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A triangle wave is a waveform that can be obtained by subtractive synthesis by integrating (lowpass filtering) a square 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 (A2). |
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Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave, and so its sound is smoother than a square wave and is nearer to that of a sine wave.
It is possible to approximate a triangle wave by additive synthesis, by adding odd harmonics of the fundamental, rolling them off with frequency faster than with a square wave. The infinite series will converge to a triangle wave.
This infinite Fourier series converges to the triangle wave:
- <math>x_{triangle}(t) = \sum_{k=0}^\infty \frac{\cos (2k+1)t}{(2k+1)^2}<math>
Note that its peak amplitude is exactly <math>\pi^2/8<math>.
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