DSP Robotics Support

When using multi-sine setup, we can create intervals or chords (3-tone or more). Now - I'm trying to find some formulas, that allow to calculate a vibrations/beating, that happens in chords and intervals. There is a lot of data in the internet about ratios and tunings in general, but I can't find anything useful on that slight harmonic beating (not "binaural beating", when two freqs are very close together). Some parts I have figured out experimental, but still don't know if there is a way to generalize it. Harmonic series and "fractions" are involved.

On an example.

1) Let say, that you have an interval of two tones. 100Hz and 204Hz. If you see the waveform of that mix, you will notice, that there is 4Hz beating (second tone differs by 4Hz from H2 harmonics of the first one).

2) Let say that you have a CEG chord of three tones. To start with - 200Hz, 250Hz and 300Hz. You could say, that this is "harmonic chord" of CEG (1/1.25/1.5 ratios) because it rather does not vibrates. But if you shift the mid tone, let say to 248 or 252Hz, then you get an audible vibration of 2Hz (250Hz is the harmonic zero-point).

My question - how to calculate this stuff more general?
And also for more than 3 tones, and around various harmonic zero-points (various depths of vibration)?

I think I figured out...

There seem to be two things that interfere on that level of my interest. First is something that internet calls "displacement frequencies", and second - interval points from harmonic series.

As for displacement freqs, I found that name following a math formula that emerged from practical experiments. They show something like this. If you have two sine sounds, f1 and f2, then you have a (f1+f2)/2 frequency point, that will vibrate with third f3 sound, close to that point. So if you make f1 = 1000Hz and f2 1200Hz, then mid freq is 1100Hz. If you play a third tone let say f3 = 1101Hz, then you get a 1Hz vibration.

As for intervals - If you have two sounds, f1 and f2, then f2 will produce such vibration at N*f1 points (harmonics; have to figure out to which N it is useful). But... Both, f1 and f2 can be considered as Nth harmonics of something lower else, so now I have to check to which degree it is usefu. I suspect, that 4 harmonics down (which produces 0.25 fractional harmonic steps) will be enough.

These two combined with each other, if they cross, they probably produce even stronger effect, or they change the nature of effect (consonance or something like that). Did I miss or over-complicated something?

Anyway - now it should be easier to create something that calculates these harmonic flows.