Triangle wave

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A bandlimited triangle wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).

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, demonstrating 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).


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)[citation needed], 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:


Sine, square, triangle, and sawtooth waveforms

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

where the symbol represent the floor function of n.

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

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

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

A simple equation with a period of 4, with . 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:

or, a more complex and complete version of the above equation with a period of , amplitude , and starting with :

The function (1) is a specialization of (2), with a=2 and p=4:

An odd version of the function (1) can be made, just shifting by one the input value, which will change the phase of the original function:

Generalizing the formula (3) to make the function odd for any period and amplitude gives:

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

Note: sin y = cos x

Arc Length[edit]

The arc length per period "s" for a triangle wave, given the amplitude "a" and period length "p":

See also[edit]