FM

Modulation at audio rate showed that moving a carrier's pitch at audio rate adds sidebands either side of it, spaced by the modulator's frequency. FM synthesis builds an instrument on that effect. It starts with two sine waves and creates new frequencies by modulating one with the other, and it changes the timbre over each note by changing the index with an envelope. Subtractive synthesis instead starts with many harmonics and removes some with a filter.

The first tool below plots the two sine waves in three lanes over the same few milliseconds. The top lane is the modulator. The middle lane is the carrier's frequency, which rises while the modulator is above zero and falls while it is below it, by an amount set by the index. The bottom lane is the carrier's waveform drawn over a plain sine at the carrier's own frequency: its cycles bunch together where the frequency is high and spread apart where it is low. With a ratio of a few hundredths the modulator runs at a few hertz and the movement is heard as vibrato. As the ratio is raised towards 1, the same movement repeats too fast to follow as a change in pitch, and it is heard first as roughness and then as a change in timbre.

The composer and researcher John Chowning found the technique at Stanford University in 1967 and published it in 1973 as "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation", in the Journal of the Audio Engineering Society. Stanford licensed it to Yamaha, and the Yamaha DX7, released in 1983, made FM one of the most widely heard kinds of synthesis of that decade. The next tool is the simplest FM voice: a modulator and a carrier, both of which are sine waves, with an envelope on the index and another on the level.

Ratio

The modulator's frequency is set as a ratio of the carrier's, so the relationship holds across the keyboard. The sidebands fall at the carrier's frequency plus and minus whole multiples of the modulator's. When the ratio is a whole number, every sideband lands on a harmonic of the carrier, and the result has a clear pitch. A ratio of 1 gives every harmonic, as in a sawtooth, and a ratio of 2 gives only the odd harmonics, as in a square wave, although the levels of the harmonics differ from those waveforms and change with the index. When the ratio is not a simple fraction, such as 1.41 or 3.14, the sidebands fall between the harmonics, and the result is inharmonic and bell-like.

Index

The index is the frequency deviation, which is how far the carrier's frequency moves either side of its centre, divided by the modulator's frequency. At index 0 there is no modulation and the carrier is a plain sine. As the index rises, energy moves from the carrier into the sidebands, and further sidebands appear. The amount in each one follows a set of curves called Bessel functions, which is why the spectrum does not simply grow brighter: some sidebands rise, fall back and rise again as the index increases, and the carrier itself drops to silence near an index of 2.4. The spectrum in the tool above is calculated from those functions for the index the envelope has reached.

An envelope on the index changes the brightness over each note, as a filter envelope does in a subtractive voice, but without a filter. A fast decay from a high index gives a bright attack that settles, which suits electric pianos and bells. A slow rise gives a swell into brightness, which suits brass.

Operators and algorithms

In an FM synthesiser each sine oscillator, with its own envelope, is called an operator. The DX7 has six operators per voice and 32 algorithms, which are fixed arrangements of which operators modulate which and which are heard. A modulator can itself be modulated, several modulators can feed one carrier, and one operator can modulate itself through feedback. Many synthesisers since have used the same technique, sometimes under the name phase modulation, which moves the oscillator's phase instead of its frequency and gives the same spectrum for a steady index. The DX7 itself works by phase modulation.