Binaural sluggishness refers to the binaural system's inability to follow fast changes in the interaural configuration of the incoming sound stream. Several studies have measured binaural sluggishness by measuring signal detection in conditions of binaural unmasking when the interaural configuration of the masker is changed over time. However, it has been shown that, in conditions of binaural unmasking, binaural sluggishness also affects the perception of temporal changes in the properties of the signal (i.e., its frequency or level) and not just in the interaural configuration of the masker. By measuring the temporal modulation transfer function for sinusoidally modulated noise presented in conditions of binaural unmasking, the first experiment of the current study showed that, due to binaural sluggishness, the internal representation of binaurally unmasked sounds conveys little or no information about envelope fluctuations with rates within the pitch range (i.e., above 30 Hz). The second experiment measured the masked detection threshold for musical interval recognition in binaurally unmasked harmonic tones and showed that, in conditions of binaural unmasking, pitch wanes when the harmonics become unresolved by the cochlear filters. These results suggest that binaural sluggishness precludes temporal pitch processing based on envelope cues in binaurally unmasked sounds.