of Microprocessors

22 MUSICAL ApPLICATIONS OF MICROPROCESSORS
As mentioned earlier, any natural sound waveform is really a combination
of sine waves originating from vibrating spring-mass systems. However,
in the 17th century, a French mathematician by the name ofJoseph Fourier
proved mathematically that any waveform, regardless of origin) is actually a
mixture of sine waves of different frequencies, amplitudes, and phases. Furthermore,
he showed that if the waveform repeats steadily, then the frequencies
of the component sine waves are restricted to being integer multiples of the
repetition frequency of the waveform. Thus, if the frequency of repetition is
100 Hz, then the component sine waves must have frequencies of 100 Hz,
200 Hz, 300 Hz, etc., up to infinity, although any components above 20
kHz will not contribute to the sound, since they are inaudible. Of course,
some of them may have zero amplitude, but in general it can be assumed that
all of them exist in the audible frequency range.
These component sine waves are called overtones or harmonics, with the
latter term preferred. The component having the same frequency as the
overall waveshape is termed thefundamental frequency, which is also the first
harmonic. The component having a frequency twice the fundamental is
termed the first overtone or second harmonic. The third harmonic has a
frequency three times the fundamental and so forth. Since the frequencies of
the component waves are fixed, each one can be characterized by giving its
amplitude and its phase angle either as an absolute quantity or w.ith respect
to the fundamental.
Figure 1-6 shows how harmonic sine waves can be combined together
to produce a waveshape that is about as far as one can get from a curvy wave.
Combining two waveforms really means adding them point by point as
shown to get the combined result. A squared-off waveform such as this
actually has an infinite number of harmonics with nonzero amplitudes. A
practical synthesis of the waveform from harmonic components has to stop
somewhere, of course, and leave aU of the higher-frequency harmonics with a
zero amplitude.
As can be seen, each additional harmonic gives a closer approximation
to the desired rectangular-shaped wave. With the first 32 harmonics represented,
the apprOXimation is getting quite close with steeply rising sides and
reasonably flat top; however, there is still a significant amount of overshoot
and ringing. These imperfections ate mainly due to using the set ofharmonic
amplitudes designed for an infinite series and stopping the series abruptly at
the 32nd harmonic. A modification of the amplitudes taking into account
the fact that no harmonics above 32 are allowed produces a visibly superior
rendition of the desired shape.
The significance of Fourier's theorem can be realized by noting that all
of the acoustically important aspects of the shape of a waveform can be
specified with a comparatively small number of parameters. For example, a
1,OOO~Hz waveshape, no matter how complicated, can be specified by 20
amplitudes and 20 phase angles corresponding to the 20 audible harmonics.