About | Home | BESSIE | Concerts

Launch Bessie

• Info
• History of FM
• Sound Basics
• FM & Vibrato
• Unit Generators
• Simple FM Theory
• Modeling Timbres
• Amplitude Envelope
• FM Parameters
• Bibliography
• Acknowledgements
• Future Bessie

BESSEL FUNCTIONS

Bessel Functions are mathematical functions which can be graphed as you see on the right. Look at the top, left most graph (captioned: "Zeroth Order Bessel Function") first.

Notice we have an X and Y axis. The X axis represents INDEX values ranging from zero to 25 (we just chose that range because it seemed the most practical). The Y axis represents relative amplitude of the "Kth" order sideband. Phew, what does this mean ?

Remember that FM causes spectral components to be created which center around F(c). These components are spaced at whole number integers times F(m).

Remember our formula for the location of sidebands: F(c) +/- K*F(m)

In this formula, K represents integers ranging from zero on up
(0, 1, 2, 3, 4, etc.). Since K*F(m) is both added to AND subtracted
from F(c), each value of K is responsible for TWO additional partial components being added to our sound (except for K=0, in which case the result is F(c) itself !). So it's easier to refer to both components as the "Kth" components (i.e. when K=1,
we have the 1st set of side bands added above and below F(c)).
We can now refer to the Kth set of components of our sound.

It turns out that the relative amplitudes of the "Kth" components are controlled by various "orders" of Bessel Functions, or the "Kth" order of Bessel Functions. Thus, in our first graph above on the left, the "Zeroth Order of Bessel Function" refers to what amplitude you will get when K=0 in our formula: F(c) +/- K*F(m).

Now when K=0, you really don't get ANY "sideband" components, in fact, you get only the Carrier itself. However, as K increases, these graphs become much more meaningful !

Let's take one more example which might be more clear.

Look at the graph captioned "1st Order Bessel Function".
This chart should be used to determine what the amplitudes of the side bands will be when K=1. This is really our first set of side bands that appear to the left and right of F(c) on the frequency axis. We can see that the STRONGEST amplitude that can be obtained for this first set of side bands is about .6, and that's when the INDEX VALUE (X axis) is set to about 2.

This means that when INDEX = 2, the relative amplitude of the two new spectral components: F(c) + (1)*F(m) AND F(c) - (1)*F(m) will be about 60% of the maximum possible value, and thus will contribute greatly to coloring the sound.

HEY, WHAT ABOUT INDEX VALUES THAT CAUSE A SIDEBAND TO HAVE A NEGATIVE VALUE ?????

Ah, now this part is a bit tricky, but it's the final step in understanding simple FM.

Let's go back to the first set of side bands, when K = 1. We saw that when the Index was set to a value of 2, we got a maximum relative amplitude for our first order side bands of .6 (or 60%). But what if the INDEX had been set to 5 ? Then what ?

Looking again at the graph above ("1st Order Bessel Function" ), if you trace along the X axis to an INDEX value of 5, you'll see that the graph goes into the negative region of the X axis. In fact, it dips below -.2 ! What the heck is a negative relative amplitude ?

Well, it turns out not to be too important. In fact, FM theory works out that all negative relative amplitudes can be read as positive values, so with our INDEX set at 5, we arrive at an answer of -.2 or lower for the relative amplitude of our spectral components. We simply "ignore" the negative sign and say that the relative amplitudes will be a + .2 or about 20% of max. The ignoring of negative values, and thinking of them as positive values is called: "taking the ABSOLUTE VALUE of a number. This is a common procedure in mathematics.

Generally speaking, we can take the "absolute value" of all the values on the Y axis of the Bessel charts. Sometimes, however, if you have a rather high INDEX value set, or are using complex FM (as opposed to our "simple" FM), then these negative values may just cancel out other positive values of the same frequency being caused by some other set of side bands. For now, just take the ABSOLUTE VALUE of the relative amplitudes for the various Kth order of side bands and use BESSIE at various INDEX settings, while referring to the graphs above, and try and GUESS what the sounds will be like. You can quickly develop a strong intuition as to how certain FM parameter settings in BESSIE will affect the resultant sound. Experiment, guess and HAVE FUN !

If you're still reading this ..... here's a little bonus for you:
see the graph at the bottom of the charts above ? The one
captioned "0-25th Order Bessel Functions...etc." ? This is a three dimensional graph that shows ALL values of INDEX from zero through 25, and ALL orders of side bands from zero through 20 and ALL relative amplitudes of those side bands. This chart is really too small to read accurately, but it's fun just to see what one of nature's functions looks like in 3D !