User blog comment:Vel!/Music/@comment-160.94.192.220-20141231173154/@comment-2033667-20141231182930

It's not entirely clear to me how free improv is analogous to diagonalization. However, I have considered what "diagonalization" would mean in the context of non-improvisational composition, and it ties in to your last sentence.

Consider this example: pick a random work in the Western classical music tradition, from Bach up to maybe Schoenberg or so, and look at how diatonic it is throughout its development. Chances are that if you were to graph this quantity, it would be rather "flat" -- in general you'll see that works don't suddenly change from diatonic to nondiatonic or vice versa. To "diagonalize over diatonicism" would be to take the opposite approach -- to slide over all possible levels of diatonicism in a single piece, using this musical parameter to its fullest extent. This could be a piece that starts like a Renaissance counterpoint piece and slowly turns into Webern-like atonality, or any other change over time that uses the full range. I like to think of this as "modulation" -- like some imaginary knob on a synthesizer that I can use to control diatonicism. This can apply to any other musical parameter such as tempo, pitch range, mood, genre, and so forth.