# Triangle wave

A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

Triangle wave
A bandlimited triangle wave[1] pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).
General information
General definition${\displaystyle x(t)=4\left\vert t-\left\lfloor t+3/4\right\rfloor +1/4\right\vert -1}$
Fields of applicationElectronics, synthesizers
Domain, Codomain and Image
Domain${\displaystyle \mathbb {R} }$
Codomain${\displaystyle \left[-1,1\right]}$
Basic features
ParityOdd
Period1
Specific features
Root${\displaystyle \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} }$
DerivativeSquare wave
Fourier series${\displaystyle x(t)=-{\frac {8}{{\pi }^{2}}}\sum _{k=1}^{\infty }{\frac {{\left(-1\right)}^{k}}{\left(2k-1\right)^{2}}}\sin \left(2\pi \left(2k-1\right)t\right)}$

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).