Editing Triangle wave (section)

===Harmonics===
[[Image:Synthesis triangle.gif|thumb|upright=1.6|right|Animation of the additive synthesis of a triangle wave with an increasing number of harmonics. See [[Fourier Transform|Fourier Analysis]] for a mathematical description. ]]

It is possible to approximate a triangle wave with [[additive synthesis]] by summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by {{pi}}) and multiplying the amplitude of the harmonics by one over the square of their mode number, {{math|''n''}} (which is equivalent to one over the square of their relative frequency to the [[fundamental frequency|fundamental]]).

The above can be summarised mathematically as follows:
<math display="block">
x_\text{triangle}(t) = \frac8{\pi^2} \sum_{i=0}^{N - 1} \frac{(-1)^i}{n^2} \sin(2\pi f_0 n t),
</math>
where {{mvar|N}} is the number of harmonics to include in the approximation, {{mvar|t}} is the independent variable (e.g. time for sound waves), <math>f_0</math> is the fundamental frequency, and {{mvar|i}} is the harmonic label which is related to its mode number by <math>n = 2i + 1</math>.

This infinite [[Fourier series]] converges quickly to the triangle wave as {{mvar|N}} tends to infinity, as shown in the animation.