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), and so its sound is smoother than a square wave and is nearer to that of a sine wave.
One simple definition of a triangle wave is
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 ), 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:
It is also possible to approximate a triangle wave with abs() and floor():
Or with modulo:
Patent No. 7,729,145 Issued on June 1, Assigned to Shindengen Electric Manufacturing for Power Converter, Method, Triangle Wave Generation Circuit (Japanese Inventor)
Jun 02, 2010; ALEXANDRIA, Va., June 4 -- Toyotaka Takashima, Hanno, Japan, has developed a power converter and method, and triangle wave...