On Synthesizers: Waveforms, Spectra, And Oscillators, Part II

Synth pioneer Bob Moog's original column for Keyboard explored issues affecting musicians and technology developers.
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Continuing our celebration of 40 years of Keyboard, we are presenting Bob Moog's orginal "On Synthesizers" columns in their entirety.

In last issue's column, I talked about two ways of describing a steady, pitched sound: by its waveform (graph of the vibration in time), or by its spectrum (a list of the strengths of the harmonics). Synthesizer oscillators produce simple waveforms whose spectra turn out to be musically useful.

The sawtooth is one of the more popular synthesizer waveforms. Here are the sawtooth waveform and spectrum. Three cycles of the waveform, and the first eight harmonics of the spectrum, are shown (check out the graphics in the scanned version of article).

The sawtooth wave rises linearly, then drops abruptly, once every cycle of vibration. This gives a spectrum in which all of the harmonics are present. The higher the harmonic number, the weaker is that particular harmonic; no one harmonic predominates. The sound of a sawtooth without further processing is bland and bright, vaguely similar to that of an amplified bowed string instrument.

Here is one example of a rectangular (or pulse) waveform.
Rectangular waves rise abruptly and stay level, then fall abruptly and stay level. The percentage of the time of a complete cycle, which is occupied by the up portion of the wave is called the width, or duty cycle. The above sketch shows a rectangular wave with a duty cycle of 1/4. Now, notice the spectrum. Every fourth harmonic is missing! In fact, a rectangular wave whose duty cycle is 1/N will have a spectrum where every Nth harmonic will be weak or absent, and the harmonics in between will be stronger. Rectangular waves sound generally bright. Those with very short duty cycles are fuzzy and reedy, while those with longer duty cycles (around 1/4 or 1/3) sound rich and vocal.

When the duty cycle of a rectangular wave is exactly 1/2 (the top part is the same width as the bottom part), it is called a square wave:

Here, every other harmonic is missing; only odd harmonics are present. The absence of even harmonics (octaves of the odd harmonics) gives the square wave a characteristically hollow, woody quality, like a flute or clarinet.

Another popular synthesizer waveform is the triangular:
The triangular wave rises linearly, then falls linearly at the same rate, once every cycle. Unlike either the sawtooth or the rectangular, the triangular has no abrupt jumps. Its spectrum contains only odd harmonics, and these harmonics are weak. In fact, 8/9 of the sound energy of a triangular wave is in the fundamental frequency, while all of the overtones account for only 1/9 of the sound energy. The triangular wave produces a mellow, muted sound, which is useful for flutelike or whistling tones, or to beef up brighter waveforms with additional fundamental pitch sound energy.

The four waveforms that I discussed, sawtooth, rectangular, square, and triangular, together with the sine wave that I mentioned in the last column, illustrate some general rules that you can apply to predict the sound of a simple waveform, once you know its shape. Here they are:
1.A waveform with one or more sharp edges will have a high harmonic content, and will sound bright.
2.A waveform with one or more flat spots will have a spectrum where certain harmonics are weak and other harmonics are strong. Such a waveform will sound bright and distinctive.
3.A waveform with sharp corners but no sharp jumps will have a small amount of harmonic content, and will not sound as bright as the sawtooth.
4.A symmetrical waveform (one whose bottom half is the mirror image of the top half) will have no even harmonics, and will therefore sound hollow and woody.

Of course, synthesizer waveforms travel through many circuits before we ever hear them. The most important of these for determining tone color are the filters, which are the subject of my next column.

See also:

On Synthesizers: Waveforms, Spectra and Oscillators, Part I

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