Lately I've been listening to jazz pianists in a slightly different way than I usually do: I've been focusing much more on what they play (or don't) with their left hands.
You can learn a lot like this. For instance, you'll find that the left hand parts, particularly in respect to rhythm, define a jazz pianist's "style" more than anything else does. The LH rhythms even influence their RH phrasing much more than previously suspected.
Here's just one example:
We know that many "modern" jazz pianists use rootless chord voicings in their left hands. Many pianists even use the identical chord voicings as each other. But some, like the great Tommy Flanagan and Hank Jones, play traditional comping rhythms underneath their RH solos. On the other hand, pianists like Keith Jarrett and Chick Corea are often looked to as examples of pianists who phrase in a freer way than traditional beboppers. This may be true, but if you play a transcription of a Jarrett solo in your RH while playing traditional bebop rhythms with your left, the phrasing will still sound traditional and less "free." A close listen to Jarrett's and Corea's left hand parts reveals why this is so: they play free rhythms in the LH too. Jarrett in particular will often play nothing at all with his LH in trio settings, leaving his RH free to play asymmetrical phrases in a way reminiscent of saxophonist Sonny Rollins' pianoless trio recordings. Neither Jarrett nor Corea will play repetitive or predictable rhythms with their left hand, as even the greatest beboppers will do. This greatly influences both what they play with their right hands and how we as listeners hear those right hand parts.
In other words, the left hand rhythms influence the right hand solos much more than we might realize. And by extension, the left hand is really helping to define the overall feel or style in a huge way. Much more than the melodies or harmonies that a given pianist uses.
See for yourself. Listen to the left hands parts that are played by a wide variety of jazz pianists. Then you can apply what you learn to your own playing. What style do you want to play in? Choose an appropriate left hand technique, and you'll be that much closer to playing jazz piano the way you want to play.
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