# Triangle wave

A bandlimited triangle wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).
 Triangle wave sound sample Sorry, your browser either has JavaScript disabled or does not have any supported player. You can download the clip or download a player to play the clip in your browser. 5 seconds of triangle wave at 220 Hz Problems playing this file? See media help.
 Additive Triangle wave sound sample Sorry, your browser either has JavaScript disabled or does not have any supported player. You can download the clip or download a player to play the clip in your browser. After each second, a harmonic is added to a sine wave creating a triangle 220 Hz wave Problems playing this file? See media help.

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 with cycle frequency f over time t:

\begin{align} x_\mathrm{triangle}(t) & {} = \frac {8}{\pi^2} \sum_{k=0}^\infty (-1)^k \, \frac{ \sin \left( 2\pi(2k+1) ft \right)}{(2k+1)^2} \\ & {} = \frac{8}{\pi^2} \left( \sin (2\pi ft)-{1 \over 9} \sin (6\pi ft)+{1 \over 25} \sin (10\pi ft) - \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:

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