Jazz lesson

Chris Bakriges was kind enough to give Xiao Li and I an informl Jazz improv lesson on the last day of IPWMT. Very interesting insights into the mind of a Jazz improviser/teacher:
i. start by experimenting with modes to colour the dominant (or, thinking conversely, what mode is implied by a certain colour note)
ii. jazz improv basically involves moving from mode to another (much like modulation and modal inflections in tonal practice, I suppose)
iii. in harmonizing a melody Jazz-style, the particular chord structure relates to the indicated chord without necessarily sounding the root at all! e.g. Bb7 rendered as Ab-Bb-D (perhaps, think of the indicated Jazz chords as signifying the underlying harmonic movement in the Rameau sense of a fundamental root movement, i.e. surface bass line can be quite different from what the chord indications literally stipulate).
iv. he prefers to think linearly rather than simply move from one Jazz chord to another, and he sees the linear rendering as the musical fingerprint of a Jazz improviser
v. in spacing his chords, seems that he is equally conscious of both the effects of the harmonic verticalities as well as linear connections

Points (i) and (ii) remind me of Bach's alternative harmonizations of his chorales whereby different tonal interpretations--both in terms of keys and tonicizations--are explored for the same chorale tune.