Editing Triangle wave (section)

{{Short description|Non-sinusoidal waveform}}
{{Infobox mathematical function
| name = Triangle wave
| image = triangle-td and fd.svg
| imagesize = 400px
| imagealt = A bandlimited triangle wave pictured in the time domain and frequency domain.
| caption = A [[Bandlimiting|bandlimited]] triangle wave<ref name="bandlimited-synthesis">{{cite conference |title=LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-filtered Waveforms |last1=Kraft |first1=Sebastian |last2=Zölzer |first2=Udo |date=5 September 2017 |book-title=Proceedings of the 20th [[International Conference on Digital Audio Effects]] (DAFx-17) |pages=255–259 |location=Edinburgh |conference=20th International Conference on Digital Audio Effects (DAFx-17) |conference-url=http://www.dafx17.eca.ed.ac.uk/}}</ref> pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220&nbsp;Hz (A<sub>3</sub>).
| general_definition = <math>x(t) = 4 \left\vert t - \left\lfloor t + 3/4 \right\rfloor + 1/4 \right\vert - 1</math>
| fields_of_application = Electronics, synthesizers
| domain = <math>\mathbb{R}</math>
| codomain = <math>\left[ -1, 1 \right]</math>
| parity = Odd
| period = 1
| root = <math>\left\{ \tfrac{n}{2} \right\}, n \in \mathbb{Z}</math>
| derivative = [[Square wave (waveform)|Square wave]]
| fourier_series = <math>x(t) = -\frac{8}{{\pi}^{2}}\sum_{k=1}^{\infty} \frac{{\left( -1 \right)}^{k}}{\left( 2 k - 1 \right)^{2}} \sin \left(2 \pi \left( 2 k - 1 \right) t\right)</math>
}}

{{Listen|filename=220 Hz anti-aliased triangle wave.ogg|title=Triangle wave sound sample|description=5 seconds of triangle wave at 220 Hz|format=[[Ogg]]}}
{{Listen|filename=Additive_220Hz_Triangle_Wave.wav|title=Additive Triangle wave sound sample|description=After each second, a harmonic is added to a sine wave creating a triangle 220 Hz wave|format=[[Ogg]]}}

A '''triangular wave''' or '''triangle wave''' is a [[non-sinusoidal waveform]] named for its [[Triangle|triangular]] shape.  It is a [[periodic function|periodic]], [[piecewise linear function|piecewise linear]], [[continuous function|continuous]] [[function of a real variable|real function]].

Like a [[Square wave (waveform)|square wave]], the triangle wave contains only odd [[harmonic]]s.  However, the higher harmonics [[roll-off|roll off]] much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).