In the auditory cortex of awake animals, a substantial number of neurons do not respond to pure tones. These neurons have historically been classified as "unresponsive" and even been speculated as being nonauditory. We discovered, however, that many of these neurons in the primary auditory cortex (A1) of awake marmoset monkeys were in fact highly selective for complex sound features. We then investigated how such selectivity might arise from the tone-tuned inputs that these neurons likely receive. We found that these non-tone responsive neurons exhibited nonlinear combination-sensitive responses that require precise spectral and temporal combinations of two tone pips. The nonlinear spectrotemporal maps derived from these neurons were correlated with their selectivity for complex acoustic features. These non-tone responsive and nonlinear neurons were commonly encountered at superficial cortical depths in A1. Our findings demonstrate how temporally and spectrally specific nonlinear integration of putative tone-tuned inputs might underlie a diverse range of high selectivity of A1 neurons in awake animals. We propose that describing A1 neurons with complex response properties in terms of tone-tuned input channels can conceptually unify a wide variety of observed neural selectivity to complex sounds into a lower dimensional description.