I'm sure that others have a different point of view but I just felt like sharing mine as I've had some experience in this area. In addition to recording and releasing a CD of jazz improvising on solo alto saxophone [entitled "SOLO JOURNEY", on New Artists Records], I also did an entire concert of jazz improvising on the solo alto saxophone.
This is my feeling on the subject: first of all, we actually do spend a lot of time playing solo, and, if our focus is right, this can really translate to the solo concert experience. My point here is that solo playing is not totally alien to the saxophonist.
Secondly, in my experience and as a result of my study with Connie Crothers, I feel that the thing to do is to improvise in a relaxed and intuitive manner based on the melody [yes, I hear all the chordal and scalar players reacting, but this is my feeling]. Improvising from melody will take the soloist beyond what playing on chords could! The melody [along with the lyrics, as Lester Young pointed out], IS the tune; it also implies the chords and the rhythm. Neither the rhythm nor the chords can do this. Note that you can only copyright a melody! [This debate is as old as the "Hawk vs. Pres" debate in Jazz.]
For practice, try spending some time just playing the melody in a loose, relaxed way, with a metronome, over and over. Really 'sing' it and let it stretch out and whatever happens, happens! This is really a profound scene. Can you imagine where one might get to doing that for a lifetime?
If you listen to some really amazing linear playing, such as Lennie Tristano's "LINE UP" or any track from his recently released CD on Jazz Records, "NOTE TO NOTE", it may or may not sound as intense as it really is upon first hearing. But if you try to sing the original melody of the tune along with it, you can hear how the solo stretches out from the tune in a way that playing from the chords could never inspire, stretching out the harmonies, rhythms, and, above all, the melody.
This is what as known as linear playing, although I do not expect to be able to really describe it in one posting, especially since I have been stretching out with it for the 16 years that I have been studying with Connie, and going after it for the more than 10 years I was trying to play jazz before I met Connie.
At my concert and on my CD, some tunes were based on standards and some were free form improvising. "Free form means playing without a fixed chord progression; without a time signature; without a specified tempo", as Lennie Tristano wrote in the liner notes of his famous and innovative Capitol sides; but it's "...not haphazard or hit-and-miss". It is playing from pure intuition and feeling. I also played some of the solos that I love to play, like Bird and Pres, and improvised after that on the same tune, and then repeated the solo. It really was fun!
So, to sum it all up, improvise from the melody [on tunes] and let your hearing and feeling spontaneously guide you; don't try to "give the listener something to relate to" If you really get into it, they'll dig the depth of it a whole lot more than if you're worrying or trying to give them something to hang on to.
By the way, I feel that this is the same way one improvises solo, duo, trio, quartet, whatever! The difference in playing comes, not from thinking or trying, but from hearing and feeling what is happening and letting the music be expressed.
[c] 1996 Richard Tabnik