Q: Why have a double sharp or flat ever? Wouldn't it just be better to make it the note that's already in the time signature?
A: key signature, yes. but, picture this: you're in the key of Bmaj and something is happening around the G#min key center which gets a lot of melodic action on the G and G# notes. Imagine you are on the music staff........ what a headache it would be to switch back and forth constantly between G and G# with accidentals. So you'll wanna put it on different letters, so you can just put the noteheads in the 2 neighboring line/spaces. otherwise they'd all be on the same line/space, with alternating accidentals stuffed in between each one. what a headache that would be. Sure, you may be wishing we'd been calling this whole situation in Abmin..... in which case i hope you like Cb and Fb, because we'd be coming from the key of Cb (featuring lots of Fb's of course). Now, suppose you are saying, wait no, how about i can have my cake and eat it too, we'll call it the key of B, which contains the note Ab and the key of Ab minor. Well then guess what, your regular scale is going to have Ab, Bb, B. so then every melody is going to be chock full of interplay between Bb and B. which would of course have to be stuffed on the same line with alternating accidentals to make your eyeballs explode. Ok sure, fine, i know what you're thinking, why can't we have the key of B with A#, then Ab below that. Well, ...... congratulations, now the regular 6 and 7th scale degrees of your melody are always going to have to be stuffed on the same line, again, with alternating accidentals lodged in between them. So as you can see, this is a lot like getting a wrinkle out of a spherical blanket..... which reminds me....... the circle of fifths does indeed have a wrinkle. you can chase the wrinkle around the circle as far as you want but it will always be somewhere. of course you could do it without the wrinkle, but if you go genuinely around 12 steps you will actually be slightly off from the original key. so indeed the connectivity of the 12 step program is a cheat