Introduction to Computer Music: Volume One

6. Principles of Audio-rate Frequency Modulation | page 12

Finding the 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.

  3:2 sidebands 1:2 sidebands
n=0 300 100
n=1 100, 500 |-100|, 300
n=2 |-100|, 700 |-300|, 500
n=3 |-300|, 900 |-500|, 700

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 page at: http://www.sfu.ca/~truax/fmtut.html

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