Bebop styles were characterized above as exploiting the harmonies by choosing scales with a lot of color tones, whereas modal playing was characterized as emphasizing the basic chord tones. Both of these approaches still use chord/scale relationships in the traditional manner of choosing a scale that implies the sound of the chord to some degree, and playing mostly within that scale. Another approach is to maintain the sense of chord progressions but play lines that lie largely outside the associated scales. This is sometimes called chromaticism. Eric Dolphy used this approach when playing with Charles Mingus and on some of his own albums such as Live At The Five Spot and Last Date. Woody Shaw and Steve Coleman are also chromatic players.
You have by now probably played some outside notes, say an Ab against a Cmaj7 chord, possibly by accident. These notes may sound wrong when played in the context of an otherwise inside melody. By playing a melody derived from a scale, you establish a particular sound, and one wrong note will sound out of place. However, when playing a melody that lies mostly outside the scale, the same notes may fit in much more logically. That is to say, non-scale tones used melodically can often sound consonant (the opposite of dissonant).
The aforementioned musicians often play very angular melodic lines, meaning they consist of large or unusual intervals and change direction often rather than being primarily stepwise and scalelike. This often seems to establish a sound in which wrong notes sound perfectly natural. Interestingly, the opposite approach works as well: lines that contain a lot of half steps often sound right even though they consist of many wrong notes. These lines are sometimes called chromatic.
You can continue to use your knowledge of chord/scale relationships when playing chromatically. For example, you know that a Db lydian scale is not normally an appropriate choice to play over a Cmaj7 chord, and you probably have some idea why. These same wrong notes, however, if used melodically over the chord, create a sound that is not all that dissonant and has a harmonic richness that is very modern sounding. In fact, even simple melodic ideas like arpeggios and scales can sound complex in this context.
You can practice these ideas with Aebersold albums, or Band-In-A-Box, or your fellow musicians, although you should be prepared for some strange looks. It has been said that there are no wrong notes, only wrong resolutions. This certainly explains why passing tones and enclosures sound consonant, but I feel it still places too high a value on playing the notes suggested by the standard chord/scale relationships. I would restate this; the only wrong notes are notes you didn’t intend to play. Any note you play is right if it is in a meaningful context and it does not sound like an accident. There is even value in making mistakes. The trick is in forming a coherent whole.