I've just picked up a Pilgrim Banjo and a Banjolele to play about with, I'm fascinated by all kinds of instruments and the banjo is one I've somehow sadly avoided up to now.
The Pilgrim plays really nicely, a lovely sound. However the Banjolele is giving me a bit of a headache. It's fine for the most part, but I get a horrible out of tune sound (like two strings really close, but out of tune) most noticeable on the 2nd string, but only certain notes. The string is definitely in tune so I'm guessing this is some kind of resonance conflict between the string and the head.
Now, I haven't as yet digested all the internet knowledge of banjos, but I have read a little about tuning the head. I'm usually pretty good with a wrench and a hammer so I've played about a bit, I've evened the tension to my feel and gone from fairly loose to reasonably tight, but I can't get rid of this duff grrr that makes my ears curl!
I find that damping the head on my banjolele produces a much nicer sound, removing those jangly overtones. If you have a dowel stick, a piece of sponge wedged between the dowel stick and the head works pretty well - move it around until you like the sound best.
I find the "afterlength" of the strings, between the strings and tailpiece, is more often the source of those resonances than is the head. First thing I'd do is mute those with a little strip of felt or something, and see what that does.
Anything which can vibrate will have its resonant frequency, of course, and sometimes you just have to over the whole banjo and make sure everything is fastened together tightly.
Something else... when you have gut or synthetic strings which are tied on at the tailpiece, there's an end of each one exposed somewhere. If that end is touching the head, it can do all kinds of odd things to your sound.