Brumby was born in Melbourne and educated at the Glen Iris State School, Spring Road Central School, and Melbourne Boys' High School. He studied at the University of Melbourne Conservatorium of Music, from which he graduated in 1957 with a diploma in education. In 1953 he was a finalist in the Australian Youth Aria competition, eventually winning the Lieder Award. He was organist at St. Oswald's Glen Iris from 1950 to 1953. Before travelling to Europe in 1962 he taught in Queensland schools and was for a time the head of music at Kelvin Grove Teacher's College. He went to Spain to study advanced composition with Philipp Jarnach, and to London to study with Alexander Goehr. On his return to Australia, he joined the staff of the Music Department at the University of Queensland, and was based in Brisbane ever since. He became Associate Professor with the University of Queensland, from which he retired in 1998. He, along with Philip Bračanin, are two Brisbane-based composers who have attained an international reputation, beginning in the 1970s, and joined more recently by composers such as Gerard Brophy, Stephen Cronin, Robert Davidson, Kent Farbach, Stephen Leek, Peter Rankine and Nigel Sabin who have attained similar renown. Brumby was Musical Director of the Queensland Opera Company from 1968 to 1971.
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