The Shape of a Wave

Every classic synthesizer waveform is two pictures of the same object. In time, it is a shape — a curve the speaker cone traces, five volts up, five volts down. In frequency, it is a recipe — a stack of pure sine harmonics, each with its own strength. Neither picture is more real than the other; a trained ear hears the recipe, an oscilloscope shows the shape. Quiver’s Vco serves the four classics — sine, triangle, saw, square — and below, both pictures are computed from the same rendered buffer, so whatever you do to the shape you do to the recipe, instantly.

The oscillator is also where digital synthesis meets its oldest enemy. A mathematically perfect saw edge contains harmonics beyond any sample rate, and everything the samples cannot hold reflects back into the audio band as aliasing. Quiver bandlimits its edges with PolyBLEP (and its corners with PolyBLAMP). The toggle below turns that protection off — look at what floods the spectrum floor, then press ▶ hear it.

The small dial above the wave is the oscillator’s entire inner life. A Vco stores exactly one number — a phase \( \varphi \) that runs around a circle once per cycle — and each waveform is just a different function of where on the circle the phase currently is. The dial spins in slow motion (about one turn every two seconds, nowhere near audio rate) while the violet dot traces the matching point on the waveform: phase runs around the circle, and the shape is what the wave does along the way.

Pitch here is not a number in hertz — it is a voltage on the voct input. Drag it: 0.00 V. Every volt doubles the frequency — 1 V per octave, with 0 V = C4 (261.63 Hz) — so a five-octave keyboard is just a 5 V span of CV. The blue trace is the waveform in volts; the blue spectrum underneath is that exact buffer passed through an FFT, and the dashed yellow lines mark the integer harmonics \( n \cdot f_0 \) — with bandlimiting on, every peak sits on one. The hollow yellow rings are the prediction, not the measurement: the closed-form Fourier amplitude of each harmonic for the current shape, anchored to the measured fundamental so theory and FFT share a reference. Watching the stems land exactly on the rings is watching a 200-year-old theorem pass a live test. The square wave has one more knob the others lack: its pulse width, pw = 0.50, the fraction of each cycle spent high.

Things to try

1. Pick sin. The spectrum is a single spike at \( f_0 \) — a sine is one harmonic and nothing else. Every other waveform on this page is built out of copies of it.
2. Cycle saw → sqr → tri and read the recipes: the saw has every harmonic falling off as \( 1/n \) (a straight −6 dB/octave staircase); the square keeps only the odd harmonics, also at \( 1/n \); the triangle keeps the odd ones but at \( 1/n^2 \), which is why its spectrum plunges and it sounds so much mellower than the square whose harmonics sit at the same frequencies.
3. Pick sqr and drag the pulse width off 0.50: the even harmonics fade back in — at exactly 0.5 they cancel perfectly. Park it near 0.10 for the thin, nasal pulse timbre every string-machine patch is built on. Hear it while you drag.
4. Pick saw, drag the pitch up to +4.00 V, and switch bandlimited off. The floor fills with hash that lands between the yellow lines — harmonics above Nyquist reflected to frequencies that are not multiples of \( f_0 \). Press ▶ hear it and flip the toggle back and forth: the naive saw rings with inharmonic, metallic junk.
5. Still naive, scrub the pitch slowly upward while listening: the true harmonics rise, but the aliases sweep downward. Partials that bend the wrong way under a pitch change are the unmistakable fingerprint of aliasing.
6. Drop to −2.00 V, still naive. The damage nearly vanishes — the offending harmonics are weak and few down here. This is why naive oscillators almost get away with bass lines and fall apart the moment you play a lead.
7. Pick sqr and watch the phasor: the square’s value is just which part of the circle the phase is in — inside the shaded slice the wave sits at +5 V, outside at −5 V. Drag pw and the slice and the wave’s duty cycle move together. Now picture the saw the same way: its value is simply how far around the circle am I, climbing from −5 V back to −5 V once per lap.
8. With bandlimited on, every measured stem lands on a hollow ring — the FFT agreeing with Fourier’s closed-form recipe, live. Switch it off at +4.00 V and the stems miss and smear off their targets: the theory did not fail, the sampling did. (Drag pw on the square and the rings themselves migrate — the recipe below is the pw = 0.5 special case of the general pulse wave.)

What you just saw

The spectrum panel is not a decoration next to the waveform — it is the waveform, written in a different basis. Fourier’s theorem says any repeating wave is a sum of sines at integer multiples of the fundamental, and the classic shapes have closed-form recipes. The sawtooth uses every harmonic:

\[ \mathrm{saw}(t) = \frac{2}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(2\pi n f_0 t) \]

The square and triangle use only the odd ones, at \( 1/n \) and \( 1/n^2 \) respectively:

\[ \mathrm{sqr}(t) = \frac{4}{\pi} \sum_{n\ \mathrm{odd}} \frac{1}{n} \sin(2\pi n f_0 t) \qquad \mathrm{tri}(t) = \frac{8}{\pi^2} \sum_{n\ \mathrm{odd}} \frac{(-1)^{(n-1)/2}}{n^2} \sin(2\pi n f_0 t) \]

Those coefficients are exactly the stem heights you saw: \( 1/n \) is a −6 dB/octave slope, \( 1/n^2 \) is −12 dB/octave. The pitch voltage feeds the exponential V/Oct law, \( f_0 = 261.63 \times 2^{V} \) — one volt, one octave — which is why equal drags of the scrub felt like equal musical intervals.

The sums above are infinite, and there lies the problem: a sampled system at rate \( f_s \) can only represent content below the Nyquist frequency \( f_s / 2 \). A harmonic at \( f > f_s/2 \) does not disappear — it folds back to \( f_s - f \), which is almost never a multiple of \( f_0 \): inharmonic hash. PolyBLEP exists precisely for this. Instead of a perfect instantaneous edge (an infinite series), it splices in a two-sample polynomial rounding whose spectrum falls off steeply before Nyquist — the wave you see with the toggle on is microscopically “wrong” in time and audibly right in frequency.

The Quiver code

The widget’s bandlimited math is Vco::tick, line for line. In a real patch the same oscillator is four lines of graph setup — its inputs are voct, fm, pw, sync, and fm_lin; its outputs are sin, tri, saw, and sqr, all ±5 V audio:

use quiver::prelude::*;

let sample_rate = 44100.0;
let mut patch = Patch::new(sample_rate);

// Add the oscillator and an output module.
let vco = patch.add("vco", Vco::new(sample_rate));
let output = patch.add("output", StereoOutput::new());

// Patch the sawtooth straight to the left channel. The unpatched `voct`
// input sits at 0 V, so the VCO free-runs at C4 (261.63 Hz) — exactly the
// widget's default. (StereoOutput normals the right input to the left.)
patch.connect(vco.out("saw"), output.in_("left")).unwrap();

// Compile once, then tick: each call advances the graph one sample.
patch.set_output(output.id());
patch.compile().unwrap();
let (left, _right) = patch.tick();

Swap "saw" for "sin", "tri", or "sqr" to pick a different recipe, and patch a CV source into "pw" to sweep the square’s pulse width the way you just scrubbed it.

Go deeper

• Tutorial: Subtractive Synthesis — patch this VCO into a filter and envelope and make the recipe move.
• Reference: Oscillators — every port of Vco, AnalogVco, Supersaw, Wavetable, and the rest of the sources.
• Concepts: Signals — the voltage conventions this page leaned on: ±5 V audio, 1 V/octave pitch, 0 V = C4.
• Next explorable: Sculpting the Spectrum — you just met a wave as a stack of harmonics; next, carve that stack with a filter.

Next: Sculpting the Spectrum