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:
Note that its peak amplitude is exactly π2 / 8.