# Triangle wave

A bandlimited triangle wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).
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A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

Like a square wave, the triangle wave contains only odd harmonics, due to its odd symmetry. 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).

## Harmonics

Animation of the additive synthesis of a triangle wave with an increasing number of harmonics. See Fourier Analysis for a mathematical description.

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 π), 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) t \right)}{(2k+1)^2} \\ & {} = \frac{8}{\pi^2} \left( \sin ( t)-{1 \over 9} \sin (3 t)+{1 \over 25} \sin (5 t) - \cdots \right) \end{align}

## Definitions

Sine, square, triangle, and sawtooth waveforms

Another definition of the triangle wave, with range from -1 to 1 and period 2a is:

$x(t)=\frac{2}{a} \left (t-a \left \lfloor\frac{t}{a}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{t}{a}+\frac{1}{2} \right \rfloor$
where the symbol $\scriptstyle \lfloor n \rfloor$ represent the floor function of n.

Also, the triangle wave can be the absolute value of the sawtooth wave:

$x(t)= \left | 2 \left ( {t \over a} - \left \lfloor {t \over a} + {1 \over 2} \right \rfloor \right) \right |$

or, for a range from -1 to +1:

$x(t)= 2 \left | 2 \left ( {t \over a} - \left \lfloor {t \over a} + {1 \over 2} \right \rfloor \right) \right | - 1$

The triangle wave can also be expressed as the integral of the square wave:

$\int\sgn(\sin(x))\,dx\,$

A simple equation with a period of 4, with $y(0) = 1$. As this only uses the modulo operation and absolute value, this can be used to simply implement a triangle wave on hardware electronics with less CPU power:

$y(x) = |x\,\bmod\,4 - 2|-1$

or, a more complex and complete version of the above equation with a period of 2π and starting with $y(0) = 0$:

$y(x) = \left|4\left(\left(\frac{x}{2\pi} - 0.25\right)\,\bmod\,1\right) - 2\right|-1$

In terms of sine and arcsine with period p and amplitude a:

$y(x) = \frac{2a}{\pi}\arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right)$