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Moore was born in Cutchogue Long Island, New York. In 1917, he graduated from Yale University. He then served in the Navy as a lieutenant, after which he went to Paris where he studied with Nadia Boulanger. In 1921, Moore went to Cleveland as Director of Music at the Cleveland Museum of Art, during which he studied with Ernest Bloch at the Cleveland Institute of Music, and performed in plays at the Cleveland Playhouse. In 1926, Moore joined the music faculty at Columbia University, where he remained until his retirement in 1962.