Like a square wave, the triangle wave contains only odd harmonics. 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).
E.g., for a triangle wave with amplitude 5 and period 4:
A phase shift can be obtained by altering the value of the term, and the vertical offset can be adjusted by altering the value of the term.
As this only uses the modulo operation and absolute value, it can be used to simply implement a triangle wave on hardware electronics.
A triangle wave with period p and amplitude a can be expressed in terms of sine and arcsine (whose value ranges from −π/2 to π/2):
The identity can be used to convert from a triangle "sine" wave to a triangular "cosine" wave. This phase-shifted triangle wave can also be expressed with cosine and arccosine:
Expressed as alternating linear functions
Another definition of the triangle wave, with range from −1 to 1 and period p, is:
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 summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by π) and multiplying the amplitude of the harmonics by one over the square of their mode number, n (which is equivalent to one over the square of their relative frequency to the fundamental).
The above can be summarised mathematically as follows:
where N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), is the fundamental frequency, and i is the harmonic label which is related to its mode number by .
This infinite Fourier series converges quickly to the triangle wave as N tends to infinity, as shown in the animation.
The arc length per period for a triangle wave, denoted by s, is given in terms of the amplitude a and period length p by
^Kraft, Sebastian; Zölzer, Udo (5 September 2017). "LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-filtered Waveforms". Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17). 20th International Conference on Digital Audio Effects (DAFx-17). Edinburgh. pp. 255–259.