Moore was natural within Cutchogue Long Island, New York. He went to the Hotchkiss School, in which he met Archibald MacLeish. Inside 1917, he graduated from either Yale University. He so served in the Navy as a lieutenant, fallowing which he attend Paris where he studied by having Nadia Boulanger. Within 1921, Moore attend Cleveland as Director of Music at the Cleveland Museum of Art, during which he exposed by using Ernest Bloch at a Clevel& Institute of Music, and performed around plays at the Cleveland Playhouse. Within 1926, Moore joined a music faculty at Columbia University, where he remained until his retirement within 1962.