I am currently working on FM synthesis and while I do understand the roles of carrier frequency, modulator frequency and modulator amplitude, I've been having problems with understanding how lots of software (I suppose also hardware) synthesizers handle these parameters.
For example, here is a basic FM setup in NI FM8:
In this case, operator E modulates operator F "fully", so on the scale of 0 to 100 provided by FM8, the FM modulation level/amount is set to 100. My question is, what does this level parameter actually mean in terms of modulator amplitude, behind the scenes? From my understanding, the modulator amplitude controls the variation in frequency of the carrier signal. However, when I set the FM level of operator E to 100 as shown here, I can hear and see that this doesn't mean that operator E then has an amplitude of 100 and consequently modulates the frequency of operator F by 100 Hz. From some approximations I made the modulator amplitude is actually in the range of about 6000 when I play a 440 Hz sine wave on my keyboard.
Similar situation with Ableton's Operator synth:
Here there's no percentage or modulation amount from 0 to 100 as in FM8's case, but rather you set the level of the modulator in dB, from 0 to -inf. From what I understand, if the dB level shown here would be the modulator amplitude, having the knob at 0.0 dB would mean the modulator amplitude would be at 1 and thus the change in frequency of the carrier would be ± 1, but that's not the case. Here too, for a 440 Hz tone the modulator amplitude seems to be around 6000 (same as having the level at 100 in FM8's case). I still don't know what the actual change in frequency of the carrier is given these parameters in Ableton's interface.
I know of the modulation index and it being defined as β = modulator amplitude / modulator frequency, so I thought maybe in the case of FM8, the background calculation is modulator amplitude = β * modulator frequency and the level from 0 - 100 actually controls the value of beta, but then for 440 Hz that would mean the modulator amplitude is 44000 at a level of 100, and that doesn't make sense either.
It seems I'm missing some knowledge that the creators of FM8 and Ableton's Operator share and use.
So, my question is how do synth makers control the modulator amplitude and how does the interface of the synth and the controls from 0 to 100 change the modulator amplitude in the background?
I would be very grateful for any help or insight.