Lyapunov was born in Yaroslavl in 1859. After the death of his father, Mikhail Lyapunov, when he was about eight, Sergei, his mother, and his two brothers (one of them was Aleksandr Lyapunov, later a notable mathematician) went to live in the larger town of Nizhny Novgorod. There he attended the grammar school along with classes of the newly formed local branch of the Russian Musical Society. On the recommendation of Nikolai Rubinstein, the Director of the Moscow Conservatory of Music, he enrolled in that institution in 1878. His main teachers were Karl Klindworth (piano; a former pupil of Franz Liszt), and Sergei Taneyev (composition; a former pupil of Pyotr Ilyich Tchaikovsky and his successor at the Conservatory).

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