In 2008, Lou Reed (of The Velvet Underground renown) said that Leonard Cohen was in the "highest and most influential echelon of songwriters." Notice that Reed didn't have much to say about Cohen's guitar-playing ability, which consists mainly of slow strumming; the majority of his songs are composed of a few relatively simple chords.
Yet in a melancholy song like "Famous Blue Raincoat," the minimalist set-up works perfectly. Before Cohen even begins singing, the slow strumming of the guitar sets a somber and sedate mood. It's almost as though we hear Cohen sitting down and thinking about whether or not to write the letter. The music is so calm that one imagines it could simply fade out before the song begins.
As Cohen moves to the second line, "I'm writing you now just to see if you're better," a group of female backup singers slowly fades in and accompanies him until the end of the stanza. They seem to somehow push the singer over the hump, to give him the momentum he needs to sing about the emotionally fraught friendship that he describes in the song.
Cohen's voice is a high baritone (between bass and tenor) that's soft and beautiful enough to seem like an instrument all by itself. For much of the song, it stays low and sedate. It's only at a few pivotal points that he pushes his voice up an octave, notably in the famous chorus: "And Jane came by with a lock of your hair..." Later, Cohen again raises his voice for the poignant line: "Well, I see you there with the rose in your teeth." In part because these moments are so rare, they are especially affecting.
As the song moves on, violins also come in to create the feeling of a slightly fuller atmosphere. In general, though, the music is very demure. Whereas many singers try to craft lyrics that will align easily with guitar loops, Cohen is a poet and the music shapes itself around his words.