FM Synthesis,
Properly Explained

What FM synthesis is

Take one oscillator, the modulator. Instead of sending its output to speakers, route it to the pitch input of a second oscillator, the carrier. Run the modulator fast enough and the carrier's pitch changes so rapidly that you stop hearing wobble and start hearing new frequency components appear: sidebands. That's FM.

Unlike subtractive synthesis, FM creates harmonics from the mathematical interaction of two frequencies. Change the ratio between carrier and modulator and you get completely different harmonic structures (pure sine to dense bell to near-noise) without touching a filter.

In FM, oscillators are called operators: an oscillator plus an envelope. On a modulator, the envelope controls modulation intensity. On a carrier, it controls volume. The DX7 has 6 operators, each of which can be a carrier, modulator, or both depending on the algorithm.

Basic FM Signal Flow:

  MODULATOR oscillator (frequency M)
        │
        │ amplitude controlled by modulator envelope
        │
        ▼
  CARRIER oscillator (frequency C)  ← modulator output shifts C's pitch
        │
        │ amplitude controlled by carrier envelope
        │
        ▼
  AUDIO OUTPUT → speakers

Modulation index: the brightness control

Modulation index Iis the brightness knob. Low index: pure sine. High index: spectrally dense and complex. On the DX7 there's no "index" knob. Instead, you set the output level of the modulator operator. Higher level = higher index = brighter. The modulator envelope moves that level over time, which is why FM timbre evolves.

Audible effect at each index range

Enveloping the modulation index

When the modulator is loud, I is high and the sound is bright. As the modulator decays, I falls and the sound darkens. This is not a filter sweep but a completely different mechanism.

The DX7 electric piano: modulator peaks at I≈2.5–3 at the strike, decays fast (20–60ms) to I≈0.8–1.2 for the body. Bright brittle click to warm and round in under a second. That's the Rhodes character.

Key scaling

Keyboard level scaling reduces modulation index as you play higher notes. Set it on the modulator and the index drops up the keyboard, keeping timbre consistent, because acoustic instruments naturally have less harmonic energy in the upper register.

C:M ratios: the most important concept in FM

The ratio determines where sidebands fall in the spectrum, which is what decides whether your sound reads as tonal, metallic, or noise-adjacent. It's probably the most impactful single variable in FM, more so than modulation index, operator count, or algorithm choice.

Integer vs non-integer ratios

Integer ratios (1:1, 1:2, 2:3, 3:1) place sidebands on exact multiples of the fundamental, and your ear fuses them into a single pitched sound, so they come out tonal.

Non-integer ratios (1:1.4, 1:1.5, 1:2.7) scatter sidebands where no harmonic series exists, which is what gives FM its metallic, bell-like, percussive character. The more irrational the ratio, the more inharmonic the result, and at high index values, non-integer ratios can get pretty close to noise.

Key ratios in detail

1:1: sidebands partially cancel even harmonics, giving a hollow square-wave-like spectrum. I=2–3 with a fast-decaying modulator is the DX7 electric piano.

1:1.4 (≈1:√2): every sideband lands at an irrational multiple of the carrier. Fully inharmonic, no harmonic series possible. Metallic shimmer even at I=0.5. Standard bell patch: I starts at 6–8, decays to near 0 over 2–5 seconds. Add a second pair at a slightly different non-integer ratio and detune for cymbal beating.

2:3 / 3:2: fifth-interval harmonic structure. Musical and resonant. 3:2 has a metallic edge at high index: bells, gongs.

1:14: very wide sideband spacing, noise-adjacent at high I. Envelope the modulator from I=6 to 0 in 10–30ms and layer with a tonal carrier for a click transient.

DX7 algorithms: configuring the operators

An algorithm is a routing diagram: signal flow between 6 operators. Operators feeding audio output = carriers. Operators feeding another operator's frequency input = modulators. An operator can be both at once. The DX7 ships with 32 preset algorithms.

Algorithm configurations (simplified representations):

  Algorithm 1 — Maximum stacking, single carrier:
    Op6 → Op5 → Op4 → Op3 → Op2 → Op1 (carrier)
    Six operators in a series chain.
    Op6 modulates Op5, which modulates Op4... Op1 outputs audio.
    Most complex single timbre. Complex sidebands of sidebands.
    Use for: dense complex tones, aggressive textures.

  Algorithm 5 — Two parallel 3-op stacks:
    Stack A:  Op6 → Op5 → Op4 (carrier)
    Stack B:  Op3 → Op2 → Op1 (carrier)
    Two independent timbral sources summed to audio.
    Useful for two-layer sounds (e.g., attack layer + sustain layer).

