Triangle wave

A triangle wave is a waveform that can be obtained by subtractive synthesis by integrating (lowpass filtering) a square wave.

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, and so its sound is smoother than a square wave and is nearer to that of a sine wave.

It is possible to approximate a triangle wave by additive synthesis, by adding odd harmonics of the fundamental, rolling them off with frequency faster than with a square wave. The infinite series will converge to a triangle wave.

This infinite Fourier series converges to the triangle wave:

$x_{triangle}(t) = \sum_{k=0}^\infty \frac{\cos (2k+1)t}{(2k+1)^2}$

Note that its peak amplitude is exactly π² / 8.