(This, and everything marked before May 2018, was originally written as part of "Matt Riggen's Jazz Blog", a blog which I kept on-and-off as a college student. Some of it is edited for style, namely the presence of gendered language and for clarity in the writing.)
Say someone calls you up and asks you to play over chords/chord changes. I feel like there’re kind of two approaches to use. You can mix and match ideas, or have one that’s pretty predominant across your playing—I certainly know I’ve got one—but they’re essentially completely different philosophical positions.
The first is seeing harmonies as being derived from melodies (vertical thought being compressed horizontal thought), and the second is seeing melodies as being derived from harmonies (horizontal thought being elongated vertical thought).
Let me explain.
The first way is where people like Miles or maybe Woody Shaw are coming from. You get this idea from their playing of diatonicism, of playing to the tune rather than the harmony that’s occurring at the moment. For Miles, this diatonicism is (almost always) that of the tune he’s playing (a whole bunch of F Ionian over If I Were A Bell off Relaxin’, tons of G Dorian over Milestones), and for Woody, the diatonicism is related to the key of the tune (eg. the rising passage in G Major that opens up his solo on If I Were A Bell off a live bootleg). This comes across pretty strongly in the tunes that they write and choose to play, too—So What, Milestones, In Case You Haven’t Heard, and The Moontrane are all kind of characterized by protracted key areas. (Ornette also tends to play this approach, obviously—his horizontal thought by necessity dictates the vertical harmony, and if Ornette plays a flattened third or a natural third, then that changes the harmonic sound at that moment.) Coltrane also does this but at a super-speed that means that the ear perceives the scale as a harmony (check out the cadenza on Russian Lullaby off Soultrane for a solid example).
To play like this, it makes more sense to think of tunes in terms of the natural guide tones that propagate themselves through the tune, and in terms of the horizontal scales each harmony represents. I’m definitely more of a horizontal kind of thinker—when I’m presented with vertical harmony (eg. a dominant 7th chord), I think of all the scales that could fit over that harmony—whole-tone, HW dim, Mixolydian—and not necessarily of the chord tones I can alter.
The second way is the way Bird thinks. Bird is almost always playing a realization of a harmonic structure instead of a chord-scale, and when he makes an alteration or harmonic decision it’s built on the tones of the chord or explicitly on its extensions. The Biddy Fleet story (wherein Bird discovers that by playing on the upper structure of harmonies, he can create a more personal sound) belies a mind that’s already thinking of melodies as being inherently and inextricably linked to chord tones. The contrafacts he writes also use melodic lines that emphasize the harmonic qualities of the chords beneath them (eg. the last two bars of Ornithology, the opening to Confirmation, et.al). In fact, even when presented with relatively horizontal tunes (eg. the blues), Bird finds ways to alter them so they become far more harmonically rich (Blues for Alice being an excellent example).
To play like this, it becomes necessary to realize the harmonic content of each chord as an arpeggio or a series of leaps. When altering the chord, only the notes one wishes to alter will become evident and not the implied chord scale (C7#5 as opposed to C whole tone). Coltrane also does this as well, but at a speed where the arpeggiated line is perceived as a moving harmony.
(As an aside, I feel like this is one of the reasons a free jazz group I play with named OGC works so well. Matt Babineaux, the alto player, has a bunch of Charlie Parker vocabulary completely under control, and I have spent much time with the First Quintet-era Miles Davis, and with Science Fiction-era Ornette. When I’m playing free, I’m thinking of scale collections and horizontally—I am not trying to generate a harmonic structure. When Matt plays free, to me it sometimes feels like he is improvising chord changes that he can fit a melodic structure to. This sort of yin-yang approach strengthens the ensemble overall and lets both of us cover what we like to do.)