Digital Sound & Music: Concepts, Applications, & Science, Chapter 6, last updated 6/25/2013
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What if you wanted a frequency of 390 Hz? Then the increment would be
, which is not an integer. In cases where the increment is not an integer, interpolation must
be used. For example, if you want to go an increment of 1.04 from index 1, that would take you
to index 2.04. Assuming that our wavetable is called table, you want a value equal to
[ ] [ ] [ ]
. This is a rough way to do interpolation. Cubic
spline interpolation can also be used as a better way of shaping the curve of the waveform. The
exercise associated with this section suggests that you experiment with table-lookup oscillators
in MATLAB.
An extension of the use of table-lookup oscillators is
wavetable synthesis. Wavetable synthesis was introduced in digital
synthesizers in the 1970s by Wolfgang Palm in Germany. This was
the era when the transition was being made from the analog to the
digital realm. Wavetable synthesis uses multiple wavetables,
combining them with additive synthesis and crossfading and
shaping them with modulators, filters, and amplitude envelopes.
The wavetables don't necessarily have to represent simple
sinusoidals but can be more complex waveforms. Wavetable
synthesis was innovative in the 1970s in allowing for the creation of
sounds not realizable with by solely analog means. This synthesis method has now evolved to the
NWave-Waldorf synthesizer for the iPad.
Additive Synthesis 6.3.3.2
In Chapter 2, we introduced the concept of frequency components of complex waves. This is
one of the most fundamental concepts in audio processing, dating back to the groundbreaking
work of Jean-Baptiste Fourier in the early 1800s. Fourier was able to prove that any periodic
waveform is composed of an infinite sum of single-frequency waveforms of varying frequencies
and amplitudes. The single-frequency waveforms that are summed to make the more complex
one are called the frequency components.
The implications of Fourier’s discovery are far reaching. It means that, theoretically, we
can build whatever complex sounds we want just by adding sine waves. This is the basis of
additive synthesis. We demonstrated how it worked in Chapter 2, illustrated by the production
of square, sawtooth, and triangle waveforms. Additive synthesis of each of these waveforms
begins with a sine wave of some fundamental frequency, f. As you recall, a square wave is
constructed from an infinite sum of odd-numbered harmonics of f of diminishing amplitude, as in
A sawtooth waveform can be constructed from an infinite sum of all harmonics of f of
diminishing amplitude, as in
( )
A triangle waveform can be constructed from an infinite sum of odd-numbered harmonics of f
that diminish in amplitude and vary in their sign, as in
Aside: The term
"wavetable" is sometimes
used to refer a memory
bank of samples used by
sound cards for MIDI sound
generation. This can be
misleading terminology, as
wavetable synthesis is a
different thing entirely.
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