Bend one sine wave with another and rich metallic timbres appear out of two oscillators. A carrier at fc whose instantaneous frequency is pushed by a modulator at fm, with modulation index I, does not smear — it splits into a comb of sidebands at fc ± n·fm, and their amplitudes are the Bessel functions Jn(I). Down the center, data flows: the two oscillators go in, the engine expands the spectrum, the sidebands come out. The blue team builds and defends it; the red team tries to break it.
source J. M. Chowning, The Synthesis of Complex Audio Spectra by Means of Frequency Modulation, JAES 21(7), 526–534 (1973) — aes.org/e-lib · elib=1954. Rendered, not quoted.
The spectrum is not memorized; it falls out of one trigonometric identity:
sin(θ + I·sin φ) = Σn=−∞∞ Jn(I)·sin(θ + n·φ)
So the FM signal is a sum of pure sines, one per sideband: a line sits at every fc + n·fm, and its height is Jn(I). Live sidebands for the current index:
| n | freq (Hz) | Jn(I) |
|---|
The neighbour is additive synthesis / Fourier: to build a bright timbre you stack many oscillators, one per harmonic. Chowning's 1973 move gets the same richness from just two — a modulator bending a carrier into a whole Bessel sideband series.
One knob, the index I, opens and closes the spectrum in time. That efficiency — rich spectra, two operators — is the sound of the DX7. Each sphere is the next one's premise.
The blue team's live check: synthesize the FM time signal directly and, separately, from the sideband sum, then measure the worst-case difference over the waveform. It must be ~0. If red swaps FM for AM, this badge is where it shows.
Three numbers define the sound: fc the carrier frequency (the pitch), fm the modulator frequency (sets sideband spacing), and I the modulation index — the peak frequency deviation divided by fm, i.e. how hard the modulator bends the carrier.
The ratio fm/fc decides the character: a simple integer ratio gives a harmonic (pitched) tone; an irrational ratio gives an inharmonic (bell / metallic) one. Feed these into the panel below.
Every value below is computed on the spot from Jn(I) — never looked up.
━ waveform ▮ spectrum (sideband magnitudes)
What the machine produces, proven: FM puts energy exactly at fc ± n·fm with amplitudes Jn(I); total power is conserved (Σ Jn² = 1 — index redistributes, it does not add); index 0 gives a pure carrier (J0(0)=1, Jn(0)=0); and the audible bandwidth grows as ≈ 2(I+1)·fm (Carson's rule, ~98% of power).
The blue team's witness (left) reconstructs the waveform live; the red team (right) tries to make the spectrum lie.
And when fm/fc is irrational the sidebands are inharmonic — a bell, not a note — which is a feature only if you wanted one. FM is not the synthesis; it is one brilliant shortcut through spectrum space.
"FM creates new frequencies out of nothing." Cut. The sidebands sit at fully predictable places, fc ± n·fm — the engine lists every one before you hear it.
"Turning up the index makes it louder." Cut. Σ Jn(I)² = 1 for all I: the index redistributes power into more sidebands, it never adds power.
"The DX7 does frequency modulation." Kept, corrected. It does phase modulation — identical spectra for a steady index, but the honest name differs, and the two diverge when I changes in time.
The red team's move: modulate the carrier's amplitude instead of its frequency — AM, not FM. AM has only three lines (fc, fc ± fm); the Bessel series above n=1 vanishes. The witness (window 7) is watching.
Swap to amplitude modulation and the sideband-at-fc±n·fm reconstruction breaks: the AM spectrum has no n≥2 lines, so it can no longer rebuild the true FM waveform. The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.