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We present DanceAnyWay, a generative learning method to synthesize beat-guided dances of 3D human characters synchronized with music. Our method learns to disentangle the dance movements at the beat frames from the dance movements at all the remaining frames by operating at two hierarchical levels. At the coarser "beat" level, it encodes the rhythm, pitch, and melody information of the input music via dedicated feature representations only at the beat frames. It leverages them to synthesize the beat poses of the target dances using a sequence-to-sequence learning framework. At the finer "repletion" level, our method encodes similar rhythm, pitch, and melody information from all the frames of the input music via dedicated feature representations. It generates the full dance sequences by combining the synthesized beat and repletion poses and enforcing plausibility through an adversarial learning framework. Our training paradigm also enforces fine-grained diversity in the synthesized dances through a randomized temporal contrastive loss, which ensures different segments of the dance sequences have different movements and avoids motion freezing or collapsing to repetitive movements. We evaluate the performance of our approach through extensive experiments on the benchmark AIST++ dataset and observe improvements of about 7%-12% in motion quality metrics and 1.5%-4% in motion diversity metrics over the current baselines, respectively. We also conducted a user study to evaluate the visual quality of our synthesized dances. We noted that, on average, the samples generated by our method were about 9-48% more preferred by the participants and had a 4-27% better five-point Likert-scale score over the best available current baseline in terms of motion quality and synchronization. Our source code and project page are available at https://github.com/aneeshbhattacharya/DanceAnyWay.