Triangle wave

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A triangle wave is a non-sinusoidal waveform named for its triangular shape.

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

Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse), and so its sound is smoother than a square wave and is nearer to that of a sine wave.

One simple definition of a triangle wave is


\begin{align}
x_\mathrm{triangle}(t)  = arcsin(sin(t))
\end{align}


It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every (4n−1)th harmonic by −1 (or changing its phase by π), and rolling off the harmonics by the inverse square of their relative frequency to the fundamental.

This infinite Fourier series converges to the triangle wave:


\begin{align}
x_\mathrm{triangle}(t) & {} = \frac {8}{\pi^2} \sum_{k=1}^\infty \sin \left(\frac {k\pi}{2}\right)\frac{ \sin (2\pi kft)}{k^2} \\
& {} = \frac{8}{\pi^2} \left( \sin (2\pi ft)-{1 \over 9} \sin (6 \pi ft)+{1 \over 25} \sin (10 \pi ft) + \cdots \right)
\end{align}
Animation of the additive synthesis of a triangle wave with an increasing number of harmonics
Animation of the additive synthesis of a triangle wave with an increasing number of harmonics

Triangle wave sound sample

5 seconds of triangle wave at 1 kHz
Problems listening to the file? See media help.

[edit] See also

Sine, square, triangle, and sawtooth waveforms
Sine, square, triangle, and sawtooth waveforms