She was born in Belfast, where she received her early musical training. She studied piano at the Trinity College of Music in London, with voice as a second subject. She was initially a Mezzo-Soprano, and sang as such in the Ambrosian Chorus; her fellow altos included Jean Allister, Pamela Bowden, and Helen Watts. She also sang in the BBC Chorus, before retraining as a soprano with Professor Frederick Husler and Yvonne Rodd-Marling, authors of "Singing: The Physical Nature of the Vocal Organ". 
Her professional debut came in 1954 in Macbeth at the Oxford University Opera Club. From 1956 to 1975, she was a member of the English Opera Group. She is noted for her performance of Elsa in Wagner's Lohengrin, the title role in Strauss's Arabella, Ellen Orford in Britten's Peter Grimes, and the Governess in Britten's The Turn of the Screw. She appeared at Covent Garden, Bayreuth, San Francisco and the Metropolitan Opera (Contessa Almaviva in Le nozze di Figaro and in Peter Grimes). She also sang Charlotte in Massenet's Werther, for San Francisco Opera. She was a regular guest at the Teatro Colón in Buenos Aires where she sang many roles such as Margherite (Gounod's Faust), Arabella, Antonia (Tales of Hoffman), Vitellia (La Clemenza di Tito). Her farewell performance in Buenos Aires was as Ellen Orford in Peter Grimes. In 1975 in Kingsway Hall London, she was soprano in Verdi's Requiem, directed by Carlos Païta with the Royal Philharmonic Orchestra.
Harper has also had an extensive concert career, including singing in the premiere of Britten's War Requiem in 1962, substituting for Galina Vishnevskaya on 10 days' notice. In 1965 she was the soprano soloist in only the second UK performance (and only the fourth performance in the work's history) of Delius's Requiem, in Liverpool, under Charles Groves. She sang in it again in 1968 in London under Meredith Davies, and made the world premiere recording with the same forces. At the Belfast Last Night of the Proms in 1985, she gave the world première of Malcolm Williamson's song-cycle Next Year in Jerusalem to international critical acclaim.
Her recordings include Peter Grimes in both audio and video formats, as well as the War Requiem (Chandos). She took part in 1957 performances (conducted by Antony Hopkins) of sacred works by Michel-Richard Delalande, recorded in LP format on the L'Oiseau-Lyre label; these pieces had never previously found their way to disc. Superb renditions in the 1970s (conducted by Sir Georg Solti) are now available, notably Mahler's 8th Symphony with Watts, Minton, Popp, Kollo, Shirley-Quirk and Talvela (Chicago Symphony Orchestra, 1972, Decca) and Strauss's Die Frau ohne Schatten with Dernesch, Hesse, King and Berry (Covent Garden, 1976, Fiori). More recently, a live concert performance of Britten's Our Hunting Fathers has been issued on the London Philharmonic Orchestra's own label.
- Obituary for Meredith Davies, Telegraph, 2 April 2005.
- Philip Reed, Obituary for Meredith Davies. The Guardian, 30 March 2005
- Andrew Clements, "Britten: Peter Grimes, Royal Opera". The Guardian, 3 October 2003.
- Andrew Clements, "Keynotes". The Guardian, 23 April 1999.
- Andrew Clements, "Wagner: Orchestral Excerpts from Die Meistersinger, etc.: London Philharmonic/ Tennstedt". The Guardian, 29 April 2005.