Some Variations on FM
Many things can be done to create more complex spectra with FM. The DX-7 was built around the idea of both double-carrier FM, in which a single modulator controls two carriers, tuned differently. This allows the creation of formant areas not possible with single FM. Also, stacks of modulators, where a modulator was itself modulated, could either produce wildly complex spectra if tuned inharmonically or produce weighted spectra, which could create a more realistic bass. This helped with one of FM's greatest drawbacks--the strength of the upper and lower sidebands are equal, but our human hearing requirings more energy in the lower frequencies to be considered as equally loud as the higher frequencies. Therefore, single FM always seemed weighted to the treble, particularly at higher values of I. Another interesting idea is to modulate the modulation index itself, providing a rapid timbral shift. or to low-frequency modulate the modulator or carrier, changing the C:M ratio and therefore the frequencies of the sidebands for some very nice effects.
Suggested Listening Examples
To hear audio-rate FM used with a high level of artistry, there can be no better source than the works of John Chowning himself. Highly recommended are Stria (1976), Turenas (1972) and Phoné (1981). Barry Truax was another pioneering FM composer with Arras, Androgyny, Wave Edge, Solar Ellipse, and Sonic Landscape No. 3.
Finding the FM C:M Ratio Normal Form
The concept of the normal form for a C:M ratio has been used for a long time. It is useful for predicting which C:M ratios will produce the same sidebands, but it is not useful for predicting their relative strengths or phases. If the value of M in a ratio is less than twice the value of C, it is not in normal form, but can be reduced to normal form by applying the operation: C = |C - M|. What this means is that you subtract M from C (ignoring any minus sign) and treat the result as the new C value. You keep doing this (often several times) until the ratio satisfies the normal form criterion.
For example, take the C:M ratio of 3:2. Take 3 - 2 and get 1. That is the new value of C (keep the old value of M), so the new ratio will be 1:2. How is this possible—how can 3:2 produce the same sidebands as 1:2? Let's try it out with 300:200 Hz as our 3:2 ratio and 100:200 Hz as our 1:2 ratio.
You can see they produce the same frequencies, but with sidebands of different orders and different reflections. Therefore, the way these frequencies react to changing values of I will be completely different. But some interesting things can be deduced using normal form. A C:M ratio is in normal form when the carrier is the fundamental in the spectrum it produces, as in our 1:2 example above—100 Hz is the fundamental. Harmonic normal form ratios are always of the type 1:N where N is an integer, and inharmonic ones aren't. For a much more detailed treatment of normal form, visit Barry Truax's FM theory page.
Suggested FM Theory Reading
J. Chowning, "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation," Journal of the Audio Engineering Society 21(7), 1973; reprinted in Computer Music
Journal 1(2), 1977; reprinted in Foundations of Computer Music, C. Roads and J. Strawn (eds.). MIT Press, 1985.
B. Truax, "Organizational Techniques for C:M Ratios in Frequency Modulation", Computer Music Journal, 1(4), 1978, pp. 39-45; reprinted in Foundations of Computer Music, C. Roads and J. Strawn (eds.). MIT Press, 1985.
C. Dodge and T. Jerse, Computer Music, 2nd Ed., Schirmer Books, 1997.
F. Richard Moore, Elements of Computer Music, Prentice Hall, 1990.