  Algorithm 32 — Maximum carriers, minimal modulation:
    All 6 operators are carriers.
    No inter-operator modulation (just Op1 self-feedback).
    Pure additive synthesis — 6 independent sine waves.
    Useful for organ simulation and as a starting point for understanding.

  Algorithm 4 — Three parallel 2-op pairs:
    Op5→Op6 (pair 1), Op3→Op4 (pair 2), Op1→Op2 (pair 3, with feedback)
    Three independent FM voices summed.
    Good for multi-timbre pads and stacked textures.

Series (stacked) vs parallel operators

Series: modulator → modulator → carrier. Each stage adds sidebands of sidebands, which makes the result exponentially denser. A 3-op series stack at moderate index is denser than a single pair at maximum index.

Parallel: multiple pairs feeding audio independently. Multiple FM timbres summed. Classic use: inharmonic pair for the attack transient, harmonic pair for the sustained body.

Feedback: self-modulation

One operator per algorithm feeds back into its own frequency input. FM's waveform selector: low = sine, medium = square/saw character, high = noise and dirt. Most DX7 bass patches use carrier feedback for grit a pure sine can't get.

Velocity and key scaling per operator

Velocity on a modulator: harder strikes increase I, making the sound brighter dynamically. Velocity on a carrier: harder strikes are louder. Dial both independently to control brightness and volume response in whatever proportion you want.

FM in electronic music

DX7 electric piano

1:1 ratio, fast modulator decay taking I from ~2.5 at the strike to 0.8–1.2 on the sustain. Bright glassy click, warm and round as it settles. Rhodes character, harder transient than the real thing. Velocity to modulator level: soft playing = warm barely-modulated tone, hard playing = bright harsh attack.

FM bells

Non-integer ratio (1:1.4 or 3:2), I starts at 6–8 and decays to near 0 over 2–5 seconds alongside the amplitude envelope. Dense inharmonic cluster at the strike, near-pure carrier at the tail.

FM bass

Carrier at 50–100 Hz, 1:1 or 1:2 ratio, modulator that decays in 20–80ms. I spikes at the note onset then falls to near zero. That click at the front is the point: pure sine bass gets lost at club volume, but the spectral flash doesn't.

FM percussion

A classic FM snare uses one pair at 1:1 or 1:2 for the body, and a second pair at 1:7 or 1:14 with high index and fast decay for the wire noise. No noise oscillator or sample needed. The Digitone is built around this approach; its “sweep” parameters are essentially pre-patched modulation index envelopes.

Techno

High index, non-integer ratios, feedback pushed: metallic industrial clang from stabs, buzzing sawtooth density from leads. Digitone and FM8 are the usual tools here.

Modern FM synthesis tools

Sound design recipes

1. FM Electric Piano (DX7-style)

FM Electric Piano — 2-operator FM
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Ratio:           C:M = 1:1
Carrier freq:    Root note (e.g., C4 = 261.6 Hz)
Modulator freq:  1× carrier (same frequency)

Carrier Envelope (controls volume):
  Attack:    ~5ms
  Decay:     1200ms
  Sustain:   0% (piano-style, full decay)
  Release:   400ms

Modulator Envelope (controls brightness):
  Attack:    ~2ms (near-instant)
  Decay:     80–150ms   ← KEY: controls "pluck" brightness
  Sustain:   15–25%     ← warm body
  Release:   200ms

Modulation Index range:
  At attack peak (full modulator):    I ≈ 2.5–3.0
  At sustain (modulator at 15–25%):   I ≈ 0.5–0.8

Velocity routing:
  Modulator output level → velocity sensitive (high sens)
  Carrier output level   → velocity sensitive (moderate)

Feedback: 0–1 (none or very slight)

Key adjustments for character:
  Shorter mod decay (40ms) → more glassy, digital character
  Longer mod decay (200ms) → warmer, more Rhodes-like sustain
  Higher sustain level (40%) → nasal, more organ-like
  Detune carrier slightly (+/- 3 cents) → chorus-like spread

2. FM Bell

FM Bell — 2-operator inharmonic FM
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Ratio:           C:M = 1:1.4   (inharmonic — irrational ratio)
Carrier freq:    400–800 Hz (bell register)
Modulator freq:  1.4× carrier

Carrier Envelope (controls volume):
  Attack:    ~2ms
  Decay:     2000–5000ms (long bell decay)
  Sustain:   0%
  Release:   500ms

Modulator Envelope (controls inharmonic brightness):
  Attack:    ~1ms (instant)
  Decay:     1500–4000ms (MUST track carrier decay closely)
  Sustain:   0%
  Release:   400ms

Modulation Index:
  At attack:   I = 5–8   (dense inharmonic cluster)
  As it decays: falls to near 0 as amplitude decays

The relationship:
  As mod env decays, I → 0, spectrum → near-pure carrier
  The transition from inharmonic (attack) to tonal (tail)
  is the defining bell characteristic.

