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Born in County Westmeath, Ireland, to Welsh parents, he went to Hereford Cathedral School then studied medicine in London, but had also pursued the study of music from an early age. He studied composition with Alan Rawsthorne and Patrick Hadley. His most successful early works included a setting of T. S. Eliot's The Hollow Men (1939), several ballets, including A Mirror for Witches, with choreography by Andrée Howard (1952) and Blood Wedding (1953), and an opera Yerma (1954).
Around 1960, he began composing in a serialist style. He was influenced in this by Edward Clark, a conductor, former BBC music producer, student of Arnold Schoenberg and husband of Elisabeth Lutyens. This continued until the late 1980s, when he returned to diatonic composition.
ApIvor is particularly well known to guitarists, as he made a major contribution to the repertoire of their instrument. Solo works include Variations (1959), Discanti (1970), Saeta (1972), and ten serial pieces included with his book Serial Composition for Guitarists (1982). He also wrote a Concertino for guitar (1954), Liaison for guitar and keyboard (1976), and Cinquefoil for flute, guitar, and viola (1984).