The two main competing factors which dictate a certain tip speed for maximum efficiency are: tip compressibility drag rise, and mass flow through the propeller disk.
We are generally stuck with a given propeller shaft speed, say 2700 RPM.
That means diameter + a little adder for forward speed, times a constant = mach. Greater diameter produces a higher mach number. So, smaller is better.
However, thrust is produced by grabbing air and pushing it rearward. The bigger the diameter, the greater the mass, the less slippage and the higher the efficiency. So, bigger is better.
These two factors compete directly. So, one can rock these factors back and forth on a computer and find the sweet spot for efficiency and maybe feed in other practical requirements as well.
As I was reading Craig Catto's autobio a couple of years ago. I felt a kinship and above all an admiration for his background, knowledge and ability. I'm sure he is a couple of steps beyond the basics I have outlined.
Data point: Cessna 182, 230 HP 2600 RPM 82" prop 948 FPS ~.86 Mach tip speed.
Ron