I'll move this thread to ballistics, then try to sort out a few mistaken notions for you.
You are correct that shorter twist is faster because it makes the bullet rotate faster for any given muzzle velocity.
The number of inches given for a full rotation of barrel twist is called the pitch of the twist, though you usually only see the word "pitch" used in ballistic formulas; most folks have shortened "rifling twist pitch" to just "twist", when they speak. Your 9¼" twist makes one complete 360° rotation in the bore for every nine and one quarter inches of barrel length. It does not directly tell you what bullets you can shoot.
All bullets have a minimum twist they need for a given muzzle velocity or they will tumble and keyhole in targets and fail to shoot accurately. They will also do well with a certain amount of extra spin rate up to a point. Thus, each bullet has a range of barrel twists it can work with. As Mike and Kdub said, commercial rifle barrels have twist rates chosen to work with most of the commercial jacket and core bullet weights available in the caliber, tending to be near minimum for the longest and heaviest and near maximum for the lightest and fastest. This is what gives Cvc944 the impression you don’t need to worry about twist rate choice with common jacket and core bullets, but the wrong twist will throw them off, too. An example:
A .222 Remington has a 14" twist barrel as standard. These will shoot most jacketed bullets about 55 grains or lighter with excellent accuracy. Try to shoot an 80 grain match bullet, though, and you are out of luck. It will hit the paper sideways if it doesn't miss it completely. Change to an 8" twist barrel, and the 80's will shoot fine. This is why, in the .223 Remington, rifles for light varmint bullets can have a 12” twist or even the older 14” twist, while match rifles for longer range, like an AR match rifle, will have an 8”, 7 ½”, or even a 7” twist barrel to shoot long heavy bullets. The bullet makers will warn you that you need at least a “x inch” twist for their heavier target bullets. Berger, for example, lists twist rates with his bullets.
What twist does, of course, is determine how fast the bullet spins for any particular muzzle velocity. That rate of spin determines the gyroscopic stability of the bullet. When a bullet flies, air pressure at the nose constantly tries to turn in. It has to spin like a gyroscope to resist being turned. If a bullet doesn't spin fast enough, it will be turned, causing it to tumble and move off its intended trajectory. If it doesn’t miss the target completely, it often strikes sideways or partly sideways in random locations. In paper targets that produces an oblong “keyhole”, instead of a round hole.
If a bullet is made to spin too fast, three things can go wrong: Any tiny imperfection in its weight distribution around the spin axis makes it wobble and the faster it spins, the bigger the wobble; accuracy is deteriorated by wobble. If it is a jacketed bullet and you try to drive it too fast for the twist in your barrel, the rotational acceleration can cause the jacket to slip free of the core and actually spin faster than the core. This is called core stripping and it results in reduced accuracy. A bonded core bullet should not do it nearly as easily as a common pressed core. In the worst case, a bullet can be made to spin so fast it flies apart on its way to the target.
The longer a bullet is, all else staying the same, the more twist it needs. Bullet length is the lever arm for the air pressure trying to turn the bullet. Length is the most critical single factor in choosing twist rate, because a small increase in length can cause a large increase in spin requirement. A solid base wadcutter, being blunt, needs less twist than a pointed bullet the same weight and same density. That is because the pointed shape is longer. For any particular shape, longer bullets are also heavier, so they get to lower muzzle velocities and spin RPM. Thus, the faster twist they require doesn’t normally force them to spin so fast they suffer the bad effects outlined in the last paragraph. In some overbore guns, they may, though.
The lighter a bullet is, length staying the same, the faster the barrel twist it needs. What makes a gyroscope hard to turn is its spinning mass. The lighter the mass, the faster it has to spin to be equally hard to turn. Thus, a less dense bullet, like a Barnes solid, needs
more twist than a denser conventional bullet the same size and shape. Tungsten bullets, for example, are more dense than lead, have very high BC's, and need less spin to be stable. Depleted uranium bullets even less.
Being less dense, a Barnes solid is also longer than same-weight bullets of the conventional jacket and core construction. That fact has the most effect on its need for a faster twist. However, if you have a twist that will stabilize regular jacketed bullets that are about 6% longer than the Barnes bullet, it will then stabilize the Barnes bullet, too. Remember, we are dealing with a twist range here, and not a fixed number.
Finally, atmospheric conditions affect twist requirements. With a given bullet and velocity, air that is more dense pushes harder against the bullet to try to turn it. It takes more spin to resist that. So, any condition that makes air more dense calls for faster barrel twist and vice-versa. Lower altitude, lower temperature, and lower humidity all call for faster barrel twist. Higher altitude, temperature, and humidity all call for slower twist. The military has sometimes used faster twists than commercial makers (10” instead of 12” for .30-06, for example) for same size and weight bullets, just to be sure they would still be stable in extreme climate conditions.
How much twist do you need? Ballisticians speak in terms of the gyroscopic stability factor, G.S., or more often just
s. This is a number specific to your bullet, figured from its physical shape, weight, spin rate, and velocity. It is calculated such that when
s=1.0 or higher, the bullet is stable in flight. Anytime
s is less than one, the bullet is unstable and will tumble. How big
s can be without causing too much wobble depends on the opinion of the ballisticians consulted, but a Sierra ballistician told me he sees best accuracy with their bullets comes when
s=1.3 to 3.0, IIRC?
That information forms the basis for a recommended twist range. A twist that gives the longest and heaviest bullets you will shoot nothing less than
s=1.3 and the shortest and lightest no more than
s=3.0 should have you good to go. If you add in your extremes of weather conditions (coldest, driest, highest barometer reading for the long bullets, vice versa for the short bullet) it will necessarily narrow the range of choices.
If you want to pick an optimal twist rate for a set of atmospheric conditions, Harold Vaughn recommends s=1.4 as optimal. Don Miller recommends s=1.5 as optimal. Lower numbers don’t let a bullet recover from in-barrel tilt and bullet jump as quickly in flight, affecting short range accuracy adversely. Faster numbers get more wobble from imperfections in the mass symmetry of the bullets, affecting accuracy at all ranges. The Vaughn and Miller numbers are where they believe the best compromise lies. I believe them both, so I split the difference and use
s=1.45.
The formula that Old Grump linked you to is the Greenhill formula, devised by George Greenhill in the late 1800’s for use with artillery shells. That it happens also to give working numbers for many supersonic bullets is a happy coincidence, but it is not really letting you optimize twist by providing an
s value.
At the other extreme are Robert McCoy’s comprehensive analytical equations. They are beyond the scope of persons not versed in calculus and differential equations.
The best tool I’ve seen is Don Miller’s revision of the Greenhill formula to include the influences of air pressure and temperature and the effect of velocity continuously (and not just at a break point of 2800 fps, as with some Greenhill variants). His version gives you the value of
s for your result. The only thing it does not do is compensate for the increased drag in the transonic range, but if you stay above 1400 fps or below 1050 fps at sea level, it’s not a concern.
You can use a free on-line version of the Miller calculator at the
JBM calculator site. That site also has a growing
list of bullet lengths to use. You can also go to the
file repository I have and download an Excel version I created. I added a couple of features. It lets you enter the
s you want and also the barrel twist you have or are looking at buying and it gives you both the twist you would need to get the desired
s, and also the
s you will get from the twist you entered. It also has an atmospheric barometric pressure estimator for altitude. I also put in a second worksheet page that calculates
s from the angle of the diagonal stringing caused by wind on actual targets, so you can see what you’ve really got? If you don’t have Excel, I have tested it to be sure it runs in Calc, which is the spreadsheet program that comes in the
free Open Office Suite.