FMSynthesis

Quora Answer : What is the basic formula for FM synthesis?

May 18, 2020

Seems that anon gave you a mathematical formula.

Is that what you were looking for?

I can give you the basic idea ... in very simple layman's terms.

Normal synthesis, you have an oscillator, producing some kind of wave. And then you process, filter that wave in some way.

FM synthesis, you get an ordinary wave. Then you get a second oscillator to change the frequency of that wave. In other words, instead of that first wave being at a constant frequency or pitch, you use the second wave, to wobble the frequency of the first wave up and down a bit.

If your second wave is very slow, the effect is like a police or ambulance siren going wee-oww wee-ow up and down. But if you speed the second wave up until it gets closer to an audio frequency, all the ups and downs blur into each other and you end up with what sounds like a new wave at a single pitch with a more complex timbre or sonic texture.

This can sound completely harsh and "unmusical" or it can sound like, say a metallic instrument like a gong, bell or glockenspiel, or a piano string etc.

A typical trick, when making an instrument sound with FM synthesis, is to define the second modulating wave's frequency in some kind of ratio to the first, the "carrier" wave's frequency. In other words, as you change the pitch of the underlying wave, you also change the frequency of the modulating wave.

This can cause even more chaos, but when done well, can lead to sweeter more "musical" sounding synthesized tones.

So ... FM synthesis has two oscillators, one modulating the frequency of the other. Which is where the name comes from FM synthesis stands for "Frequency Modulation" synthesis.

The formula for calculating the sound, then, will include parameters for both the frequency of the carrier oscillator and the frequency of the modulating oscillator. Or it may, instead, include some kind of ratio to calculate the frequency of the modulating oscillator from the frequency of the carrier.

That's what you see in the first formula in anon's answer.

fc is the frequency of the carrier. And fm is the frequency of the modulator. But you'll notice there is no fm. Instead, the value of y(t) ie. the output at any time t, is a function of a sin of both the carrier frequency AND second term which is the change in that carrier frequency which is somehow calculated from that frequency and that term m, which I'm guessing represents a way to define the ratio between the carrier and the modulating wave.

Of course, I have to say that the way people write maths is appallingly confusing. Indeed, probably all those formulae say similar things. But they don't spell out what all the letters mean, in fact they even explain the meaning of variable names like I which aren't even in the formula. (Why, maths people, why are you so sloppy about this kind of thing?)

I'm not a maths person, so I couldn't really read those formulae. Despite knowing what they are meant to be saying.