Welcome to the universe of FM sounds … really … it’s a new universe, if you are beginning to explore it and it will stay a new universe, even if you have much experience in sound design
To start a FM sound from scratch successfully, it’s very important, to have a basic understanding of how FM-synthesis works in general and what the basic methods are.
But first things first … don’t be scared, because there is no reason … it’s only new and unfamiliar. You are right that there are some “mathematical” rules, but believe me, it’s as easy as any other 101
Second important thing … FM synthesis is hard to force to do everything we might imagine. It’s more the other way around. We have to embrace the possibilities of FM synthesis and accept, what is possible and what is not. As an example. Often it’s hard to create FM sounds, which imitate certain classical mechanical instruments accurately. The reason is that the sound creation process of FM is very different compared to the generation of mechanical vibrations.
One benefit of FM is that we can create sounds that are not only classic synth sounds, based on saw-tooth or square wave oscillators, but also bell- or metal-like, otherworldly, very complex, and beautiful sounds. Last but not least. Even a simple pair of two operators can create much more and much more complex sounds, rather than most subtractive methods.
Now some hints …
- There are two main parameters we can use to define a FM sound. Those are the frequency ratio between modulator and carrier and - to no surprise - the intensity of modulation.
- To find nice harmonic sounds, which are good for melodic use, we can use ratios like: 1/1, 1/2, 1/3, etc. 2/1, 2/2, 2/3, etc. 3/1, 3/2, 3/3 etc. Most digital FM synths support to define those ratios directly like this. There is no frequency-math needed.
- Bellish- or metallic sounds are created by ratios, which do not devide as integers like 2/3 or 3/2. Keep in mind that bellish sounds have a certain ADSR shape. There is a very quick attack, no sustain and some release. A 3/2 ratio wouldn’t sound like a bell, if the envelope is more like a flute would generate, which is having a short attack, a sustain and no release.
- Ratios like 1/2 and 2/1, or 2/3 and 3/2 create quite different sounds. The numbers are not commutative.
- We can modulate the modulation intensity, which will keep the frequency ratio in place, but changes the sound, sometimes significantly.
- We can modulate the frequency ratio too and this will change the sound dramatically.
My advice would be, start by using a pair of operators only, use sine-waves (or triangle-waves, but nothing more complex) and study the influence of frequency ratio and modulation intensity. This would be best supported by an oscilloscope, if at hand, because it helps to see, how the waveshapes are changing.
- There are ratios, which generate saw-tooth or triangle like waveshapes, others create square-wave like shapes, and otheres very unique shapes for scratchy or noisy sounds etc.
- By modulating the modulation intensity, this is how far the FM modulation will change the frequency of the carrier from the original frequency, we can create different harmonic content, which changes according to the modulation depth.
- By modulating the frequency ratio step by step with a sequencer, we can create melodic and un-melodic sounds per step with only one pair of operators in use. This could be a base drum and a high-hat, or a flute and a scratch, or any other combination of sounds.
- A short frequency ratio modulation at the beginning of the sound can easily imitate a noise, like the breath noise of a flute. Or it may sound like a broken bell
Typical digital FM synthesizer have at least one ADSR envelope per operator. This can be used to modulate the frequency ratio and the modulation intensity. I don’t have the Digitone and can’t tell you exactly, how it’s implemented there. The creative use of the envelopes is the entrance door to the magic of FM synthesis.
The so called “algorithms” are combinations of operators, which are building blocks and define, which operator can modulate others and how. The more operators are used, the more complex the sound can become.
Example … if we use two pairs of operators, we have two completely different sound sources, which we can layer in many ways.
Here is a brief introduction by Gordon Reid from SOS magazine:
The complete “Synth Secrets” series is a great read too: