Gabriel Yared's most famous score, The English Patient was a bit of a disappointment for a lack of any substantial thematic/melodic material although the film was a huge success and won nine Academy awards.
His best effort to date is Betty Blue. The collaboration Beneix/Yared produced a 'classic' which still is very enjoyable to watch nearly twenty years after its first release.
At times, it seems, there is a mysterious alchemy at work when the vision of a director blends perfectly well with a composer's world, and where the result is more than the sum of the parts. To name but a few: Hitchcock/Herrmann,  Leone/Morricone, Fellini/Rota collaborative works come to mind at once.

In Betty Blue, Gabriel Yared has managed to create a score which totally belongs to Beneix's vision (or may be this is the other way round).
Betty Blue is a tale of love and destruction where the main character - larger than life Betty Blue can't stand a world  - ours - where love can't transcend all the pettiness and mediocrity of our daily lives. The film depicts her gradual disillusionment and her final brutal escape into madness and ultimately death. This is a love story of nearly mythical proportions and through the use of his thematic material and orchestrations Yared has cleverly driven this point to the audience.
The main theme " Betty and Zorg" is used many times during the film, played with synthesizers or with the melody given to a sax. It is a soft, romantic tune, a little bit sad. 

The second important theme appears at a crucial moment of the film. In the piano shop, in the dark, Zorg starts to play a somehow disjointed pattern/groove and discreetly Betty joins in with slightly off key repeated note which complements the first pattern to create a haunting piece. The slightly awkward musical elements, not particularly interesting on their own, make sense when they are put together. That is exactly how these two characters are. From the point of view of society, they seem to be two slightly maladjusted persons or even two misfits but, when they are together, they form a marvellous couple.

In the third key track Gabriel Yared uses the barrel organ very cleverly.

In this film the barrel organ has two important connotations:
In France, a similar tune was used for many years as a signature tune for an extremely popular cine-club television program with a strong emphasis on the films made before the war (Renoir, René  Clair, etc...)*
So, in using such a piece of music, Beneix placed his story in a context which is not realistic, but belongs to the realm of legend or of popular modern tale. This is a statement aiming at convincing us that, we, the spectators, are going to watch one of these popular epics with characters larger than life as we could find in Carné's films (les Enfants du paradis, Hotel du Nord, etc...). In fact the story is told  - as an immense flashback by one of the main protagonists, Zorg, who is a writer.

In France as in England, the barrel organ is linked with the street story tellers, singers who used to give , during the XIXth century and in the XXth century until the war, street renditions of popular never ending serial stories.
Obviously this tradition is at the source of the barrel organ cinematic tradition.