Subtractive synthesis starts out with a harmonically rich sound and 'subtracts' some of the harmonics, whereas additive synthesis does almost the exact opposite. It adds together sine waves of different frequencies to produce the final sound. Because large numbers of parameters need to be controlled simultaneously, the user interface is usually much more complex than that of a subtractive synthesizer.
3.4.1 Theory: additive synthesis
Additive synthesis is based on the work produced by Fourier, a French mathematician from the nineteenth century. In 1807, Fourier showed that the shape of any repetitive waveform could be reproduced by adding together simpler waveforms, or alternatively, that any periodic waveform could be described by specifying the frequency and amplitude of a series of sine waves.
The restriction that the waveshape must repeat is imposed to keep the mathematics manageable. Without the restriction it is still possible to convert any waveform into a series of sine waves, but since the waveform is not constant, the sine waves that make it up are not constant either.
One useful analogy is to think of trying to describe writing to someone, who has never seen it, over the telephone. You might start out by describing how the words are broken up into letters and these letters are made up out of lines, dots and curves. This works perfectly well as long as the words you might try to describe stay fixed, but if they change, then you would have to keep updating your description. You could still convey the information about the shape of the letters that make up the words, but you would have to provide lots more detailed description as the letters change.
The simplest example of synthesizing a waveform using Fourier synthesis is a sine wave. A sine wave is made up of just one sine wave, at the same frequency! In terms of harmonics, a sine wave contains just one frequency component, at the repetition rate of the fundamental.
More complicated waveshapes can be made by adding additional sine waves. The simplest method involves using simple integer multiples of the fundamental frequency. So, if the fundamental is denoted by f, then the additional frequencies will be 2f, 3f, 4f, etc. These are the frequencies that occur in some of the basic waveshapes - sawtooth, square, etc., and are known as harmonics. Because the numbering of the harmonics is based around their position above the fundamental or first harmonic, with a frequency of f, then the second harmonic has a frequency of 2f. The second harmonic is also sometimes called the first overtone (Table 3.4.1).
Table 3.4.1 Harmonics, Frequencies and Overtones