There is no difference between ML & modern bbls when it comes to how they react to vibration; the primary difference is with the operating pressure and bearing surface of the projectile.
The number of bbl to stock attachment points does have a direct affect on the frequency and amplitude shifts just as does the exterior profile and variations in the bbl wall thickness along with contact or lack thereof with the stock wood including its unique properties; sight type, weight, location and installation method.
For one to sit and calculate a solution of X for an rifle and claim that it is "optimum" is total horse pucky because it will only be "optimum" for a single given set of parameters - in other words the alleged optimum would require specific tolerances on: the bbl from the alloy to the manufacturing; powder burn rate; powder charge; bearing surface of the bullet; bullet alloy; parameters of the stock material and not to mention the ambient environmental conditions like temperature and atmospheric pressure.
Starting from the beginning, the first condition you have to deal with is vibration and ignoring all external factors and causes, let's assume that it starts when the sear releases (fact of the matter is that ambient noise, thermal energy, air movement, gravitational disruptions, electromagnetic variations and so forth have already set-up vibration within the gun long before the sear trips anyway). As soon as the lock components begin moving, they are transmitting vibration through the action, stock and barrel. Firing pin contact with the primer or flint contact with the frizzen starts a whole new set of vibrations as does ignition of the primer/pan charge, ignition of the main charge, first movement of the projectile and subsequent travel of the projectile down the bore. Then you have to figure in the annular pressure waves created as the primer and main powder charges ignite and burn along with the corresponding variations of the burn process in relation to the friction of the projectile that cause a multitude of individual annular pressure waves. There is no one single event but rather an almost infinite number of individual events that combine and cancel each other based simply on random chance as to where they happen to meet/pass and at what amplitude they happen to be at when they pass.
The easiest way to describe this is with a guitar. If you pluck the string at various points between the anchors, you'll immediately note the change in tone from low to high or high to low depending up where and when you initiate the next set of vibrations. There is no difference between that string and a gun barrel because everything changes with every shot and also changes during the duration of a single shot. The talk of "optimum" and "harmonics" is not a science but rather tuning that based on averages just as tuning a musical instrument. For any given "note" on say a piano, the exact frequency of that particular note is going to vary each and every time the string is struck with the hammer because it is impossible to repeatedly create the exact same strike. The same applies to a gun because despite how carefully you weigh each powder charge or select each component, every single one of them is slightly different and thus it becomes nothing more than playing a game of averages so the best thing you can do is completely forget about the word "optimum" as it relates to barrel length or profile concerning vibration (the exception is optimum length for velocity vs. burn time and so forth but not concerning accuracy as related to vibration)
When it comes to the alloy used for the barrel itself, yes, every alloy will show distinct differences in the manner in which they react to vibration with harder alloys being better conductors of vibration than softer alloys. The hardness of the alloy also affects the average vibration amplitude with harder alloys maintaining a higher residual amplitude than softer alloys. Now, before the "soft alloy" folks start with the "see, I told you so's" ... the softer the alloy, the higher the amplitude of the annular pressure waves and the lower the amount of pressure that is required to reach the same amplitude for barrels with a similar wall thickness and profile so forget the hard vs. soft argument because it becomes a moot point when examining the complete picture. While we're on this, let's dispel the rumor that leaded alloys are better - leaded alloys still transmit vibration and annular pressure waves and the amount of lead in the alloy isn't sufficient to cause a significant change in anything other than how easily the alloy cuts with tooling - the extremely low lead content acts as nothing more than a cutting edge lubricant. Thus, if you look at something like a metal bar xylophone, you will see the mfg's tend to use the same alloy but vary the length and width of the bars to achieve the desired average tone for the given bar based upon the average impact force, duration and bounce produced by the mallet. In effect, if one so chose to do so, the perceived tone can be duplicated using bars of the same size and shape but being constructed from different alloys and utilizing different contact and impact points.
Yes, there are both fixed and variable "node points" on any barrel; fixed being a stock/sight anchor point and variable being those set up by the particular parameters of the induced vibration and these points vary not only with the load but from shot to shot and in correlation to the ambient conditions at the time the shot is fired.
The stock itself has its own unique affect of barrel response as well because five stocks cut from the same tree will have five different sets of parameters to them just as cutting a thousand barrels of the same type to exacting parameter will still show a thousand different harmonic profiles with each one being unique unto itself.
