I'd like to keep this short, but probably won't. I submit the subject of gyroscopic stability in spin stabilized projectiles is poorly understood by most folks. I shall not endeavor to bestow the full dialog appropriate to the subject as it would make a book. Several in fact. Highlights follow and I will be happy to discuss any aspect in following posts if there are questions.
It may sound terribly simplistic, but the purpose of stabilizing bullets and ball is consistency. Consistency is the basis of accuracy, or the foundation for doing the same thing repeatedly with reasonable expectation of the same result. To that end, see the definitions below:
Center of Aerodynamic Pressure (CP): The sum of all atmospheric forces applied to the bullet while in flight, measured on the axis of rotation.
Center of Gravity (CG): The mass balance point of a bullet on the axis of rotation.
Pitching Moment: The linear displacement on the axis of rotation between the CP and CG of a projectile.
The CG of conical bullets is, in nearly all cases, aft of the CP. This is the reason that conical bullets with no (smoothbore) or relatively slow rotational speed, will tumble when they leave the barrel. One exception is a Forster slug, such as commonly loaded for shotguns. Other similar designs may be dynamically stable, but for the most part, conical bullets do not have dynamic stability.
Round balls have, for the purpose of this discussion, a co-located CP and CG. This assumes the ball does not deform when shot, and in the case of smooth bore guns and hard alloy, it is the case. Without such displacement there is reason to wonder why muzzle loaders using round balls have slow twist rifling in the barrels. Sometime after events unfolded at Concorde, the British discovered an elementary fact regarding such benefit, much to their chagrin.
British forces used smoothbore muskets and to some extent, subcaliber patch and ball to facilitate rapid reloading with heavily fouled bores. Given their style of warfare it made sense. Their volley fire was sufficient when facing ranks of opposing forces armed in similar fashion, even though accurate fire by individuals was problematic beyond 50 yards or so. The upstart Colonials however, did not have the manpower to waste in this fashion, and to large degree had previously adopted rifled barrels for many sporting arms. You see, someone over on the Continent had discovered that grooves in the barrel greatly assisted accuracy. Even the initial application which used straight grooves helped a great deal. The reason was simple. If one set aside the need for rapid reloading, which might be appropriate for hunters, it was discovered that the ball no longer left the muzzle with random rotations on an infinite varieties of axis. Spheres in rotation will generate aerodynamic lift on one side, thus causing the ball to swerve in that direction. Vary the axis from one shot to the next and you have random dispersion...or the accuracy of a Brown Bess. The Colonials were hunters first, warriors second. By the time of the Revolution, it had been discovered that helical rifling was even better than straight and the Long Rifles of the day were fully capable of striking man sized targets consistently at ranges beyond 100 yards. The Colonial Mothers did not raise fools and their sons understood standoff was a good thing. It was expedient to let others die for their country and although the British thought this very tawdry, they let their sense of propriety lead them to defeat.
There is a simple reason for the development of conical bullets. They are more streamlined and they carry more weight or mass per caliber as compared to a round ball. Streamlined means they carry energy further (higher BC) and weight means they penetrate more (SD). There is some debate of who first ventured into this realm, many think of a Frenchman named Claude Etienne Minie'. It is pronounced "Men-nay". He was not the first, but his bullet design caught on for the armies of the world and thus he is known for this. His bullets were pointed and hollow based designs. Initially there was a plug in the bullet base intended to obturate the sub-caliber bullet (quick to load) to ensure it would engage the rifling. Since the moment in this design was very short, it took little if any change in twist to accommodate this bullet and the armies of Europe and subsequently America, had the first practical stabilized conical bullets for their infantry forces. It's that standoff thing again. Somewhere in that rapid evolution, some efficiency expert found the base plug was not necessary. Another struck upon the idea of paper cartridges. The freight train was rollin'........but there was much to learn.
Now we get to the meat of it. A flurry of experimentation around the world started the formal study of ballistics. There are three disciplines, Interior, Exterior and Terminal. Different folks studied different aspects in the mad race to military supremacy. In this case, we talk about exterior ballistics, or that point from the rifle's muzzle to target impact.
The aforementioned benefits of conical bullets guaranteed their permanence in history. Round balls do not have endearing traits from an aerodynamic standpoint. Conical bullets do. Case closed. However, to harness the benefit of conical bullets it was necessary to develop the science and that took a lot of work. In the mid 1800's a fellow named James Forsyth was in the thick of this and wrote a book entitled "The Sporting Rifle and its Projectiles". It is available through Google Books online. He was not the discoverer of most of what he reports but was one of the first to make the information widely available via the press. The book is an interesting reference and is mostly valid by today's standard. And you thought our generation was the fount of all knowledge huh? Read the book and find out otherwise.
