Vocal communication evokes robust responses in primary auditory cortex (A1) of songbirds, and single neurons from superficial and deep regions of A1 have been shown to respond selectively to songs over complex, synthetic sounds. However, little is known about how this song selectivity arises and manifests itself on the level of networks of neurons in songbird A1. Here, we examined the network-level coding of song and synthetic sounds in A1 by simultaneously recording the responses of multiple neurons in unanesthetized zebra finches. We developed a latent factor model of the joint simultaneous activity of these neural populations, and found that the shared variability in the activity has a surprisingly simple structure; it is dominated by an unobserved latent source with one degree-of-freedom. This simple model captures the structure of the correlated activity in these populations in both spontaneous and stimulus-driven conditions, and given both song and synthetic stimuli. The inferred latent variability is strongly suppressed under stimulation, consistent with similar observations in a range of mammalian cortical regions.