Humans have the remarkable intuitive ability to use their voices to imitate any sound they hear. For example, imagine conveying the sound of an engine to a mechanic, a birdcall to a friend, or a strange new synthesizer to a musician. Vocal imitation is fundamental to human communication, and may even form the basis of early language learning. But it is also mysterious: how do we use the limited affordances of a vocal tract to communicate novel sounds far beyond its reach? We propose that the answer lies in recursive social reasoning: we present a computational model of vocal imitation that extends the Rational Speech Acts framework with a simulated vocal tract for the speaker, and a feature-based model of auditory perception for the listener. Without fitting any parameters, our model accurately predicts the types of phones human speakers choose when imitating a variety of real-world sounds (R^2 = 0.809).