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Data-driven approaches to automatic drum transcription (ADT) are often limited to a predefined, small vocabulary of percussion instrument classes. Such models cannot recognize out-of-vocabulary classes nor are they able to adapt to finer-grained vocabularies. In this work, we address open vocabulary ADT by introducing few-shot learning to the task. We train a Prototypical Network on a synthetic dataset and evaluate the model on multiple real-world ADT datasets with polyphonic accompaniment. We show that, given just a handful of selected examples at inference time, we can match and in some cases outperform a state-of-the-art supervised ADT approach under a fixed vocabulary setting. At the same time, we show that our model can successfully generalize to finer-grained or extended vocabularies unseen during training, a scenario where supervised approaches cannot operate at all. We provide a detailed analysis of our experimental results, including a breakdown of performance by sound class and by polyphony.