The original underscore tends to be of a broad brush (Antoine Duhamel), typical of the time in French and Italian films. Not written in sync, it is often cut in for its specific, own affect, the diagetic sound being taken out to focus. This technique has come back in a big way for the last two years in a row, Oscar winning scores by Gustavo Santaolalla. This technique of marrying music to picture is often employed by directors of extremely strong mind (Kubrick). It leaves little chance for what the composer might bring to, get from the picture. In this film, Godard is so sure of his musical control that he starts and stops it rather obviously and abruptly as if to remind you he is in control.
The best thing about the film's music, in my opinion, is the very skilled intertwining of the musical theatre type of numbers. One where we meet a character who always hears a song in his head, which because it reminds him of his first love's telling him she did not love him, is making him crazy. We hear the music like he does, but Pierrot/Ferdinand, for the moment firmly in the diegesis, can't, and we hear how the character's slight mumbling and moaning along fits into the jazz lounge piano accompaniment that follows him around. There is also a wonderful song and dance number between Pierrot and Marianne as they walk through a woods in southern France, she singing about her fate line (in her palm) and he singing about her thigh line. The songs are credited in the film (although not at Imdb) to Boris Bassiak, a very curious fellow, indeed!
I just have to like a film where the characters sit and read Céline (another Ferdinand).