When I was first starting to learn jazz, I read something where Chick Corea said, "If you want to play fast, practice slow... a lot!"
This is true, but counterintuitive to how we usually approach fast playing. But if you want proof, here it is:
Ask any young kid who's taking piano lessons to sit down and play a slow piece they've learned well. If they've practiced it enough, it shouldn't be a problem. They'll play it with ease and it will sound pretty good.
Now ask them to play it as fast as they can. 30 years of piano teaching experience has shown me that 99% of the kids would be able to play it at light-speed, even if they've never played it fast before. In fact, they can often play the slow pieces faster than the pieces they're trying to play fast!
Why is this? It's not that the slow pieces are "easier." On the contrary, a slow piece played very fast may even be technically more difficult than a pice designed to be played rapidly, which may contain scales and such.
The difference is that by repeatedly playing the music slowly, the notes, sounds, and hand positions have become imprinted in the player's physicality and music ear. The fingers will remember where to go.
It's the same reason that sprinters don't train by sprinting all day. After all, who could? They'd tire themselves out too quickly. Instead, a fast sprinter will start by doing slow, gentle stretches, perhaps followed by some strength training, cross-training, and finally, by actually sprinting. For athletes too, a great amount of slow work transitions into speed when the time is right.
Another way of looking at it is: "If it works for Chick Corea, one of the fastest and nimble-fingered jazz pianists of all time, it'll work for you too!"
Practice your jazz improvising slowly and thoroughly. Then, when you're ready, "go for it!" You'll play with more natural ease, fluency, accuracy and musicality than if you only try to play faster and faster all the time.
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