The shorter antennas with a base loading coil are (at the design frequency) electrically the same as a full length 1/4 wave antenna. There are two things to keep in mind with this sort of antenna (and BOY is that a simplification). One is the length of the vertical element, and how much inductive reactance the base coil furnishes the antenna to make it both "resonant" and physically shorter, and to make the input impedance at least close to the "standard" 50 ohms impedance.
The part of the base coil that has the tap going to ground does the impedance matching. The rest of the coil above that tap point is the loading coil part and makes the vertical element electrically longer. There can be a seemingly infinite combinations of those two characteristics (plus the length of the vertical element) to make a usable antenna. So, if you want to change the resonant frequency/band of a typical base loaded antenna just changing the length of the vertical element only works over a very narrow range. To make drastic freq/band changes that base coil will have to be changed too.
"There's gotta be an easier way! Right?", yep there is. But you will definitely pay for it. The simplest way is to make that base coil variable. Now, stop and think that through for a minute. A 'screwdriver' type antenna makes for a variable loading coil. So, do the same thing for that impedance matching coil, Ta Da!! At this point 'simple' and 'inexpensive' have disappeared over the horizon... and are gaining speed all the time. Wanna try it? Hey! Be my guest, but take pictures and keep track of the cost, would you?
Oh, and then there's the "how well does it perform?" part. If you are looking for best performance, stick with the full length 1/4 wave. There doesn't have to be a huge difference, but there certainly will be a difference inversely proportional to over-all length...
