Automated, data-driven construction and evaluation of scientific models and theories is a long-standing challenge in artificial intelligence. We present a framework for algorithmically synthesizing models of a basic part of human language: morpho-phonology, the system that builds word forms from sounds. We integrate Bayesian inference with program synthesis and representations inspired by linguistic theory and cognitive models of learning and discovery. Across 70 datasets from 58 diverse languages, our system synthesizes human-interpretable models for core aspects of each language’s morpho-phonology, sometimes approaching models posited by human linguists. Joint inference across all 70 data sets automatically synthesizes a meta-model encoding interpretable cross-language typological tendencies. Finally, the same algorithm captures few-shot learning dynamics, acquiring new morphophonological rules from just one or a few examples. These results suggest routes to more powerful machine-enabled discovery of interpretable models in linguistics and other scientific domains.