Conical bullets are stabilized by application of angular momentum, that being imparted by rifling in the barrel. It requires less that 1/2% of the energy produced by the powder charge and only requires that the momentum be sufficient to overcome the overturning or pitching moments created by the difference between CP and CG. Stability is defined by what is known as gyroscopic stability factor (Sg) and in theory a value of 1.0 is unity and the projectile will achieve stability. In the practical world, it requires a minimum Sg of 1.1 and more realistically about Sg 1.3. I will point out here that bullet weight has absolutely nothing to do with the twist required to stabilize a bullet. Bullet Length is the litmus. Recall that round balls have co-located CG/CP. When the form is modified to a conical shape the displacement of CP is forward (with a few exceptions noted) The further the form is stretched, the more angular momentum is required to overcome the pitching moments, ergo, you need a quicker twist to achieve the required Sg. A .25 caliber bullet 1.25" long will require a 7" twist. A DG bullet on approximately the same length requires much less, perhaps a 16" twist. The reason is found in the larger radius of the bullet which provides for more velocity at the circumference and thus more angular momentum. Gyroscopes and bullets have a common aspect, that being what is known as rigidity in space. It is this momentum force that resists upset by outside forces such as aerodynamic forces, gnats and global warming. (sorry....) When a bullet or gyroscope is acted upon by an outside force it will precess 90* to the axis of rotation in the direction or rotation. Every time. The degree to which it precesses is regulated by the Sg. When a bullet is upset in this manner, the precession occurs in the form of nutations, or a wobbling motion that will, given sufficient Sg, stabilize or null in a new rotational axis. Bullets go through this process when affected by wind, imbalance, gravity and any other of the myriad of forces experienced in flight. That includes gnats. A bullet may experience as many as 3-5 modes of precession. They WILL experience a minimum of two. Save that discussion for later if there is interest.
It is important that bullets be properly stabilized through the range of their intended application. The aerodynamic forces at play during a bullet's flight are variable. The highest drag is found at Mach 1 and this is where the .22 RF HV bullet lives. Secondarily, the flight regime for the .22 RF and most black powder bullets is in the transonic range of Mach .7-1.3. This is the place where the pitching moments are strongest for RF bullets. For those asking the obvious question, in absolute terms, drag is higher at higher velocities but most conical forms exhibit a curious reduction in coefficient of drag as Mach number increases and the absolute increase in drag is thus not of linear nature. So, where do we go from here?
An English fellow named Greenhill is credited with a formula which defines the required twist rate for a given length of bullet. You can read a bit more here:
The formula's age has little to do with its applicability. Those who twiddle with high velocity guns will opine a different constant is appropriate for their bullets and that is true. Others dealing with very low velocity projectiles will use yet another constant. The constants change when one has a particular agenda in mind. The basic formula is not unlike the way gun builders apply more or less standard twist rates for particular chamberings. It works for practical purposes. The formula shows the path to gyroscopic stability for conical bullets. If you need to stabilize a bullet of X length, Y rate of twist is required.
In the world of the .22 RF there are many bullet designs and few twist rates. The 1:20 will stabilize bullets typical of the .22 Short and 1:16 will do the same for Long Rifle bullets...or shorter. This leads to another subject: what twist do I need and why do they only make a certain twist for .22s today? What twist you need is defined by the longest bullet you intend to shoot in your rifle. The day's standard for production .22s is 1:16". It will stabilize all .22 RF bullets with the possible exception of the Aguila 60 grain SS round. I say possible because some report acceptable performance with the 16" twist. I say also, there are variables at play and you never quite know until you try it. As an example, I have a .358 Win. w/ a 16" twist that stabilizes 250 grain spitzers very well. It is a rare exception, most require at least a 14" twist for that and some opt for 12" twist rates. So, is there any adverse effect found in "over stabilizing" a bullet, such as using a 16" twist for shooting .22 shorts? While a bench rest competitor may be able to ascertain the answer to that, I cannot for short range work, ie. less that 600 yards. The theory is that rifling imparts deformation on bullets, and the faster the twist, the greater the deformation. I will go a step further and state this is a fact. What is difficult to evaluate is just how much impact this has. Hardcore CF bench competitors will shoot bullets and barrels matched to each other, with the minimum twist required to stabilize the bullet from muzzle to target and not an inch further. If they do their calculus for competition in Denver, Co. their gun will not shoot for sour grapes at sea level. I don't know the converse to be true. When science does not provide solid answers, superstition steps to the fore. So far as practical application, the difference between a 16" and 20" twist for shooting shorts is not apparent to me. My T/C carbine shoots consistently in the MOA range out to 50 yards with a 16" twist. So far as I'm concerned, case closed. A friend has a Rem. Model 24 (chambered for shorts only) that shoots nearly as well and I believe it has a 20" twist. He is a pragmatic fella...and cares not a whit what the twist rate is.
A word of caution. It is a common misconception that one may compensate for slow twist by using higher velocity. The concept is valid only to very small degree. Very small. On the other hand, a great deal of benefit is gained by quickening the twist rate a small bit. No, I'm not going to do the math, but you can. A 12" twist at 3000 fps vs. a 10" twist at 3000 fps....pull out your calculator and do the math to determine RPM. Once that is done, figure the difference between a 3200 fps bullet w/12" twist and compare to the 3000 fps bullet at 10" twist. I think after you've done this exercise you'll get over the higher velocity theory of bullet stability. It works, but the gain is VERY small.
What all that means is that it won't hurt to "over stabilize" a bullet, but it will be a small disaster to do it the other way around. If you want to walk off the beaten path, do a lot of homework before you order that barrel with an odd twist rate. The large gun manufacturers do what they do to achieve certain economies of scale and production in order to make a profit. Their approach may not always be the absolute best for a given application, but they do pretty well for a range of applications. They do well and you might do better...but you'll have to do a lot of research, or rely on another's expertise if life without guard rails appeals to you.
Well, that constitutes a scratching of the surface without a lot of painful math. Accept it or do your own research. If you want further discussion on the subject, fire away with the questions. I might not know the answer, or you might not agree with the answer. It's a DEEP subject though, this thing called Exterior Ballistics; Subchapter B: Spin Stabilized Projectiles.......