Neural sequence-to-sequence TTS has achieved significantly better output quality than statistical speech synthesis using HMMs. However, neural TTS is generally not probabilistic and the use of non-monotonic attention both increases training time and introduces "babbling" failure modes that are unacceptable in production. This paper demonstrates that the old and new paradigms can be combined to obtain the advantages of both worlds, by replacing the attention in Tacotron 2 with an autoregressive left-right no-skip hidden Markov model defined by a neural network. This leads to an HMM-based neural TTS model with monotonic alignment, trained to maximise the full sequence likelihood without approximations. We discuss how to combine innovations from both classical and contemporary TTS for best results. The final system is smaller and simpler than Tacotron 2, and learns to speak with fewer iterations and less data, whilst achieving the same naturalness prior to the post-net. Unlike Tacotron 2, our system also allows easy control over speaking rate.