Actually, I just finished watching 1971's Best Picture. It's unlikely I can add any new critical insights to a well-regarded, thoroughly analyzed film that's nearly 40 years old, but one thing struck me; the music sounds like it belongs to a different kind of movie, perhaps a Hammer horror film, something with Ingrid Pitt and Christopher Lee.
The music is pretty minimalist - just an eerie, sustained tone or two whenever there's a need to build suspense. Just an unsettling "Eeeeeeeeeeeeeeeeeee..." as Gene Hackman hunts through an abandoned building with his gun drawn, or when he spots someone suspicious across a crowded room. It sets up a disturbing dissonance, and I can't decide if it's a matter of directorial intent or an artifact of the times that doesn't play today the way it did in the early 1970s.
It's possible that the unsettling music is meant to comment on Popeye Doyle's mental state. He's an obsessive type, who doesn't seem to care how many innocents are hurt during his relentless pursuit of...I was going to say justice, but Doyle's motivations are never really fleshed out. He pursues criminals because he's a policeman, seemingly without any thought of higher ideals. When he accidentally kills a fellow policeman near the end of the film, he expresses no remorse whatsoever. When he regards suspicious citizens, he looks at them with the eyes of a predator, and the music seems to paint Doyle as a dangerous creature.
And yet, this film doesn't attempt to reverse the typical good/evil dynamic. The cops might be rough around the edges, but the villains cause enough mayhem to justify (in the mind of the audience) the brutal police response.
The music doesn't seem so out of place when you consider the world in which the characters live. It's a dangerous, dirty place, full of world-weary cops and ruthless criminals. Ordinary citizens barely figure into the film at all, and when they do, they are almost always victims.
Perhaps The French Connection is a horror movie after all...