Any sound is a stack of pure sine waves. Add them one harmonic at a time — amplitude 1/n — and the pile climbs toward a square or a sawtooth. It never quite lands: right at the edge the sum overshoots by a stubborn ~9% (Gibbs), no matter how many partials you throw at it. Rendered here, not quoted: the sines are summed live.
source J. Fourier, Théorie analytique de la chaleur (Firmin-Didot, Paris, 1822) — the trigonometric series; realised as an organ in the Hammond additive drawbars (1935). No DOI; historical — archive.org scan. Gibbs overshoot: Gibbs, Nature 59 (1899). Marked AMBER.
A periodic wave is a sum of harmonics k = 1,2,3… of the fundamental ω. Each is a pure sine sin(kωt) with its own amplitude.
Square (odd only): (4/π)·Σk odd sin(kωt)/k
Sawtooth (all): (2/π)·Σk≥1 (−1)k+1 sin(kωt)/k
The 1/k weighting is the whole game. It is what makes the pile converge.
Straight out of the-fourier-series: the same trigonometric sum Fourier used for heat in 1822, turned inside-out into a musical instrument. Where the-fourier sphere analyses a wave into partials, THE ADDITIVE SYNTHESIS runs it backwards — stack the partials to build the wave. It inherits the family flaw intact: the 9% Gibbs overshoot at every edge.
Live re-check of the CENTER engine: is the square reconstruction still converging (does mean-squared error fall as harmonics are added, and is the Gibbs overshoot still ≈9%)?
If window 6 drops the 1/n weighting, this flips red within the frame — the sum stops converging.
A target waveform and a harmonic budget N. That is all additive synthesis takes — no wavetable, no sample.
The blue trace is the running partial sum Σ sin(kωt)·ak; the faint teal is the ideal target. Watch the edge: the ripple never leaves.
The stacked sines are the waveform — and the MSE to the target shrinks with every partial, while the 9% overshoot at the discontinuity refuses to.
WALL "Add more sines and you'll get a perfect square — a flat top and a clean vertical edge."
No. The pointwise sum never converges uniformly at a jump. The overshoot peak stays at ~1.0895× the step forever; more harmonics only make the spike narrower, never shorter. A truly flat, vertical edge needs infinite bandwidth you do not have.
"The 1/n weighting is just a volume taste — any amplitudes will do."
↳ False. Drop 1/n (equal amplitudes) and the series diverges — the sum does not approach the square at all. The weighting is required for convergence.
"Gibbs overshoot is a rounding bug."
↳ It is a theorem, not a glitch — ≈8.95% of the jump, the constant (2/π)∫₀^π sinc.
Force equal amplitude on every harmonic (delete the 1/n). The pile stops converging — window 7 catches it live.