You can hear an example of contrapuntal improvisation below. I am playing very realistic sounding bass in my left hand and soloing in my right hand. Two figures are moving against -- and complimenting each other. The inter-relationship is based on harmonic relationships. For example, let's say I'm playing a ii V | I progression. As a trained musician, the pallette of notes will be obvious to me. My left hand (playing bass) will outline the harmonic bass root movement of the tune. (This is one type of melody, and its guided by simple, but very specific rules.) Meanwhile, my right hand will be playing notes derived from the same harmonic pallet as well.
Note that counterpoint is not limited to two instruments. The most obvious example in classical music would be the Fugues of J.S. Bach, and in traditional Jazz, the competing, but complimentary melodies of Dixieland band; however, counterpoint occurs constantly, even when we're not aware of it. For example, suppose I play a chord progression. Each note in the chords I play represent a different voicing,. Chords, in essence, are "push button" devices that allow us to harmonize many notes at once without having to thinking of them one by one. By analyzing chords, we can even go so far as coming up with multiple melodic variations of a tune on the spot. (Please be sure to take a lesson with me to see this concept analyzed in action.)
This leads to my methodology -- basically, regardless of instrument, I want my students to be able to sit down at the piano and play LH against the melodies and chords the are experimenting with. This turns improvisers into true composers. To boot, the methodology is not so difficult, and an end benefit is also a very practical form of ear training: Let's say, for example, you're a saxophone player or a vocalist. If you're a true professional, your ears will be glued to the bass player, not just the score as you perform. Naturally, it will be a lot easier to understand what's going on if you yourself can play lines like that yourself. You'll be able to hear the chord progressions even if you don't have perfect and melodic pitch. The walks will give everything away.
Incidentally, there's an added bonus of studying with me: regardless of your instrument I can accompany you on the spot in a very realistic sounding way. As you get used to playing with me, playing out in real sessions will become easier than you could ever imagine... and as you get used to controlling music from the bottom up you'll become a much stronger performer and leader.
(audio: coming soon)