3.4.2 Harmonic synthesis
So far, additive synthesis seems to be based around producing a specific waveform from a series of sine waves. In practice, the 'shape' of a waveform is not a good guide to its harmonic content, since minor changes to the shape can produce large changes in the harmonic content.
Conversely, simple changes of phase for the harmonics can produce major changes in the shape of the waveform. In fact, although the human ear is mainly concerned with the harmonic content, the relative phase of the harmonics can be very important at low frequencies. For frequencies above 440 Hz, you can change the phase of a harmonic and thus alter the resulting shape of the waveform, but the basic timbre will sound the same. Control over phase is thus useful under some circumstances and is found in some additive synthesizers.
The harmonic content of waveshapes is a useful starting point for examining this relationship between shape and perception. Mathematically and harmonically, the 'simplest' waveshape is the sine wave. Sine waves sound clean and pure, and perhaps even a little bit boring. Adding in small amounts of oddnumbered harmonics produces a triangular waveshape, which has enough harmonic content to stop it sounding quite as pure as the sine wave (Figure 3.4.1).
FIGURE 3.4.1 (i) A triangle waveform constructed from six sine wave harmonics is very different from a sine wave, even though the fundamental is by far the strongest component. (ii) A combination of equal amounts of the first 12 harmonics produces a waveform which looks (and sounds) like a type of pulse waveshape.
A square wave contains only odd harmonics. It has a characteristic 'hollow' sound, and the absence of the second harmonic is particularly noticeable if a square wave is compared with a sawtooth wave (Figure 3.4.2).
FIGURE 3.4.2 (i) A square waveform constructed from six sine wave harmonics has a close approximation to the ideal waveshape. (ii) Changing the phase of the third harmonic radically alters the shape of the waveform.
A square wave that has been produced with a phase change in the second harmonic no longer looks like a 'square' wave, and yet the harmonic content is the same (Figure 3.4.3).
A sawtooth wave contains both odd and even harmonics. It sounds bright, although many pulse and 'super-sawtooth' waveshapes can contain greater levels of harmonics. Again, a sawtooth wave with a phase change in the second harmonic does not look like a sawtooth, although it still sounds like one to the ear (Figure 3.4.3).
FIGURE 3.4.3 (i) A sawtooth waveform constructed from 12 sine wave harmonics has a close approximation to the ideal waveshape. (ii) Changing the phase of the second harmonic radically alters the shape of the waveform.
Pulse waves contain more and more harmonics as the pulse width narrows (or widens) from square. A 10% pulse has the same spectrum as a 90% pulse and it also sounds the same to the ear. One special case is the square wave, where the even harmonics are missing completely. Pulse widths of anything other than 50% include the second harmonic, and this can usually be clearly heard as the pulse width is varied away from the 50% value.
Finally, there is the 'even harmonic' wave. If a sawtooth contains both odd and even harmonics and a square wave contains just the odd harmonics, then what does a wave containing just the even harmonics look like? Actually, it is just another square wave, but one octave higher in pitch, and with a fundamental frequency of 2f!
In practice, adding together sine waves produces waveforms that have some of the characteristics of the mathematically perfect ideal waveforms, but not all. Producing square edges on a square wave would require large numbers of harmonics – an infinite number for a 'perfect' square wave. Using just a few harmonics can produce waveforms that have enough of the harmonic content to produce the correct type of timbre, even though the shape of the waveform may not be exactly as expected.