There are specific techniques that you can practice to prevent strings from ringing out.
In general, the solution is to achieve finer motor control. Here's a little check-list for you. You may already be doing some of these things, in which case, you're technique will only need some minor tweaking;
1) Anchoring. Try lightly resting the nail of your pinky finger of your right hand just below the strings. This allows you to have a better sense of where your pick is in relation to the strings. If you do this though, make sure that you're pinky is relaxed and curled inwards towards your palm so that you don't put too much stress while resting on it.
2) Make sure the pick is in line with your index finger. That is to say, "pointing" in the same direction as your index finger. Doing this allows you to have a lighter grip on the pick, allowing it to tilt slightly as it sweeps across the strings.
3) Alternate your picking, moving the pick both up and down. If you're doing this as consistently as clockwork, it helps you're picking to become faster and more economical.
4) Try turning the pick at a slight angle so that it glides across the strings more gently.
5) Don't pick from your elbow! If you're wrist is stiff when you're picking, you're doing something wrong. Instead, try to relax your wrist, bend it slightly, and pick by rotating your wrist from your forearm, as if you were opening a door. This allows you to have finer motor control over the strings.
6) Try approaching the strings from roughly a 45 degree angle. This allows you to target specific strings swooping down with hawk-like accuracy, and is especially good for rakes and sweeps.
7) On your playing hand (left hand), use your spare fingers to dampen the strings after each musical phrase. This usually happens to be the index, pinky, and sometimes the thumb can be used to dampen the low E string.
Try these techniques, and with enough practice, you should be shredding flawlessly in no time.
But remember - there's no silver bullet, no quick fix, and absolutely no substitute for practicing your guitar for hours on end.
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