The musical improviser is an improviser just as much as the players -- he or she can make offers, respond to offers, must learn not to block, develop narrative, etc. This means two things: 1) it's best if the musician has an improv background, and 2) the players should learn to listen for musical and sound offers.

Which brings us to the most obvious difference about musical improvising. The offers are much subtler. Most of the time, the audience won't notice the musician at all (except occasionally during songs). Sadly, much of the time the actors won't notice the music either, at least not consciously. The down side is that it can be frustrating to make offers (whether narratively, dramatically, or whatever) that are ignored. The up side is that the musician can afford to make (occasional) BIG offers, since it still won't really block or force the story in any one direction. By big, I don't mean loud of course, but rather very significant (e.g. like playing a dramatic chord to emphasize a revelation, or switching from sad to happy music).

Something I've occasionally tried is something that doesn't apply to actors, which is to make what I call contrary offers. These are musical offers that are intended to be counter to the scene. For instance, during a scene set in a nursing home, play rock music. Or in a military camp, play a children's song. As you may gather, because of the way music complements a scene, such offers don't destroy what has been offered, but rather usually benefit a scene. Even if the players don't pick up on it (and surprisingly, they will often not notice what seems to be an incredibly blatant contrast), it may affect their play subconsciously, or at least it can affect the flavor of the scene for the audience. (A cautionary note -- I'd recommend this only for experienced musical improvisers. It's better to learn how to do "appropriate" music before doing "inappropriate" music.)

For songs, remember that most actors are not singers. The key word is simple. Simple simple simple. Simple rhythms, simple harmonic progressions, simple song structure. Whenever possible, provide clues as to what comes next in the song. Play a 7th before modulating. Provide really obvious lead-ins. Even talk out loud to the singer(s), saying "one more time!". Powers of 2 (except 12-bar blues). You can do 90% of your songs with I-IV-V (with the occasional ii). Provide dramatic contrasts between verses and the chorus. Provide accompaniment-style accompaniment, not solo style (e.g. play single notes in the left hand, and chords in the right, no funny stuff). Also, use instruments that have strong attacks (piano, guitar), so the singers can immediately hear chord changes. Remember that you are trying to make them look good, not yourself. The audience won't notice any fancy stuff you do anyway.

Warm-ups: singing requires warming up. Before each show, do a few minutes of vocal exercises, then practice improvising some songs. What I usually do these days is have a group warm up their voices by singing scales and the like, then maybe one line-at-a-time song in gibberish, then a few more line-at-a-time songs in English, then one more where they all sing their own song at the same time, bellowing the song to the back of the house. Make sure all the songs are in different styles.

Learn different styles. Analyze different styles of music, and try to identify what's unique about it. Pay particular attention to music you don't like, which you probably haven't listened to all that much. Try to identify the chord progressions, the instruments used, the rhythms, the structures, the tempos, the bass lines, the instrumental riffs. Find what's common, find how composers make theirs different. Spend a little time paying attention to the music of horror movies, 70's action-adventure TV shows, Shakespearean dramas, science fiction, samurai epics, 30's musicals, etc. Music can have a tremendous impact on the flavor of genre scenes.

Always experiment. The subtlety of musical offers means small mistakes will never be noticed, so feel free to make big mistakes. Try doing scenes with just drums as background, or abstract avant-garde randomness. Take the opportunity on stage to use a chord progression you've never used before (as long as no one's singing to it). Take a real song and deform it into something unrecognizable. Switch moods erratically. Pimp the actors. Pimp yourself.

Remember all the standard rules of improv. Listen. Make mistakes. Have fun. A good improviser is someone who makes their fellow players look good. Say yes. Don't plan. Look for endings. Try to do something you've never done before. Reincorporate. Etc.

Great musicianship isn't necessary to be a good musical improviser. I consider myself a mediocre musician. Someone who's a decent musician and a decent improviser will make a much better musical improviser than a great musician who's a bad improviser. (And by improviser, I mean theatrical improv, not musical improvisation a la jazz.)

It's very hard to play so much music in the course of an evening that the audience actually notices it, provided that the music is appropriately varied. But don't be afraid to let scenes have no music either.

Games involving music include doing an opera (in English or in gibberish), doing a musical, doing interpretive dance, having players be called upon to sing their innermost thoughts, making up a song (in a nightclub, as a commercial jingle, as karaoke), having actors change emotions based on the music, singing whenever the musician is playing music. Lots of games are improved with music, including genre switching, scenes in foreign locales, radio plays. Music can be used just to introduce scenes, or as background music between scenes. Also, any scene can potentially have the player (or players) break into song, and any scene can have background music.

Be careful to vary the music. I have found it very easy to fall into a pattern of playing music that's reflective of a narrow range of moods, or just in one tempo. Be conscious of what you've played before, and try to provide as much variety as possible (as appropriate) during the course of a show.

Work "The Girl From Ipanema" into every show at least once.