Additive

The theorem underneath

Fourier's result is that any periodic waveform, however complicated, can be written as a sum of sine waves at whole-number multiples of its fundamental. Not approximated: written, exactly, given enough terms.

That is a licence to work backwards. If every periodic sound is a sum of sinusoids, then a bank of sine oscillators with the right amplitudes can produce any of them. There is no filter, no waveform to choose, no cleverness at all. You specify the answer.

The catch is in the phrase given enough terms, and the first figure is about what happens when you do not have enough.

The corners never arrive

Drag the slider and the sum visibly becomes a square. The sides straighten, the top flattens, and the corners sharpen. What does not happen, however far you go, is the ripple beside each edge getting any shorter.

It gets narrower, and it moves closer to the edge, and its height converges to about 1.179 rather than to 1. This is the Gibbs phenomenon, the limit is exactly (2/π)·Si(π), and it is a property of the series rather than a defect in anyone's implementation. A perfect corner needs infinitely many partials, and every finite sum overshoots it by the same 18%.

Worth connecting to lesson 2. A band-limited sawtooth is exactly this: a partial sum, truncated because the rest will not fit under Nyquist. So every oscillator in this course has a Gibbs ripple on its edges, and always did. The alternative was not a cleaner square but the folded rubbish that lesson 2 measured, which is a much worse trade.

What you get for the money

Building a square out of sinusoids is a poor advertisement, because one oscillator could have done it. The case for additive is the two things it can do that no filter can, at any price.

Every partial can have its own envelope. On a real piano the high partials die away much faster than the low ones, so the note starts bright and darkens by itself as it decays. A subtractive voice cannot imitate that: it has one envelope on one filter, so its partials can only fade together under a single curve. Lesson 1 flagged this as a caveat about its own model. This is the lesson that fixes it.

The partials do not have to be harmonic. A filter can only remove what the oscillator supplied, and oscillators supply whole-number multiples. Bells, gongs and most struck metal have partials at ratios nothing like whole numbers, which is why they have a vague sense of pitch and why subtractive synthesis has never produced a convincing one.

The bell is the interesting preset

Those ratios are roughly what a tuned church bell actually has, and two of them are worth pointing at. There is a partial below the nominal pitch, called the hum, which is why a large bell feels like it is sounding lower than the note it is tuned to. And there is a partial at about 1.183, which is a minor third above the prime.

That minor third is unavoidable. It falls out of the geometry of a bell-shaped piece of bronze, founders have known about it for centuries, and it is the reason a big bell sounds sombre no matter what music is written for it. Nobody chose it. You are hearing a shape.

So why did additive not win

It has the strongest theoretical claim of any method here. Fourier says it can produce anything periodic, and the per-partial envelopes extend that to a great deal that is not. And it is a niche.

The cost is linear in partials. A bright sound wants sixty-four of them. Eight-voice polyphony then means five hundred and twelve oscillators, each with its own envelope, where a subtractive instrument needed eight oscillators and eight filters. In 1975 that was impossible. Today it is merely expensive, and the thing you would spend it on usually sounds much the same as a cheaper method.

The control problem is worse than the cost. Sixty four amplitudes and sixty-four envelopes is two hundred and fifty-six numbers per sound, and no arrangement of knobs makes that playable. Subtractive synthesis gives you a cutoff and a resonance, and those two controls move all the partials at once in a way people can hear and predict. Additive gives you everything and therefore no handle.

Where it does win, it wins by analysis. Serious additive instruments do not ask you to type in the numbers. They record a real sound, analyse its partials over time, and hand you the measured envelopes to modify. That is resynthesis, it is how the Synclavier and the Kawai K5 worked and how spectral tools work now, and it sidesteps the control problem by having a machine fill in the numbers first.