I think we are making some progress here; pulling together the comments made upstream regarding the history of both the manufacturing processes and the specifics of the weaponry, there does seem to be a logical progression.
This progression does seem to produce a logical explanation for the bore sizes of munitions.
If we start from the original smooth bore muskets, we know that they had a bore of 0.5 inch, and used a spherical ball.
The half inch ball, and the corresponding bore size, was logical and reasonable as a fraction of a standard unit of measure; ie: the inch.
However, with the increased use of muskets, and the corresponding demand for balls, it would have been logical and reasonable to try to economize on the use of lead in the production of balls.
Similarly, it would also have been logical to consider means by which to reduce the consumption of propellant.
So, it would be logical to halve the size of the ball, and the corresponding bore of the musket.
The volume of lead required to make one half inch spherical ball makes two 0.38 inch spherical balls.
Similarly, that same volume would make one dozen 0.22 balls.
Spheres of these diameters could conveniently and efficiently be manufactured in a standard shot tower.
With the introduction of cartridges, it would be logical and economical to use existing musket barrels, at least during the transition phase.
Therefore, cartridges would be designed to fit existing barrels; and cartridge manufacturing processes would be tooled to make cartridges of these calibers.
Tooling for the new rifled barrels would be created to make the new rifled barrels to correspond to the manufactured cartridges.
Once the volume of such weaponry was sufficiently large, these calibers would have been established as the defacto standards.
The other odd calibers may be simple variations on these units due to some identified manufacturing efficiency in materials or process.