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FM Index

We already know that FM is basically a computer controlling Vibrato rate and depth (again, thinking of our basic pitch or center frequency as [F(c)] and the Unit Generator that is affecting or modulating its frequency as [F(m)]. In fact, let's now use the term Carrier Frequency for F(c) and Modulator Frequency for F(m). This will place us more in step with the theory of simple FM Synthesis.

There is yet one more very important variable to understand for FM and that's the concept of INDEX.

What is INDEX ? You have the Carrier Frequency F(c) being modulated by the Modulator Frequency F(m), right ? Well, we can know how fast F(c) is being modulated above and below the given frequency by simply knowing the value of F(m), but just HOW FAR above and below F(c) is the modulation occurring ? In other words, we know the rate of the modulation but not the "depth" or "delta F" (change in frequency) of the modulation.

Let's say that F(c) is set to 440 Hz (a common tuning note for orchestras !). If F(m) was first adding 50 Hz and then subtracting 50 Hz, modulating F(c) between ......say.......... 390 Hz (440 - 50) and 490 Hz (440 + 50) then we have a delta F of +/- 50 Hz. Now if we get into large numbers, say F(m) were 3,482 Hz and delta F was +/- 6,964 Hz, things start getting pretty hairy to add and subtract.... so it turns out that it's much easier to express delta F in relation to F(c). Using the last numbers I gave:
rather than expressing delta F as
+/- 6,964 Hz, I compare it to F(m) like so:

delta F / F(m) = INDEX
6,964 / 3,482 = 2

So in this case we have an Index of 2.

It's much easier to conceptualize and talk about FM INDEX than the actual value of delta F, this way you never really have to add and subtract frequencies. You simply discuss the RELATIONSHIPS between frequencies. This gives you a much neater way of discussing FM Synthesis.

The precise value of the INDEX has a strong affect on FM sounds. Generally speaking, higher values of INDEX result in more partials (frequencies) in the sound and this means it will generally sound BRIGHTER to you. Very high values of INDEX may even cause "foldover" or "aliasing", which is heard to add low frequency components to the sound as well as high frequency components. Very high values of INDEX (say 10 and up) can be used to generate large complex sounds with lots of partials.... all over the spectrum.

The "foldover" effect might be fun, but if you are trying to imitate a known acoustic musical instrument (like a bassoon for instance), then you don't want to have too high an index and risk adding low frequencies into your sound.
Try running BESSIE with various values of INDEX . Notice that as you increase the index, the sound gets "brighter".

What happens if you select an index
of 0 ? (Try it !).

[An index of 0 means there is no frequency modulation going on so you should only get the carrier tone by itself ---- a sine tone].

What's the maximum Index you can choose with BESSIE ? Do you hear the effects of "alliasing" with any of the high index settings ? Decide for yourself which values of INDEX are musically useful and which values allow you to better immitate a musical instrument. Lower values of Index will be more useful when trying to immitate acoustic instruments (like a bassoon for example). On the other hand, perhaps you are more interested in discovering NEW SOUNDS ? In that case, large values of INDEX might be more fun.

No matter what frequency you choose for F(c), BESSIE will track it and alter delta F so that the INDEX remains constant (the relationship remains constant and that's the most important thing if you want predictable FM results).

How do you know how many spectral components will be generated by FM synthesis ? You can approximate it simply by adding 2 to the INDEX value. Example: if INDEX = 1, then you will generate approximately 3 sets of partials (3 above F(c) and 3 Below).


Just exactly where are these new components generated ? That's simple ! Sidebands (or spectral components) generally appear above and below F(c). They occur at intervals of [K * F(m)] above and below F(c) (where K are integer values from zero on up).

If you read other articles or books on FM Synthesis, you will see this formula for the location of spectral components: Fc +/- K * F(m). (See the diagram above)


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