Deep learning-based probabilistic models of musical data are producing increasingly realistic results and promise to enter creative workflows of many kinds. Yet they have been little-studied in a performance setting, where the results of user actions typically ought to feel instantaneous. To enable such study, we designed Notochord, a deep probabilistic model for sequences of structured events, and trained an instance of it on the Lakh MIDI dataset. Our probabilistic formulation allows interpretable interventions at a sub-event level, which enables one model to act as a backbone for diverse interactive musical functions including steerable generation, harmonization, machine improvisation, and likelihood-based interfaces. Notochord can generate polyphonic and multi-track MIDI, and respond to inputs with latency below ten milliseconds. Training code, model checkpoints and interactive examples are provided as open source software.