Sawtooth wave

The sawtooth wave (or saw wave) is a kind of basic non-sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw.

Related Topics:
Non-sinusoidal - Waveform - Saw

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A bandlimited sawtooth wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A2).

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The piecewise linear function y = x - floor(x) is an example of a sawtooth wave with period 1.

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A sawtooth wave's sound is harsh and clear and its spectrum contains both even and odd harmonics of the fundamental frequency. Because it contains all the integer harmonics, it is one of the best waveforms to use for constructing other sounds, particularly strings, using subtractive synthesis.

Related Topics:
Harmonic - Fundamental frequency

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A sawtooth can be constructed using additive synthesis. The infinite Fourier series

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:x_{sawtooth}(t) = rac {2}{pi}sum_{k=1}^{infin} rac {sin kt}{k}

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converges to a sawtooth wave. In digital synthesis, the series is only summed over n such that the highest harmonic, Nmax, is less than the Nyquist frequency (half the sampling Frequency). This summation can generally be more efficiently calculated using the Fast Fourier transform. If the waveform is digitally created directly in the time domain using a non-bandlimited form, such as y = x - floor(x), infinite harmonics are sampled and the resulting tone contains aliasing distortion.

Related Topics:
Nyquist frequency - Fast Fourier transform - Bandlimited - Aliasing

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An audio demonstration of a sawtooth played at 440 Hz (A4) and 880 Hz (A5) and 1760 Hz (A6) is available below. Both bandlimited (non-aliased) and aliased tones are presented.

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