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the melody will follow a scale from a root note. 99.9% of the time the bassline is playing the root note. once you have found the root you can play any chord in that root and inversions thereafter. if you want the chords to be harmonious with the melody you would probably end up following some of the harmonic intervals that the melody is playing.
inversions are sort of hard to explain.
if the bassline is playing in C and a C chord = C, E, G, then the first inversion would be E, G, C. its sort of taking the root and sticking it at the end. which makes it technically similar to an E minor chord but with a different 5th (an E minor = E, G, B). so it resolves differently. but you can play these over a melody that is improvised around a C major scale.
but in this way you can basically play a load of chords over the melody of an existing song as long as you know the root from which it is played. if your theory is solid enough you wont even need the bassline to do this since you will be able to identify where the half steps in the scale are and figure out what the scale is.
alternatively, if you know its a minor or major scale - just play 5ths from the root (in guitar books these are known as power chords or 5 chords) and add some of the notes played by the melody.
you cant really go wrong because of the tonal ambiguity of 5ths - both minor and major scales can be played over them.
in summation, this is one of those things where you can pretty much take the root and various notes of the melody and just play them as chords. most of the time it'll work. as such, thinking about the theory behind it just complicates things too much - which is pretty much exactly what ive done here 
ive been lazy recently and havent gotten around to adding the extras yet. i should really try to explain inversions and 5ths but i hope this post helped somewhat :/
Last edited by Derivative on Oct-21-2005 at 23:53
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