I've been playing All New Materials by Periphery. The intro necessitates playing a powerchord on the bottom three strings (it's drop tuned) with my third finger. If I try to hit a problem. Either I have to contort my finger in a horrible way (all bent and awful) and the other fingers are all over the place, or I try to play it like I do with my first finger and I just can't get a good sound.

So is this an issue of finger strength and independence that I can work through, or does everyone else have trouble with this? It's rather annoying and slightly painful.

My stuff

Gibson Les Paul Studio
Ibanez ADC120
Tanglewood TGRF VS
Blackstar HT20
Roland Micro Cube
All you're doing is playing a barre chord without any of the other fingers, it shouldn't be causing you any problems

You might be better off just shifting your index finger along instead of trying to use the third
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You may as well learn to barre with all 4 fingers - especially if you're regularly going to be playing songs in drop D. I guess it's an issue of finger strength (and partially also string guague) but if you work on it I can guarantee you will eventually be able to barre drop D power chords with all 4 fingers with ease.
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Yeah, it's mostly finger strength and independence. Try not to bend your finger back on itself too much - if you can keep the finger fairly curved it's better, although its not always feasible.