin (
c
n + I sin (
m
n)) = A sin (
c
n + (n)) ,
(16)
where
c
is called the carrier frequency,
m
is called the modulation frequency,
and I is the modulation index. Strictly speaking, equation (16) represents a
phase modulation because it is the instantaneous phase that is driven by the
modulator. However, when both the modulator and the carrier are sinusoidal,
there is no substantial difference between phase modulation and frequency mod-
ulation. The instantaneous frequency of (16) is
(n) =
c
- I
m
cos (
m
n) ,
(17)
or, in Hertz,
f (n) = f
c
- If
m
cos (2f
m
n) .
(18)
Figure 8 shows a pd patch implementing the simple FM algorithm. The
modulation frequency is used to control an oscillator directly, while the carrier
frequency controls a phasor~ unit generator. This block generates the cyclical
phase ramp that, when given as index of a cosinusoidal table, produces the same
result as the osc unit generator. However, this decomposition of the oscillator
into two parts (i.e., the phase generation and the table read) allows to sum the
output coming from the modulator directly to the phase of the carrier.
Given the carrier and modulation frequencies, and the modulation index, it
is possible to predict the distribution of components in the frequency spectrum
of the resulting sound. This analysis is based on the trigonometric identity [1]
x(n) = A sin (
c
n + I sin (
m
n))
= A
J
0
(I) sin (
c
n)
carrier
+
(19)