Looks like it's time for a bit of Basic Radio Wave Transmission and Propagation Theory. Unfortunately, I've forgotten more than I recall. (At least of the details and the "why.") But here goes, anyway.
For starters: The best
antenna for a given frequency is one that is tuned for that frequency's wavelength. Wavelength is inversely proportional to frequency, following the fomula:
Wavelength in meters = 300 / Frequency in MHz
Wavelength in feet = 984 / Frequency in MHz
(Good grief, I actually had to look those up
The next most efficient is quarter-of-a-wavelength lengths (3/4, 1/2, 1/4).
The next two things you need to know is, generally
, radio waves that are... um... let's say "bounced off the sky" (to cut a rather long explanation short), generally end-up horizonally polarized, even if they started out vertically polarized, and man-made noise generally has more vertically-polarized components than it does horizontally-polarized ones.
Lastly, you need to understand that the ground under your antenna forms part of the antenna system.
Putting this all together: A vertical antenna on top of the house, particularly one that has no real or virtual ground-plane and is not well-grounded, to boot, that is less than 1/4 wavelength of the desired frequency will generally perform much more poorly than a (multiple) quarter-wave or better, horizontally-polarized antenna that's over a decent ground plane (e.g.: the ground).
More than this, in this venue, is simply not practical. I used to have stacks of books on antenna and transmission line theory and design, and on radio wave propagation theory and practice.
One of the easiest antennas to hoist, that is relatively efficient at a wide variety of wavelengths, is a Windom
, cut to 1/2 wavelength for the middle of the desired band, with a single-wire feed 1/3 of the way from one end. (E.g.: If your interest is in 40 meters: A 20 meter Windom.) Use your favourite search engine to search for design details. I used one of these, cut for 80 meters, for years, along with an antenna tuner, on everything from 160 meters through 10 meters.