Harmonic content of waveforms
The ordering of waveforms on some early analogue synthesizers was not random. The waveforms are deliberately arranged so that the harmonic content increases as the rotary control is twisted.
One example: the Minimoog waveforms are arranged in the order of increasing harmonic content.
FIGURE 3.3.3 A block diagram of a typical VCO.
Arguably the simplest waveshape is the sine wave (Figure 3.3.4). This is a smooth, rounded waveform based on the mathematical sine function. A sine wave contains just one 'harmonic', the first or fundamental. This makes it somewhat unsuitable for subtractive synthesis since it has no harmonics to be filtered.
FIGURE 3.3.4 A sine waveform and harmonic spectrum and the same diagrams with actual frequencies shown.
A triangle waveshape has two linear slopes (Figure 3.3.5). It has small amounts of odd-numbered harmonics, which give it enough harmonic content for a filter to work on.
FIGURE 3.3.5 A triangle waveform and spectrum.
A square wave contains only odd harmonics (Figure 3.3.6). It has a distinctive 'hollow' sound and a very synthetic feel.
FIGURE 3.3.6 A square waveform and spectrum, with a typical clarinet spectrum for comparison.
A sawtooth wave contains both odd and even harmonics (Figure 3.3.7). It sounds bright, although many pulse waves can actually have more harmonic content. 'Super-sawtooth' waveshapes replace the linear slope with exponential slopes, as well as gapped sawtooths: these can contain greater levels of the upper harmonics than the basic sawtooth.
FIGURE 3.3.7 A sawtooth waveform and spectrum, with the spectrum also shown on a vertical decibel scale.
Depending on the ratio between the two parts (known as the mark–space ratio, shape, duty cycle or symmetry), pulse waveforms (Figure 3.3.8) can contain both odd and even harmonics, although not all of the harmonics are always present. The overall harmonic content of pulse waves increases as the pulse width narrows, although if a pulse gets too narrow, it can completely disappear (the depth of PWM needs to be carefully adjusted to prevent this).
FIGURE 3.3.8 A pulse wave and spectrum. The relative levels of the harmonics depend on the width of the pulse.
A special case of a pulse waveshape is the 50:50 equal ratio square wave, where the even harmonics are not present. Pulse width modulated pulse waveforms are known as PWM waveforms and their harmonic content changes as the width of the pulse varies. PWM waveforms are normally controlled with LFO or an envelope, so that the pulse width changes with time. The audible effect when a PWM waveform is cyclically changed by an LFO is similar to two oscillators beating together.
It is possible to adjust the pulse width to give a square by ear: listening to the fundamental, the pulse width is adjusted until the note one octave up fades away. This note is the second harmonic and is thus not present in a square waveform. See also Figure 3.3.8.
All of the waveshapes and harmonic contents shown previously are idealized. In the real world the edges are not as sharp, the shapes are not so linear and the spectra are not as mathematically precise. Figure 3.3.9 shows a more realistic spectrum with dotted lines. This is a result of the filtering process used in producing the spectrum display and does not mean that there are extra frequencies present.
FIGURE 3.3.9 Analogue waveshaping allows the conversion of one waveform shape into others. In this example the sawtooth is the source waveform, although others are possible.
Although the waveshapes are based on mathematical functions, this does not always mean that they are all produced directly from mathematical formulas expressed in analogue electronics. For example, the 'sine' wave output on many VCOs is produced by shaping a triangle wave through a non-linear amplifier which rounds off the top of the triangle so that it looks like a true sine wave (Figure 3.3.6). The resulting waveform resembles a sine wave, although it will have some additional harmonics – but for the purposes of subtractive synthesis, it is perfectly adequate. Section 3.4 on additive synthesis shows what real-world waveforms look like when they are constructed from simpler waveforms, rather than the perfect cases shown earlier.
There are two major modifiers for audio signals in analogue synthesizers: filters and amplifiers. Filtering is used to change the harmonic content or timbre of the sound, whilst amplification is used to change the volume or 'shape' of the sound. Both types of modifiers are typically controlled by EGs, which produce complex CVs that change with time.
Effects such as reverb and chorus are not normally included as 'modifiers' in analogue synthesizers, although there are some notable exceptions: For instance, the EMS (Electronic Music Studios) VCS-3 has a built-in spring-line reverb unit.