The whole purpose of horizontal polarity is to accomplish:
1- Reduction of vertical signals that interfere with communication
2- Reduction of noise, most man made noise is vertically polarized
3- Increase the far field lobe of the transmitted signal
As far as number of elements & gain go after 4 elements the amount of gain
diminishes a lot .75 for the fifth- . .5 for the sixth- .25 for the seventh.
The spacing of the reflector and the driven element affect gain more than number of
elements, the length of the boom plays an important role in a yagi.
As far as band width the diameter of the elements determine that, thicker is better.
You guys that have been in the game a while remember the Wilson Long John?
It was the biggest bad ass 11 meter antenna you could have back in the 70's
The length of the boom is what gave it tremendous gain, not the number of elements.