Another point that is often misstated, particularly in full-stock ML's, is the barrel length to stock-force relationship. If one chooses to say the "optimum" barrel length is A yet another claims the optimum barrel length is B, one must consider the facts surrounding such claims and base the conclusion of averages on the individual construction and operational parameters. If you anchor both ends of a solid round bar of say 44" and apply 2 pounds pressure to the middle of said bar, an accurate dial indicator will show a given amount of deflection opposite the direction of force. If you then shorten that same bar to say 36" and apply the same 2 pounds of force to the center, the amount of deflection will be reduced showing a deflection reduction relationship corresponding to the reduction in length. It is always interesting to watch the reactions when this is demonstrated and people see just how little force is necessary to cause considerable deflection even in what most would consider a "heavy" barrel.
Thus, as it applies to a long rifle with say a 44" barrel and you hear complaints of it shooting way off the assumed center of profile, the first thing that must be determined is if it is actually fault of the barrel or fault of the stock. Just one ounce of force caused by the stock on the center of a 44" long bbl is enough to move the POI considerably 50 or 100 yards downrange. Static force can also change in relation to the amount of binding that is taking place between the stock & bbl in that following the stresses applied during loading or firing can cause considerable variations in the amount of force being applied by the stock based simply upon where the stock just so happens to bind.
It doesn't matter it it's a modern gun or a traditional gun, the variations in dynamic and static forces applied to the bbl are in a constant state of fluctuation. The more one can average out the external influences, the easier it is to find a load combination that is willing play within the dynamic response averages. There are huge differences created every time you change any of external influences and external influences means everything from the stock to barrel interaction to the ambient atmospheric condition to the reaction of the particular load being fired. Also, the external influences also include the amount of induced and applied thermal energy which can have a profound affect on even the best piece of stress-relieved steel because the stresses being applied are dynamic and not residual static stress.
Purely from the harmonics standpoint, the particular length of the barrel does not matter because it is the parameters of the load and barrel/action interaction with the stock that become the two primary determining factors after discounting ambient external conditions not directly related to the gun itself. Testing done on a Winchester M70 featherweight chambered in .270win, a standard profile Remington M700 chambered in .300win mag and a standard profile Winchester M70 chambered in .30-'06 all produced a similar corresponding pattern of barrel movement based upon the primary determining factors. Although there were distinct differences in both the amplitude and frequency movement associated with different rifles, the basic pattern of movement associated with similar mechanical changes made to each rifle proved to be predictable to the point of becoming boring.
The amplitude and frequency shifts produced very similar patterns with the action being anchored to the stock without any bedding as they did when the action was bedded with a given compound provided the barrel remained free-floating. Despite the distinct differences in barrel profiles, bedding the barrel with a hard-setting compound produced increases in frequency of the vibrations just as did bedding the action with a hard-setting compound but the difference in amplitude direction changes proportionally when the barrel itself was bedded. Likewise, bedding only the action with a the vibration absorbing compound corresponded to reduction in both the amplitude and frequency of vibrations within the free-floated barrel.
When it comes to a full-stock like a long rifle, the almost infinite number of variables related to the barrel / stock interaction all have a direct affect on the actual consistency of accuracy however ML barrels are not exempt from physics. Two distinct advantages one sees in a tradition long rifle is that more often than not, the projectile of choice is the round ball that has the shortest bearing surface of any projectile. Subsequently, the low frequency black powder propellant combined with the PRB produces operating pressures that are considerably lower than what we commonly see in a centerfire rifle and thus the amplitude of resultant annular pressure waves is also reduced. Another benefit is the considerably higher barrel mass as compared with a modern CF rifle. However, despite the benefits of the typical long rifle, the barrel is still susceptible to both vibration and annular pressure waves that will affect accuracy.
If we discount the affects of annular pressure waves and vibration within the long rifle barrel, why then do we talk about the "sweet spot" load? Answer is simple, the "sweet spot" is that point at which the particular load ejects the ball from the muzzle at the correct time when the muzzle is in roughly the same position within the same plane - yep, same thing the CF shooters do. It is the inherent advantages of the typical long rifle design and materials that creates the sweet spot that is normally quite wide. Also, if one works with enough loads, a pattern can be established with most any long rifle where it will have two or more sweet spots for the same or different powder granulations. The number and width of the sweet spots correspond directly to all the combined factors. A 15/16" OD .50 barrel 28" long will often have numerous rather wide sweet spots just as a swamped barrel that is 42" long thus proving barrel length and profile is not a reliable factor in determining the given results of actual live firing.
In reference to your last post, if the stock is using the barrel as a reinforcement, the subsequent stress induced into the barrel changes the parameters of how that barrel will react to the dynamics of the vibration and annular pressure waves associated with firing the shot. Thus is why an independently stabilized stock combined with a set pre-load being applied to the barrel increases the consistency of barrel response. Consistency is only half the battle because in addition to consistency, the more the amplitude and frequency of the response can be reduced, the less limited the sweet spot becomes.
