Having armed myself with prerequisite knowledge of bessel functions, I made some time to read thru John Chowning’s “The Synthesis of Complex Audio Spectra by Means of Frequency Modulation”, the 1973 paper that defined FM synthesis and lead to the production of the ever-popular Yamaha DX7. Here’s an outline:
Regarding natural sounds, Chowning observes that “the character of the temporal evolution of the spectral components is of critical importance in the determination of timbre.” From this observation we understand the need for a synthesis technique that allows greater dynamic control over spectra than additive or subtractive synthesis tend to offer.
The general formula for a frequency-modulated signal is given:
- e = the instantaneous amplitude of the modulated carrier
- α = carrier frequency (rad/s)
- β = modulator frequency (rad/s)
- I = modulation index, the ratio of the peak deviation to the modulating frequency
The Fourier-Bessel expansion shows how the resulting spectra is distributed via sidebands and their amplitudes are a function only of the modulation index:
Reflected Side Frequencies
The resulting sidebands will often fall in the negative frequency domain. These below 0 Hz frequencies will be reflected and mix in with the existing positive-frequency sidebands, amplifying, attenuating, or having no effect on them.
Example: A carrier frequency of 100 Hz and a modulator frequency of 100 Hz with a modulation index of 4 will result in a like-signed reflected frequency at 100 Hz, increasing the energy at 100 Hz. An unlike-signed reflected frequency will be at 200 Hz, decreasing the energy at 200 Hz.
Harmonic and Inharmonic Spectra
If the ratio of the carrier frequency to the modulator frequency is a rational number, then the resulting spectra will be harmonic. If the ratio is irrational, it will be inharmonic.
Example: A ratio of 4/1 will have harmonic spectra whereas a ratio of 1/sqrt(2) will have reflected frequencies that fall between the positive components thus will not be harmonically related.
As the index of modulation increases, the overall bandwidth of the spectrum increases. But since the amplitudes of each component is determined by Bessel functions, each component may either increase or decrease and the bandwidth increases. Tie in the effects due to the reflected frequencies and you end up with a complex dynamic spectra.
This section discusses the implementation of FM synthesis on the DEC PDP-10 computer using Max Mathews’ MUSIC V software. Block diagrams of the circuit and its parameters are given along with some (Fortran?) code changing the oscillators behavior to allow decreasing angles.
Simulations of Instrument Tones
Techniques for simulating instruments with FM synthesis are given for Brass-like tones, Woodwind-like tones and Percussive sounds. A more sophisticated FM circuit is given to allow more control over the sound.
A look forward is given towards new classes of sounds creating using the FM synthesis technique.
Oh how right you were, Dr. Chowning. Thanks for all the sounds!