Curiously, Aristotle never proposes a tri-partite division of genres in The Poetics, despite accusations to this effect. Here are his words (well, his translator's words):
"For the medium being the same, and the objects the same, the poet may imitate by narration—in which case he can either take another personality as Homer does, or speak in his own person, unchanged—or he may present all his characters as living and moving before us."
Here's another, probably better translation:
"A third difference in these arts is the manner in which one may represent each of these objects. For in representing the same objects by the same means it is possible to proceed either partly by narrative and party by assuming a character other than your own--this is Homer's method--or by remaining yourself without any such change, or else to prepresent the characters as carrying out the whole action themselves.
So there are three modes: one is clearly dramatic: the characters are there (as actors) in front of an audience. But the others are both narrative forms: one is a pure narrative form (without dialogue) and the other is a mixed mode, which he attributes to Homer. In Homer, there are speeches of characters as well as Homer's narration.
He hardly mentions the lyric in the entire work. Of course, lyric poetry would be in one's own voice, and hence the pure narrative mode (as opposed to the mixed one), but that's not what Aristotle actually says. This was a neoclassical idea, not one of Aristotle himself, as far as I can tell.