Triangle

Triangle wave

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).

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

Triangle wave sound sample

5 seconds of triangle wave at 1 kHz

Problems listening to the file? See media help.

It is also possible to approximate a triangle wave with abs() and floor():

$\begin{align} x_\mathrm{triangle}(t) = 2 * \mbox{abs}(t - 2 * \mbox{Floor}(t/2) - 1) - 1 \end{align}$

Or with modulo:

$\begin{align} x_\mathrm{triangle}(t) = 4 (t%1)^2+2(t%2)-4(t%1) (t%2)-1 \end{align}$

See also