Most of the big time players out there have at one point in time mentioned both Nat King Cole and Ahmad Jamal as major influences. A lot of my favorite players, swing guys like Oscar and Monty and others, are big fans of those guys.
However at this point in your development as a pianist it seems like the best thing to do is learn tunes on the piano. There are a lot of things to learn about playing the piano as far as chord voicings and reharmonization that eventually start setting you apart from other players. So if you take a tune like Misty and then listen to Red Garland or whoever playing it, pick up their style on that tune, that in itself will give you a huge push.
I used to take a tune, such as Misty, and learn how a bunch of different guys would play it. I'd listen to Red, Oscar, Ahmad, Hyman, anyone who had recorded it and I'd take all the stuff I really thought was hip and try to incorporate it.
We used to have great discussions here about dealing with tunes, different approaches and such, and I learned a lot that way as well.
I agree with Scott. I learned the old standard "Blue Skies", and played it so many times, that finally I came up with my own version of it. Same with "Blackbird", after many listens to Miles. With all the preparation that you have done, you are now prepared to do the same on a play list of songs of your choosing.
Learning tunes is indeed a great idea at this point. Make sure to listen to several classic recordings in addition to checking out the original lead sheets to find the correct chord changes. Sometimes vocal versions with a piano trio can be very helpful in learning the real melody and real changes. Often you have to consult multiple sources. For a tune like "Misty" you can go right to the primary source and listen to Erroll Garner himself play it. That's one level–learning the changes.
Of course, in checking out other versions, you'll learn reharmonizations. That's a subject that you can talk about for hours. But for me, it's finding the movements in between the changes that is so interesting. And, when you reharmonize it's only for a moment, and never requires a separate sheet of chord changes for the bass player.
To start learning about the subtle inside stuff great pianists play between the chord changes check out Barry Harris's sixth-diminished scale. There are tons of videos of Barry himself demonstrating the scale and how to apply it to tunes. Just search Barry Harris and sixth-diminished scale.
Basically, the goal is to start hearing and playing lines even while you are playing chords. For example Cmaj7 is also G6/C–F6/C–Fm6/C–Fo∆7/C–G6/C. So at this point, along with the A and B voicings that you have memorized make sure you can play your diminished chords, sixth chords, and minor sixth chords in all inversions. In the long run, these small chords are extremely helpful.
Volume 1 of this educational jazz piano book contains 15 jazz piano exercises, tricks, and other interesting jazz piano techniques, voicings, grooves, and ideas Scot Ranney enjoys playing.
Volume 2 has 14 jazz piano exercises and tricks of the trade, and quite a bit of it is Calypso jazz piano related material, including some Monty Alexander and Michel Camilo style grooves. Jazz piano education is through the ears, but books like this can help.
Volume 3 contains 12 jazz piano exercises and explorations by the acclaimed jazz piano educator, pianist, author, and recording artist Tim Richards.
Volume 4 is by Jeff Brent, a jazz pianist, composer, teacher, and author of "Modalogy" and other acclaimed jazz theory and education books. In this book Jeff shares detailed analysis of transcriptions of live performances. He covers everything from the shape of the songs to the tricks and licks he uses in improvised lines to the ideas behind his lush chord voicings.
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