I am no mechanical engineer, but spent a career building/working/tweaking/improving NM and sniper rifles, so my experience with barrel harmonics is from the practical side of shooting and testing. This is another area where modern and muzzleloading rifles have much in common.
Over 23 years in that game, we "heard it all" about fluted barrels causing the barrel harmonics to improve and make the barrels more accurate than heavy round barrels. Can't tell you how many different fluted barrels we tried and tested in our super expensive test racks and found it meant nothing to improving accuracy outside of some people THINKING it made the barrels shoot better and gave them more confidence, so they shot better with them. IF fluted barrels were such an accuracy enhancement modification, then EVERYONE in bench resting and high power long range competition of many kinds and sniping would have used them for many years and no heavy round barrels would have been used.
Now modern cartridges produce much higher pressures than does black powder and most, if not all of the black powder substitutes. (I put a qualifier in there because I have not tested every kind of black powder substitute, but if the pressure was much higher than black powder, it would play Heck with percussion nipples and hammers.) You still get the effect of barrel harmonics with black powder guns, but it is not as much as with modern cartridges that have higher pressures.
Oh, the problem mentioned about loosening an action screw on a modern gun supposedly causing a barrel harmonics problem is not an accurate depiction of what is happening. The reason the rifle shoots larger or poorer groups when an action screw is loosened is that the BEDDING of the receiver is not as tight and the receiver "moves" more in the bedding that causes poorer groups.
What can be or is bad about barrel harmonics is when there is or are negative nodes of vibration that act against the wave flow of vibrations down the barrel. To get an idea of a negative node of vibration, imagine throwing a good sized rock into a pond that is calm and smooth. Waves emanate outward from the point the rock splashed into the water. This is sort of like normal barrel vibrations where the waves flow somewhat to very equally down the barrel. However, when the waves in the pond hit something like the post of a dock, you see smaller waves rebounding/bouncing off the dock post and against the larger waves made by the rock. That is a negative node of vibration that upsets the normal and somewhat smooth flow of vibrations down the barrel. How much of a disrupting influence the negative node/s of vibration are to the normal flow of vibrations in a barrel, the more it will hurt accuracy.
OK, so what sets up negative nodes of vibration in a muzzleloading barrel? Normally it could be or is the barrel retaining pins or keys if those pins or keys put a lot of pressure on the barrel in just those spots. However, most pins and keys do not put a lot of pressure on the barrel so the negative nodes are not as disrupting to barrel harmonics. Actually, a wood fore arm of a stock that was not well seasoned when the rifle was assembled, could cause more pressure from the forearm twisting as the stock dries out and set up a worse node of vibration than almost any pin or key could do.
One thing that NM shooters learned about M1903 rifles was it was normally GOOD to have four to six pounds of pressure applied to the barrel by the very front of the wood stock. That pressure actually slightly bent the barrel. It helped the barrel settle back into the same position shot after shot with the relatively thinner G.I. contour barrels. However, with larger and thicker round barrels and glass bedding, it was found it hurt accuracy to have that pressure on the barrel from the front of the stock.
Are any of you are old enough to remember Model 40X .22 cal. "Match" rifles that had two adjustment screws near the front of the stock to put pressure on the barrels to supposedly increase accuracy? We tuned them using tuning forks and it was almost as much "magic" or wishful thinking than anything else. Then after more testing, it was found those rifles were more accurate when those adjustment screws did not touch the barrel at all.
Bottom line for muzzleloaders is the super heavy "Chunk" guns with the enormous barrels and "railroad tie" stocks are the most accurate muzzleloaders with round balls. Part of the reason that Hawken and Plains rifles were so accurate was because of their heavier barrels that allowed more harmonious vibrations in the barrel. When one gets down to Longrifle barrels and only shooting at 100 to 200 yards, there is so little difference in barrel harmonics between a swamped vs straight barrel, it really means nothing to most of us.
Gus
