Triangle Wave - Harmonics

Harmonics

It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every (4nāˆ’1)th harmonic by āˆ’1 (or changing its phase by Ļ€), 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=0}^\infty (-1)^k \, \frac{ \sin \left( (2k+1)\omega t \right)}{(2k+1)^2} \\ & {} = \frac{8}{\pi^2} \left( \sin (\omega t)-{1 \over 9} \sin (3\omega t)+{1 \over 25} \sin (5\omega t) - \cdots \right) \end{align}
where is the angular frequency.

Read more about this topic:  Triangle Wave