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Burrell was born in Norwich, where her father, a schoolteacher by profession, served as an assistant organist at the cathedral, and attended Norwich High School for Girls before studying music at Cambridge University. She began her career as a viola player, but soon became well known for her compositions and became a full-time composer. One of her first compositions was for the 1980 St Endellion Music Festival. She used to attend as a viola player, but festival organizer Richard Hickox told her she "ought to write something for the festival. The result, Missa Sancte Endeliente, was a large-scale mass using Cornish and Latin texts.
Her first major orchestral piece was titled Landscape (1988). It describes the wild, windswept countryside. It was one of the winners of the "Encore" awards organised by the Royal Philharmonic Society and BBC Radio 3. Another notable orchestral work was Das Meer, das so groß und weit ist, da wimmelt's ohne Zahl, große und kleine Tiere ("The Sea is so big and wide and swarming with numerous little animals"). It describes the sounds and the atmosphere of the sea.
Burrell has written other orchestral works including concertos for viola, flute and clarinet, an opera The Albatross (1997), lots of choral works and chamber music. She likes modern architecture and her music sometimes shows this in the way it is shaped. She has also written music for young people, such as Lights and Shadows (1989) which includes a children's choir, a recorder group and lots of percussion.
She teaches at the Guildhall School of Music and Drama and became the Artistic Director of Spitalfields Festival in London in 2006, taking over from Jonathan Dove. In 2006, she was awarded a fellowship from the Arts and Humanities Research Council at the Royal Academy of Music, to compose a major series of ensemble organ works over five years. She lives in Harwich, Essex.