Have you tried Charles Austin's An Approach to Jazz Piano? It's probably the most comprehensive jazz piano/jazz theory book out there, and it starts with "rock" type harmony and moves on from there. I found "The Jazz Piano Book" assumed a lot of cultural knowledge that I didn't necessarily have at the time, when I was starting jazz piano, as well as introducing ideas in a weird order, and at the time I just needed a really thorough explanation of all the things from first principles, and ways to put them into practice.
Levine's The Jazz Piano Book is more of a textbook than a How To, so no wonder you are finding it hard work. Given that your stated aim is to get into 'improvisation for cocktail-style piano', perhaps start by obtaining jazz piano arrangements of some cocktail-style tunes; you are bound to learn something just by playing them, and the cocktail quaffers are unlikely to care whether the solo is yours, Hal Leonard's or Mulgrew Miller's (Levine is a big fan).
Mark Levine - The Jazz Piano Book.pdf
Harry Likas was the Technical Editor of Mark Levine's "The Jazz Theory Book" and also helped develop "The Jazz Piano Book." Harry spends his time teaching jazz piano online and playing solo piano gigs.
I give my students an outline that compiles the various popular modern jazz piano voicings onto one page. Think of this outline as a menu. Apply these voicings when playing a ballad by recognizing what note is in the melody on beat one and use that note to direct you to which voicing to play beneath. My arrangement of "Soul Eyes" on page 2 details the thought process.
Harry is equally comfortable teaching beginning lessons as well as advanced lessons, and enjoys the challenge of inspiring all levels of students. His experience has made him quite a versatile piano teacher, and he can cover classical, jazz, pop and blues piano styles, at any level of instruction. 2ff7e9595c
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