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He was born in Toronto, and began playing the double bass at the age of eleven in order to complete a bluegrass trio with his brothers, Paul Quarrington and Tony Quarrington. At the age of thirteen, he began to study with Thomas Monohan, who was at the time the principal bassist of the Toronto Symphony Orchestra. He also studied under Peter Madgett, who is still a member of the TSO. After he received a degree in music from the University of Toronto, he went on to Austria and Italy to study under two renowned double bass pedagogues, respectively Ludwig Streicher and Franco Petracchi.
Quarrington has won numerous music competitions, including first prize in the CBC talent festival in 1976, and second prize in the Geneva International Competition in 1978 (first prize was not awarded that year). He played as principal bass with the Hamilton Philharmonic Orchestra between 1979 and 1988, and the Canadian Opera Company between 1989 and 1991, and then proceeded to become principal bass of the Toronto Symphony Orchestra. Since September 2006, he has been the Principal Bass of the National Arts Centre Orchestra. Quarrington has played concerti with many North American orchestras, including the Winnipeg Symphony Orchestra, the TSO and the NACO with Pinchas Zukerman. In April 2006, he played the world premiere of John Harbison's double bass concerto, which was commissioned by the International Society of Bassists.
Quarrington plays on a 1630 Giovanni Paolo Maggini double bass, and is well known for his unusual practice of tuning his bass in intervals of fifths, like a cello, but an octave lower.
He has made three solo recordings, one, on CBC Records, containing a wide variety of pieces for the solo double bass with orchestra, and two others, on Naxos Records. The first Naxos release contains works for double bass and piano by Giovanni Bottesini. The second disc, also featuring the music of Bottesini, was released in 2008. For that recording Quarrington collaborated with such artists as clarinetist Jim Campbell, soprano Monica Whicher, and American bassist Hal Robinson, principal bassist of the Philadelphia Orchestra.