Sound Synthesis Theory: Oscillators and Wavetables: An Open Learning Guide

An oscillator is a repeating waveform with a fundamental frequency and peak amplitude and it forms the basis of most popular synthesis techniques today. Aside from the frequency or pitch of the oscillator and its amplitude, one of the most important features is the shape of its waveform. The time-domain waveforms in Fig. 5.1 show the four most commonly used oscillator waveforms. Although it is possible to use all kinds of unique shapes, these four each serve a range of functions that are suited to a range of different synthesis techniques; ranging from the smooth, plain sound of a sine wave, to the harmonically rich buzz of a sawtooth wave.

Oscillators are generally controlled by a keyboard synthesizer or MIDI protocol device. A key press will result in a MIDI note value which will be converted to a frequency value (Hz) that the oscillator will accept as its input, and the waveform period will repeat accordingly to the specified frequency. From here, the sound can be processed or manipulated in a variety of ways in the synthesizer or program to enrich or modify the sound further.

Note that the below diagrams have been incorrectly rendered. The x axis should be placed lower, every single signal should pass through y=0 at π radians.

As mentioned previously, the sine wave can be considered the most fundamental building block of sound. The best way to generate an oscillator which produces this waveform is to make use of an inbuilt library or function in the system concerned. Many programming languages have standard mathematics libraries with many of the trigonometric functions represented. A cycle of a sine wave is 2π radians long and has a peak amplitude of +/−1, as shown in Fig. 5.2. In a digital system, the generated waves will be a series of equally-spaced values at the sample rate.

With a sample rate of 44100 cycles per second, and a required cycle length of 1 second, it will take 44100 samples to get from 0 to 2π. In other words, we can determine the steps per cycle S from cycle length T:

Where f, in the second result, is the same result in terms of frequency. The importance of this is that it is possible to expand it into an algorithm that will be suitable for generating a sinusoidal wave of a user specified frequency and amplitude- effectively the simplest synthesizer possible! A sinusoidal wave can be generated by repeatedly incrementing a phase value by an amount required to reach a desired number of 2π length cycles a second, at the sample rate. This value can be passed to a sine function to create the output value, between the user specified peak amplitude.

The most important thing to note about this algorithm is that when the phase value has exceeded 2π it will subtract by one whole period. This is to ensure that the function "wraps" round to the correct position instead of going straight back to 0; if a phase increment oversteps 2π and resets to 0, undesirable discontinuities would occur, causing harmonic distortion in the oscillatory sound.

The square wave cannot be generated from a mathematical function library so easily but once again the algorithm is particularly straightforward since it is constructed from straight line segments. Unlike the sine wave, square waves have many harmonics above their fundamental frequency, and have a much brighter, sharper timbre. After examining a number of different waveforms it will start to become apparent that waveforms with steep edges and/or abrupt changes and discontinuities are usually harmonically rich.

(Note that the following square, sawtooth, and triangle functions are "naive"; they are equivalent to sampling the ideal mathematical functions without first bandlimiting them. In other words, all of the harmonics above the Nyquist frequency will be aliased back into the audible range. This is most obvious when sweeping one of these waveforms into the high frequencies. The aliased harmonics will move up and down the frequency spectrum, making "radio tuning" sounds in the background. A better method to produce waveforms for audio would be additive synthesis, or something like MinBLEPs. A properly-bandlimited waveform will have "jaggies" as you approach the discontinuities instead of piecewise straight lines.)

The square wave is constructed in a very similar fashion to the sine wave, and we use the same approach by cycling through a pattern with a phase variable, and resetting once we exceed 2π radians.

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Text adapted from Sound Synthesis Theory/Oscillators and Wavetables by Wikibooks contributors under CC BY-SA 4.0. Paragraphs have been selected and abridged, and formatting adjusted. This adapted text is shared under the same licence. Contributor history.