FM is both incredibly complex and shockingly simple.
Complex because it’s the literal opposite of the subtractive synthesis we’re used to. Instead of starting with harmonically dense waveforms and removing harmonics with filters, FM takes the most harmonically simple wave form (sine), and adds harmonics to it with modulation. If we try to look at it from the subtractive standpoint of the waveforms generated (AKA: the time domain, AKA: with an oscilloscope), it looks crazy!
But if we look at FM the way it wants to be looked at, from a harmonic perspective (AKA: the frequency domain, AKA: with a spectrograph), it’s shockingly simple. We can learn all there is to know about with three examples:
A sine wave is heard (at audio rates) as a single frequency:
This is a C4:
If we modulate the frequency of a sine wave with another sine wave, we add harmonics.
The level of those harmonics are determined by the depth of the modulation, and the placement of those harmonics are determined by the ratio of the modulator to the carrier.
A 1:1 ratio adds every harmonic in the series. With a fundamental of C4, we see harmonics for C5, G5, C6, E6, and G6:
A 2:1 ratio adds every second harmonic in the series. With a fundamental of C4, we see harmonics for G5, E6, A#6, and D7:
A 3:1 ratio adds every third — C6, A#6, E7, G#7, etc.
What about those slightly smaller peaks between every third??
These are aliasing-like “reflections” of higher harmonics — C5 in addition to C6, E6 in addition to E7, etc. These get more pronounced as we reach higher into the harmonic series with higher ratios. And the higher into the harmonic series we go, the more weird intervals we find. So it’s usually not worth going above a ratio of 5:1 unless we’re looking to make some crazy sounds.
These rules are recursive.
We can modulate modulators. What happens when we do? The exact same thing as when we modulate a carrier — it adds harmonics depending on its ratio and depth. But instead of adding them to the fundamental of the carrier, it adds them around the harmonics the the modulator is adding to the carrier, essentially filling in gaps.
So we can take our 3:1 modulator and modulate it with a 2:1 modulator. The result will be like adding every 2nd harmonic around every third harmonic:
Here, the green is a 3:1 mod of the carrier sine wave. The orange outline is a 2:1 mod of 3:1 mod of the carrier sine wave. We can see that every third harmonic has had a second harmonic added around it.
That’s it. That’s all there is to FM. Of course, by chaining modulators in different combinations and sweeping mod depth over time with envelopes we can make infinite soundscapes. But all follow these simple rules. So now, when looking at an FM patch, you ought to be able to close your eyes and picture exactly what its spectrogram looks like.