Transformers have drawn attention in the MIR field for their remarkable performance shown in natural language processing and computer vision. However, prior works in the audio processing domain mostly use Transformer as a temporal feature aggregator that acts similar to RNNs. In this paper, we propose SpecTNT, a Transformer-based architecture to model both spectral and temporal sequences of an input time-frequency representation. Specifically, we introduce a novel variant of the Transformer-in-Transformer (TNT) architecture. In each SpecTNT block, a spectral Transformer extracts frequency-related features into the frequency class token (FCT) for each frame. Later, the FCTs are linearly projected and added to the temporal embeddings (TEs), which aggregate useful information from the FCTs. Then, a temporal Transformer processes the TEs to exchange information across the time axis. By stacking the SpecTNT blocks, we build the SpecTNT model to learn the representation for music signals. In experiments, SpecTNT demonstrates state-of-the-art performance in music tagging and vocal melody extraction, and shows competitive performance for chord recognition. The effectiveness of SpecTNT and other design choices are further examined through ablation studies.