Sidebands from Nothing

Vibrato is a slow wobble in pitch — a modulator bending a carrier a few times a second. Speed the wobble up a thousandfold, past the point where the ear can follow the pitch moving, and it stops being vibrato and becomes timbre. Two pure sine waves, no filter anywhere in sight, and a forest of partials appears — not at random, but at positions you can predict to the hertz. That is FM synthesis, and the whole trick is below. The yellow stems are the prediction; the blue curve is the measurement of the exact buffer you can hear.

The carrier is the sine you hear, pitched at 0.00 V on the V/Oct scale (0 V = C4 ≈ 261.6 Hz). The modulator — the ghosted violet sine in the top plot — runs at 2× the carrier’s frequency and never reaches your ears directly: its only job is to bend the carrier’s frequency up and down. How hard it bends is the modulation index I = 3.0, the peak frequency deviation divided by the modulator’s frequency, \( I = \Delta F / f_m \). Scrub all three and watch the predicted stems and the measured spectrum move in lockstep.

Two smaller views close the loop. The Bessel panel graphs the first five weights \( J_0 \) through \( J_4 \) against the index, with a yellow cursor parked at the current I — the stem heights in the spectrum are literally these curves sampled at the cursor, and the dots on the zero line mark \( J_0 \)’s first two zeros, 2.405 and 5.520, where the carrier’s own stem vanishes. The index waterfall underneath records a sweep: during bloom (or instantly, via the waterfall button) each column paints the predicted spectrum at one value of I, low frequencies at the bottom — so you can watch the sidebands fan outward in Bessel order as the index rises, frozen into one picture.

One honest disclosure: the widget computes the classic phase-modulation form, \( y(t) = 5\sin(2\pi f_c t + I \sin(2\pi f_m t)) \) volts, because that is the parameterization the sideband math is written in. It has exactly the spectrum of sinusoidal linear FM with peak deviation \( \Delta F = I f_m \), which is what Quiver’s Vco does on its fm_lin jack: the module computes freq += (fm_lin / 5) · base, so a modulator swinging ±depth volts produces \( \Delta F = (\text{depth}/5) f_c \), and therefore \( I = \frac{\text{depth}}{5} \cdot \frac{f_c}{f_m} \). To dial an index of 3 at a 2× ratio you need a ±30 V modulator swing — which is why FM patches put a gain stage between the operators.

Things to try

1. Set the ratio to 1×. Every sideband lands exactly on a harmonic of the carrier — fc, 2fc, 3fc… — and the tone turns sawlike. The readout says harmonic.
2. Set the ratio to 2×. Sidebands land at fc ± 2k·fc: only the odd harmonics, and the reflected ones fold back onto the same grid. Hollow, squarelike — the same recipe a clarinet uses.
3. Set the ratio to 3.75×. Now the sidebands miss the harmonic grid and the readout flips to inharmonic: this is where bells, gongs, and metal live.
4. Scrub the index down to 0 — a lone stem, a pure sine. Now raise it slowly and watch the spectrum’s width track the Carson bandwidth readout, ≈ 2(I+1)fm: each unit of index wakes roughly one more sideband pair.
5. Find I ≈ 2.4. The center stem — the carrier itself — drops into the noise floor, because \( J_0(2.4048) = 0 \): the carrier vanishes from its own spectrum while everything around it keeps ringing. The next disappearance is at I ≈ 5.5.
6. Press bloom: the index sweeps from 0 to its target and the sidebands grow outward in order, each k-th pair waking only once I catches up to k — Bessel functions, animated. The waterfall strip freezes the whole sweep into one picture: sidebands fanning outward as the columns march right.
7. Find I ≈ 2.4 on the Bessel panel — the blue \( J_0 \) curve crosses zero exactly where the carrier disappears from its own spectrum. Park the yellow cursor on the dot and watch the center stem die; the next blackout waits at 5.520.
8. Set the index to 12, press waterfall, then flip the ratio between 1× and 3.75× and press it again: harmonic ratios print clean horizontal stripes, inharmonic ones weave a dense unaligned fabric.

