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Caia Augusta Juliette Greene (born Aarup) was a Danish pianist and composer who played a prominent role in New York's musical life during the early twentieth century. After touring Scandinavia as a concert pianist, she settled in New York, where she taught piano at the Metropolitan Conservatory of Music (later the Metropolitan College of Music), the Associate School of Music, the Brookfield Center Summer School, and private studios at Carnegie Hall. Alongside her teaching, she performed as a song accompanist and composed songs and piano works.