- Triangle wave
A

**triangle wave**is anon-sinusoidal waveform named for its triangular shape.Like a

square wave , the triangle wave contains only oddharmonic s. 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 asine wave .One simple definition of a triangle wave is

:$egin\{align\}x\_mathrm\{triangle\}(t)\; =\; arcsin(sin(t))end\{align\}$

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:

:$egin\{align\}x\_mathrm\{triangle\}(t)\; \{\}\; =\; frac\; \{8\}\{pi^2\}\; sum\_\{k=1\}^infty\; sin\; left(frac\; \{kpi\}\{2\}\; ight)frac\{\; sin\; (2pi\; kft)\}\{k^2\}\; \backslash \; \{\}\; =\; frac\{8\}\{pi^2\}\; left(\; sin\; (2pi\; ft)-\{1\; over\; 9\}\; sin\; (6\; pi\; ft)+\{1\; over\; 25\}\; sin\; (10\; pi\; ft)\; +\; cdots\; ight)end\{align\}$

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

:$egin\{align\}x\_mathrm\{triangle\}(t)\; =\; 2\; *\; mbox\{abs\}(t\; -\; 2\; *\; mbox\{Floor\}(t/2)\; -\; 1)\; -\; 1end\{align\}$

Or with modulo:

:$egin\{align\}x\_mathrm\{triangle\}(t)\; =\; 4\; (t\%1)^2+2(t\%2)-4(t\%1)\; (t\%2)-1end\{align\}$

**See also***

Triangle function

*Sine wave

*Sawtooth wave

*Square wave

*Wave s

*Sound

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