A question: Would bebop, the chromatic and scientific approach to playing jazz that involves the memorization of specific, complex vocabulary patterns, have occurred to its early progenitors without the atonal (i.e. twelve-tone) music of Arnold Schoenberg? Conceptually, there is a very large amount of overlap between the two styles. Both involve the dictated use of notes in a specific order - through the use of tone rows in twelve-tone composition, and in bebop through the use of lines or licks committed to memory in every conceivable key. In both cases what is left for the artist to freely choose is placement, primarily. The twelve-tone composer can use the next tone as part of a melody (in other words horizontally) or harmonically, or as part of a chord (i.e. vertically). Of course, the rhythms are up to the writer as well. In jazz, the notes are also very largely (if not entirely) predetermined, and the same kinds of choices (placement and rhythm) are left to the performer.

Schoenberg's music has taken a lot of critical poundings over the years, and it continues to do so. It's not to everyone's taste, I think it's fair to say, but Glenn Gould was a great admirer (and interpreter) of Schoenberg's, and so my guess is that we'll eventually catch up with it. But I am convinced that Charlie Parker et al were greatly influenced by it as a concept, and I'm not sure that their discoveries would have happened without Schoenberg's. Here's one of his early twelve-tone works, with the score included: http://www.youtube.com/watch?v=xrjg3jzP2uI. And here's Bill Evans' "Twelve Tone Tune", a tip of the cap from a jazz musician of unsurpassed learning: http://www.youtube.com/watch?v=dB78xeZ8quk.