Sound Modelling
105
Yamaha DX7
carrier/modulator frequency
ratio
nonlinear distortion
NLD
waveshaping
-20
0
20
40
60
80
0
2000
4000
6000
8000
10000
[dB]
frequency [Hz]
magnitude spectrum
Figure 10: Spectrum of a sound produced by amplitude/frequency modulation
as in (27).
Discussion
The synthesis by frequency modulation was very popular in the eighties, es-
pecially because it was implemented in the most successful synthesizer of all
times: the Yamaha DX7. At that time, obtaining complex time-varying spectra
with a few multiplies and adds was a major achievement. There was a theory
that allowed to predict the spectra given the parameter, and the bandwidth of
FM sounds could be controlled smoothly by means of the modulation index.
However, it proved difficult to obtain FM patches starting from the analysis of
real sounds, so that the most successful reproductions have been based on intu-
ition and multiple trials. Some of the parameters, such as the carrier/modulator
frequency ratio) are too critical and non-intuitive. Namely, little changes in a
modulator frequency produce dramatic changes in timbre. The modulation in-
dex itself, despite displaying a global intuitive behavior, is related to each single
partial amplitude by means of exotic functions that have no relationship with
the human hearing system.
5.3.2
Nonlinear distortion
The sound synthesis by nonlinear distortion (NLD), or waveshaping [8], is con-
ceptually very simple: the oscillator output is used as argument of a nonlinear
function. In the discrete-time digital domain, the nonlinear function is stored in
a table, and the oscillator output is used as index to access the table.
The interesting thing about NLD is that there is a theory that allows to
design the distorting table given certain specifications of the desired spectrum.
If the oscillator is sinusoidal, we can formulate NLD as
x(n) = A cos (
0
n)
(28)
y(n) = F (x(n)) .
(29)
For the nonlinear function, we use Chebyshev polynomials [1]. The degree-n