What you just saw

The entire widget is one identity. A sine whose phase is wiggled by another sine is exactly a sum of sines at evenly spaced frequencies:

\[ \sin\bigl(2\pi f_c t + I \sin(2\pi f_m t)\bigr) = \sum_{k=-\infty}^{\infty} J_k(I) \sin\bigl(2\pi (f_c + k f_m) t\bigr) \]

The weights \( J_k(I) \) are the Bessel functions of the first kind — the yellow stem heights. Each one answers “how much energy lands k steps away from the carrier at this index?” At \( I = 0 \), \( J_0 = 1 \) and every other \( J_k = 0 \): a bare carrier. As I grows, \( J_k(I) \) stays near zero until \( I \approx k \), then swells — which is why sidebands appear in order, and why the audible bandwidth obeys Carson’s rule, \( B \approx 2(I+1)f_m \). Each \( J_k \) also oscillates in I, passing through zeros — at \( I \approx 2.405 \) it is \( J_0 \)’s turn, and the carrier itself blinks out. Energy is never created, only redistributed: \( \sum_k J_k(I)^2 = 1 \) for every I. When \( f_c + k f_m \) goes negative, \( \sin(-\omega t) = -\sin(\omega t) \): the component reflects back to \( |f_c + k f_m| \), which is why low carriers with big ratios grow extra stems near the bottom — that fold is through-zero FM, the oscillator’s phase briefly running backwards.

This math belongs to linear FM only, and the Vco gives you both flavors for a reason. The linear input adds deviation symmetrically — \( f_c(1 + m\sin\omega t) \) averages to exactly \( f_c \), so the note stays in tune at any depth. The exponential fm input multiplies instead: \( f_c \cdot 2^{a\sin\omega t} \), and because \( 2^x \) is convex, the average of \( 2^{a\sin\omega t} \) is greater than 1 — the upward swings outweigh the downward ones and the perceived pitch climbs as depth grows. Lovely for vibrato at small depths, hopeless for tuned FM timbres. That is why classic Chowning FM is patched into fm_lin.

The Quiver code

The two-operator patch behind everything above — modulator sine, a gain stage to set the depth (and thus the index), into the carrier’s linear through-zero FM input:

use quiver::prelude::*;

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

let carrier = patch.add("carrier", Vco::new(sample_rate));
let modulator = patch.add("modulator", Vco::new(sample_rate));
let depth = patch.add("depth", Attenuverter::new());
// fm = 2 x fc. V/Oct is logarithmic, so a frequency RATIO becomes a
// voltage OFFSET into the modulator's pitch input: log2(2.0) = 1 V.
let ratio_cv = patch.add("ratio_cv", Offset::new(2.0_f64.log2()));
let output = patch.add("output", StereoOutput::new());

patch.connect(ratio_cv.out("out"), modulator.in_("voct")).unwrap();
// Modulator sine -> depth -> the carrier's LINEAR (through-zero) FM input.
// `fm_lin` is the sideband math above; the exponential `fm` input is not.
patch.connect(modulator.out("sin"), depth.in_("in")).unwrap();
patch.connect(depth.out("out"), carrier.in_("fm_lin")).unwrap();
patch.connect(carrier.out("sin"), output.in_("left")).unwrap();
patch.connect(carrier.out("sin"), output.in_("right")).unwrap();

patch.set_output(output.id());
patch.compile().unwrap();

The Attenuverter’s gain is level / 5 V (unity by default), so drive its level input above 5 V — an Offset works — for the >1 gains that big indices demand: index I at ratio r needs a modulator swing of 5 · I · r volts. To make the index move — the bright-attack electric-piano trick — patch an Adsr into that level input instead. The runnable version, with a sweep of ratios and depths, is examples/tutorial_fm.rs.

Go deeper

• Tutorial: FM Synthesis Basics — operator algorithms, index envelopes, and the classic DX7 recipes.
• Reference: Oscillators — the Vco’s full port map, including both FM inputs.
• Next explorable: Patch Flow — how signals actually move through a compiled patch, one tick at a time.

Next: Patch Flow