Definitions

# Triangle wave

A triangle wave is a non-sinusoidal waveform named for its triangular shape.

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

One simple definition of a triangle wave is


begin{align} x_mathrm{triangle}(t) = arcsin(sin(t)) end{align}

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 $pi$), 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=1}^infty sin left(frac {kpi}{2}right)frac{ sin (2pi kft)}{k^2} & {} = frac{8}{pi^2} left(sin (2pi ft)-{1 over 9} sin (6 pi ft)+{1 over 25} sin (10 pi ft) + cdots right) end{align}

It is also possible to approximate a triangle wave with abs() and floor():


begin{align} x_mathrm{triangle}(t) = 2 * mbox{abs}(t - 2 * mbox{Floor}(t/2) - 1) - 1 end{align}

Or with modulo:


begin{align} x_mathrm{triangle}(t) = 4 (t%1)^2+2(t%2)-4(t%1) (t%2)-1 end{align}