FM synthesis

Definition

In FM synthesis (frequency modulation) one oscillator, the modulator, rapidly varies the frequency (in practice, in most digital instruments, the phase) of another oscillator, the carrier. When the modulation is at audio frequency you don't hear a vibrato but new harmonics: the sidebands. The technique was developed by John Chowning at Stanford between the late '60s and early '70s (paper “The Synthesis of Complex Audio Spectra by Means of Frequency Modulation,” 1973), licensed to Yamaha and made famous by the DX7 (1983): 6 operators, 32 algorithms, one feedback.

The two key quantities

• C:M ratio (carrier frequency : modulator frequency). Sidebands appear at fc ± n·fm. With integer ratios (1:1, 1:2, 2:1, 1:3) the spectrum is harmonic; with non-integer ratios (1:1.41; 1:3.5) it is inharmonic: bells, metals, noises.
• Modulation index (how strong the modulator is; on the DX7 it is the output level of the modulating operator). Low index = few sidebands, almost sine-like sound; high index = many sidebands, bright sound. The amplitude of each sideband follows Bessel functions: that is why FM “changes timbre” in a less linear way than a filter.

Spectrum examples

• C:M = 1:1 → sidebands at 1, 2, 3, 4… × f: full spectrum, similar to a sawtooth.
• C:M = 1:2 → sidebands at 1, 3, 5… × f: odd only, similar to a square wave (clarinet).
• C:M = 1:1.4 → non-multiple frequencies: bell or glass.
• C = 220 Hz (A3), M = 440 Hz: sidebands at 220, 660, 1100, 1540 Hz and “reflected” frequencies (negative sidebands fold back to positive with inverted phase).

Operators and algorithms

An operator = sine oscillator + envelope. The algorithm is the connection scheme: which operators are carriers (heard) and which are modulators. Chains (op2 → op1) give complex spectra; operators in parallel sum timbres (additive FM). Feedback (an operator modulating itself) produces a sawtooth-like wave or noise if pushed.

Envelopes: the secret of FM

The FM timbre lives in the modulator's envelope: a modulator with fast decay produces a bright attack that dies away into a pure sound, like a hammer on a string.

• E-piano (the famous “E.PIANO 1” of the DX7, everywhere in '80s ballads): 1:1 carrier for the body + modulator with a high ratio (e.g. 14:1) and very short decay for the metallic “tine”; velocity on the index.
• FM bass: C:M 1:1 or 1:0.5, high index with 150–300 ms decay, moderate feedback; great in synthwave and house.
• Bells and tubular bells: non-integer ratio (1:3.5), long decays, index that decreases slowly.
• FM growl and reese (dubstep, neuro): LFO or envelope on the index, ratio 1:1 or 1:2, distortion downstream.
• FM brass: index that grows with the attack (50–100 ms), like the “lip” opening.

Instruments

DX7 V (V Collection 9), Native Instruments FM8, Dexed (free, emulates the DX7), Ableton Operator, Sytrus in FL Studio (FM/RM matrix with 6 operators), Serum and Vital (FM from one oscillator to another), Arturia Pigments.

Example in A minor

Ballad at 70 BPM: Am(add9) – Fmaj7 – C/E – G6 played with an FM e-piano (Sytrus or DX7 V): velocity → index for a brighter attack on hard hits; light chorus; EQ: cut at 200–300 Hz to avoid mud, presence at 2–3 kHz for the tine. FM bass 1:1 with 200 ms decay playing A1 – F1 – E1 – G1.

Common mistakes

• High index without an envelope: static, harsh sound. Always move the index.
• Random ratios: work with integers first, then “detune” by a few cents for beating.
• Aliasing: very high indices on high notes generate frequencies beyond Nyquist; use oversampling if available.

Exercise

1. In Sytrus or Dexed start with two operators (1 modulator → 1 carrier). Try the ratios 1:1, 1:2, 1:3, 1:1.41 at a fixed index, then move the index with an envelope.
2. Build an e-piano and an FM bass from the init patch without presets.
3. Compare an FM bass and a subtractive one on the same line: which cuts through the mix better?

Connections

“Subtractive synthesis”, “Additive synthesis”, “Envelopes, LFOs and modulation”, “The harmonic series and timbre”, “Sync and ring modulation”.