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.