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Kavafian was born in Turkey to Armenian parents. She moved with her family to America in 1956, and began studying violin in Detroit at age six. Her teachers included Ara Zerounian, Mischa Mischakoff, Oscar Shumsky, and Ivan Galamian, the last two of which she studied under while attending the Juilliard School from 1969 to 1975. Her first major exposure came when she won the Vianna da Motta International Violin Competition in Lisbon in 1973. She won the Young Concert Artists International Auditions in 1978 which led to her New York recital debut at Carnegie Hall. That same year she became a member of the Tashi ensemble with Peter Serkin, who also accompanied her for her New York solo debut.
She began performing with her sister, Ani, in 1983, when the pair played together at Carnegie Hall. In 1983-84 she toured with Chick Corea. A member of the Chamber Music Society from 1989–93 and 1996–2002, she played with the Beaux Arts Trio from 1992-98 and sporadically thereafter. She founded her own group, Opus One, in 1998, with Steven Tenenbom, Anne-Marie McDermott and Peter Wiley. She teaches at the Curtis Institute of Music, the Juilliard School and the Bard College Conservatory of Music. She plays a J. B. Guadagnini violin made in Milan in 1751, and a Moes and Moes viola, made in 1987.
Kavafian's repertory includes classical and early romantic works by Mozart, Beethoven, and Mendelssohn, as well as 20th century works of Ruth Crawford Seeger, Charles Wuorinen, and Toru Takemitsu. Takemitsu has composed a concerto for Kavafian. She is also the founder of two prestigious music festivals: the Bravo! Vail Valley Music Festival and the Music from Angel Fire.
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With Chick Corea
- Septet (ECM, 1984)
With Beaux Arts Trio
With Mark O'Connor
- Mark O'Connor String Quartets No.'s 2 & 3 (Omac 2009)