Optional: stack a second operator pair at C:M = 3:2
  with slightly different decay to create beating partials
  (characteristic shimmer of real bronze bells).

Velocity:  mod output → velocity. Hard = harsh; soft = gentle.

3. FM Bass

FM Bass — click attack + pure sustain
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Ratio:           C:M = 1:1 or 1:2
Carrier freq:    40–100 Hz (sub register)
Modulator freq:  1× or 2× carrier

Carrier Envelope (volume):
  Attack:    ~1ms
  Decay:     varies by bass style
  Sustain:   80–100% for held notes
  Release:   50–150ms

Modulator Envelope (brightness / click):
  Attack:    ~1ms
  Decay:     15–60ms    ← this controls the "click" length
  Sustain:   0–5%       ← near-zero for clean sustain
  Release:   20ms

Modulation Index:
  At attack:  I = 2–5   (click / brightness burst)
  Sustain:    I ≈ 0     (nearly pure sub-bass carrier)

Effect: the brief I spike at note onset gives the bass a
distinct transient attack that cuts through a mix at high
volumes — essential for techno/club bass that reads on large
sound systems. Without it, pure sine bass gets lost.

Tuning: detune carrier -5 to -10 cents for warmth.
Add feedback (level 2–3) on carrier for analog-like grit.

For 808-style: very slow carrier decay (3–8 seconds),
mod decay 20–40ms, add slight pitch envelope on carrier
(drops 1–3 semitones over the first 50ms).

4. FM Metallic Percussion (Cymbal / Hi-Hat)

FM Metallic Percussion
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Layer 1 — Main metallic body:
  Ratio:      C:M = 1:1.4
  Carrier:    400–600 Hz
  Mod Index:  6–8 at attack, decays to 0

  Carrier env:  Attack 1ms, Decay 80–400ms (closed vs open hat)
  Mod env:      Attack 1ms, Decay 60–300ms (tracks carrier)

Layer 2 — High-frequency noise burst (optional, for snap):
  Ratio:      C:M = 1:14
  Carrier:    800–1200 Hz
  Mod Index:  8 at attack, decays to 0 in 10–30ms
  Carrier env: Attack 1ms, Decay 20–50ms (very short)

  This layer adds the initial noise-like snap. Very short
  decay — it disappears before the body of the hit.

Sum both layers. Tune them apart by 30–80 cents for
metallic beating/shimmer (characteristic of real cymbals).

Closed hi-hat:   carrier decay 40–80ms
Open hi-hat:     carrier decay 400–800ms
Crash:           carrier decay 1500–4000ms, 3–4 layers detuned

Add a high-pass filter at 200–400Hz to remove sub content
from the carrier (cymbals have no sub-bass energy).

5. FM Plucked String / Marimba

FM Marimba / Plucked Tone
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Ratio:      C:M = 1:2   (adds warm even harmonics)
Carrier:    Root note
Mod Index:  1.5–2.5 at attack, decays to 0.3–0.6

Carrier env:
  Attack:   1ms
  Decay:    600–1200ms
  Sustain:  0% (marimba-style natural decay)
  Release:  200ms

Mod env:
  Attack:   1ms
  Decay:    100–300ms   (faster than carrier)
  Sustain:  15–25%
  Release:  100ms

Character:
  Low I = woody, hollow (pure marimba/xylophone)
  Mid I = nasal, clarinet-like
  Shorter mod decay = more percussive click, like mallet
  Longer mod decay = sustained warm tone, like bowed mallet

Feedback: 0 (clean) or 1 (very slight warmth)

FM synthesis cheat sheet

Ratio character quick reference

Modulation index guide

The DX7 algorithm summary

Envelope design rules

Controls: how long the sound lasts,
         ADSR of the audio amplitude
Does NOT control: brightness or timbre

Controls: how the sound changes color
High mod level → bright/complex
Low mod level → dark/pure
DECAY TIME is the most important parameter

Mod decay: shorter than carrier decay
This creates the bright attack that
fades to a duller sustain — instrument-like

Mod attack: slow (200–800ms)
Sound starts pure, grows complex
Combine with LFO on mod level for movement

Ratio: non-integer (1:1.4, 3:2)
Mod env: fast attack, long decay (= carrier)
I at attack: 6–8
I at tail: near 0

Carrier: 40–100 Hz
Mod env decay: 15–60ms only
Sustain: 0–5%
Creates transient click + clean sub