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).

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:

${\begin{aligned}x_{\mathrm {triangle} }(t)&{}={\frac {8}{\pi ^{2}}}\sum _{k=0}^{\infty }(-1)^{k}\,{\frac {\sin \left(2\pi (2k+1)ft\right)}{(2k+1)^{2}}}\\&{}={\frac {8}{\pi ^{2}}}\left(\sin(2\pi ft)-{1 \over 9}\sin(6\pi ft)+{1 \over 25}\sin(10\pi ft)+\cdots \right)\end{aligned}}$

Animation of the additive synthesis of a triangle wave with an increasing number of harmonics

Another definition of the triangle wave is:

$x(t)=(t-a\left\lfloor {\frac {t}{a}}-{\frac {1}{2}}\right\rfloor )(-1)^{\left\lfloor {\frac {t}{a}}-{\frac {1}{2}}\right\rfloor }$

The triangle wave can also be expressed as the integral of the square wave:
$\int \operatorname {sgn}(\sin(x))\,dx\,$

See also

Sine, square, triangle, and sawtooth waveforms

• Triangle function
• Sine wave
• Sawtooth wave
• Square wave
• Waves
• Sound