My layman's take is that TFA is probably wrong and that language started with song. Gestures are probably the origin of communication (see also: speakers of non-mutually-intelligible languages tend to instinctively make gestures, often created on-the-fly to mimic some intended meaning or desire), but language is different. Meaning in language is encoded in a mutually-intelligible format that is passed down memetically rather than genetically (which is why you see different "accents" in genetically-similar but geographically-separated populations in all of the aforementioned species). We instinctively recognize the resemblance between a given gesture and the object or action it's mimicking, but the meaning of language (like the connotation of particular elements of song, like note progression, which again varies between cultures) has to be acquired.

Again, speaking out of my a*. Feel free to tear apart.