Harmonics is about tuning the rifle so that the pellet leaves the barrel at the exact same point every time during the firing cycle. If your barrel only moved in 1 plane like a pendulum an accurate rifle would be when the pellet leaves when the pendulum stalls at an end. An inaccurate one would be where the pellet leaves in the middle of the swing where the smallest difference is seen by the shooter.
This is essentially what you're doing and it can be done with weights, altering velocity or picking different pellets. You can alter when your pellet leaves the barrel with velocity or you can change where your barrel will be when this happens, the goal is the same. Get to a point where your barrel is in a "stall" point when the pellet leaves the muzzle.
This explains it better than anyone has on this thread:
Assuming you're not using totally rubbish pellets there really shouldn't be a real reason why they can't group in your rifle. This "my barrel likes Air Arms and not JSB" thing you hear isn't what it seems. It's people have found a pellet that doesn't suit the harmonics of their gun, instead of altering their rifle though most will just switch pellets to find something that suits (until they switch batches and find the same thing again).
The downside of harmonic tuning is you cannot alter anything, if you do you're starting at step 1 again. This means you will have to invest in a large number of pellets of the same batch and you can't mess around adding silencers and strippers without starting over again.
The benefit is unlocking the true potential of your rifle.
Talk of dampening your rifle and designing things with this in mind is a mute point really to 99.999% of us won't ever do this, neither do we need to. I've also seen people (youtubers) describe harmonics as setting up your valve so that you don't get any hammer bounce on firing. This is not the same thing either.
Your rifle is going to move on firing, no amount of set up and tuning will beat physics. Equal and opposite forces and all of that stuff applies to everything. Even if your barrel was perfectly stiff you would experience some harmonic effect through the action and stock, the only difference here is now you've made it harder to tune.
This is my FT gun, the barrel is 16mm. Over this is a 2mm wall stainless steel shroud, over this is a carbon fibre sleeve all epoxied together. I cannot think how you could make an airgun barrel stiffer than this and yet in the stripper you see is a small adjustable cone that can tune the gun that weighs very little. It's amazing the effect it has, it took a couple of afternoons to get it right.
And then I had to do it all again when I altered the stabiliser...
Due to the stiffer barrel the effect was certainly less than I saw with my EV2 and the printed tuner but it was very noticeable. Groups dropped from 20mm at 50 yards to well under 10mm, I do not have the skill level shooting off my knee and outside to do any better than that.
My go to procedure for setting up a rifle now is find a batch of pellets that group the best and have no fliers (most important) and then make the rifle shoot them as well as it can.