"Für Elise" (German: for Elise) is the popular name of the "Bagatelle in A minor", WoO 59, a famous piece of music for solo piano by Ludwig van Beethoven, written in 1808.
It is not entirely certainly who "Elise" was. The most popular theory is that Beethoven originally titled his work "Für Therese", Therese being Therese von Malfatti, the daughter of a Viennese medical doctor, and at the time the focus of Beethoven's attention. When the work was published after Beethoven's death, it is thought that his famously illegible handwriting was misinterpreted as "Für Elise".
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