FM Synthesis

After basic oscillators and additive synthesis, FM introduces a new idea: one oscillator continuously changes the frequency of another oscillator.

Tip: Start with a slow modulation rate (LFO range) on a single sine wave carrier so you can visually and audibly track the pitch oscillation as basic vibrato. Then increase the modulation rate into the audio range and watch the single sine develop new sidebands and a new timbre.

1. Two operators

Operator 1 — Modulator Changes the instantaneous frequency of the carrier.
→
Operator 2 — Carrier The oscillator we actually hear.

With no modulation, the carrier is simply a sine wave. Add the modulator and its output pushes the carrier frequency above and below its centre frequency.

carrier frequency ← carrier frequency + modulator

2. Interactive FM

Sidebands above Nyquist are suppressed.
Carrier
Modulator
Approx. deviation
Perceptual region

Modulator

FM output

Simplified spectrum / sidebands

carrier and FM sidebands Nyquist / aliased components
Nyquist protection: leave this ON for normal synthesis. Turn it OFF only as a teaching demonstration: sidebands that exceed Nyquist are folded back into the audible spectrum as aliases.
y(t) = sin(2πfct + I sin(2πfmt))

The equation is included for reference. Conceptually: a sine-wave modulator changes the phase/frequency position of a sine-wave carrier. I is the modulation index.

3. Slow modulation → vibrato

If the modulator is slow — for example 5 Hz — we hear the carrier pitch moving up and down:

220 Hz centre → slightly above → 220 Hz → slightly below → repeat

This is simply vibrato. The modulation is slow enough that we perceive the movement itself.

4. Audio-rate modulation → timbre

Increase the modulator into the audible range and we stop hearing a separate pitch wobble. Instead, the carrier develops additional spectral components called sidebands.

sidebands occur around fc ± n fm

So with a 220 Hz carrier and 220 Hz modulator, components can appear around:

220 ± 220,   220 ± 440,   220 ± 660 ...

The modulation index controls how strongly the carrier is modulated. Increasing it generally produces more significant sidebands and therefore a more complex tone.

5. Ratios matter

The relationship between carrier and modulator frequency strongly influences the character of the result.

RatioGeneral tendency
1 : 1Sidebands fall on integer relationships and often produce a clearly harmonic spectrum.
1 : 2, 1 : 3...Different but still orderly harmonic structures.
Non-integer ratiosSidebands become less harmonically aligned; useful for bell-like, metallic and inharmonic timbres.
This is why FM synthesis can move from a simple sine to bells, electric-piano-like tones, metallic percussion and very complex spectra using only a few oscillators.

6. Operators do not have to be sine waves

The simplest FM explanation uses sine-wave operators because the relationships are easiest to see and hear. But an operator can begin with a more harmonically complex waveform.

Try it above: keep the carrier frequency, modulator frequency and modulation index fixed, then change only the modulator or carrier waveform. The spectrum becomes more complex immediately because square, triangle and sawtooth waves already contain harmonics before the FM interaction begins.

Conceptually:

sine operator → simple starting spectrum → FM sidebands
harmonically rich operator → existing harmonics + FM sidebands → much denser spectrum

Famous example: Yamaha TX81Z “Lately Bass”

The Yamaha TX81Z is a famous four-operator FM synthesiser associated with the preset Lately Bass. An important extension of the TX81Z design was that its operators were not restricted to sine waves: multiple operator waveforms were available.

Lately Bass is therefore a useful real-world teaching example of the broader idea: FM does not have to begin with pure sine operators. Changing the operator waveform changes the spectral material that enters the modulation process, which can produce much richer and more aggressive bass timbres.

This interactive is an educational model rather than an emulation of the exact Lately Bass patch. The sine / triangle / square / saw choices are included to make the principle immediately visible.

7. Multi-Operator Algorithms (DX Style)

Expanding beyond two operators allows complex routing arrangements called algorithms. In classic 4-operator FM synthesisers (like the Yamaha DX21, DX100, or TX81Z), operators are arranged in different signal paths:

Algorithm 1
4-Op Stack
[4] ↓ [3] ↓ [2] ↓ [1] → Out
Maximum sideband density
Algorithm 2
Branching Modulator
[4] [3] \ / [2] ↓ [1] → Out
Dual modulation sources
Algorithm 3
Dual Stacks
[4] [2] ↓ ↓ [3] [1] ↓ ↓ Out Out
Two independent FM pairs
Algorithm 4
3 Carriers Parallel
[4] / | \ [3][2][1] ↓ ↓ ↓ Out Out Out
Additive + FM mix

Key Takeaway: In FM synthesis, an operator connected directly to the output is a Carrier. An operator routed into another operator is a Modulator. Stacking operators in series creates extreme spectral brightness; placing carriers in parallel acts like additive synthesis.

8. A useful bridge from additive synthesis

Your earlier Max demonstration can connect the two lessons nicely: start with a square wave, then progressively remove its upper harmonics using a low-pass filter.

square → remove 7th, 5th, 3rd... → fundamental remains → sine-ish

This demonstrates that waveform shape depends on spectral content. Additive synthesis builds complexity by adding components; filtering can move the other way by removing components.

Then FM introduces the next idea: rather than explicitly adding harmonics one by one, modulation generates a whole new spectrum from interactions between oscillators.

9. The simple mental model

MethodWhat are we doing?
Additive synthesisAdd sine-wave components together.
Subtractive synthesisStart with a harmonically rich waveform and remove spectral components with filters.
FM synthesisUse one oscillator to continuously modulate another oscillator's frequency/phase.
FM in one sentence:
Slow modulation sounds like pitch movement; fast modulation becomes new timbre.