Let's start over with the basics.
In any application, the prop tips have to be maintained subsonic or else propulsion efficiency will suffer and noise generation will become extreme. Therefore, for any given prop diameter, there is a do-not-exceed RPM value associated with the sonic limit: big props have to turn slower, smaller ones can turn faster.
(Prop diameters are also limited by ground clearance considerations but for small planes like a Cessna 172 or a Cherokee 160, the need to keep the prop tips out of the grass can be accomodated readily.)
Next you have to match the prop with the best power operating point of the engine turning it. For 4-stroke gasoline engines of typical design this will be around 2500 RPM at which the engine is producing roughly a half horsepower per cubic inch of displacement. You then pick the maximum allowable prop diameter to keep the tips subsonic at 2500 RPM.
Next you assume the minimum number of blades for a start (2) and select the pitch angle at that diameter which will absorb the full output of the engine running at 2500 RPM. If two blades will not load the engine enough to hold it at 2500RPM at full throttle, then you add another blade and repeat the process.
For engines of less than about 180HP, the right answer is a two-bladed propeller of fixed pitch. For engines bigger than that, the right answer is generally a two- or three-bladed propeller with variable pitch.
One way to beat the sonic limit while getting more power out of the engine is to gear the engine down so it is spinning faster, and thereby generating more power, while still maintaining the prop tips subsonic. This also lets you further reduce the prop tip speed and go for a larger diameter prop if you have the ground clearance available, which will generally be more aerodynamically efficient than a smaller one- but running the engine faster shortens its service life and the cost to rebuild a geared engine has to include the cost to rebuild the gearbox itself.
For light aircraft in the homebuilt class or for ultralights, a less-expensive alternative to the geared engine is one with a rubber-belt reduction drive. This lets you use lighter and simpler 2-stroke engines which have a best power point of around 4500 RPM and still maintain the prop tips subsonic.