// engineering › Fourier Waveforms
Assemble a sawtooth from all harmonics with alternating sign; watch the 1/n amplitude decay.
y(t)=frac{2A}{pi}sum_{n=1}^{N}frac{(-1)^{n+1}}{n}sin(nomega t)
A mind behind this: Joseph Fourier 1768–1830
It contains every harmonic — 1st, 2nd, 3rd, and so on — not just the odd ones. That full, dense set of overtones is why a sawtooth sounds rich and buzzy, and why it is the favourite starting waveform for analog synthesisers emulating strings and brass.
The horizontal sweep in old CRT televisions and oscilloscopes is a sawtooth — the beam races across, then snaps back. They also drive the timing ramps in many analog circuits and the pitch sweeps in synthesisers. The slow rise and instant drop is the signature shape.
Because the sawtooth has a sharp jump once per cycle, just like the square wave. Any signal with a sudden discontinuity has harmonics that decay slowly, as 1/n. That slow decay is why both the sawtooth and square need many terms and both show the Gibbs overshoot at the jump.
The (-1)^(n+1) factor flips the sign of every other harmonic. That careful alternation is what tilts the wave into its ramp shape rather than a square. Toggle the spectrum on and increase N to see how each added harmonic nudges the line closer to a straight ramp.
Both can use all or many harmonics, but the sawtooth's amplitudes fall as 1/n while the triangle's fall as 1/n² — far faster. That is why the sawtooth sounds bright and edgy and the triangle sounds soft and flute-like. Open both tabs and compare their spectra side by side.