Moonlight Sonata
From Tardis Wiki, the free Doctor Who reference
The Moonlight Sonata was a Beethoven symphony that the Sixth Doctor attempted to play on the piano before stumbling and giving up, regretting turning down Elton John's lessons. (AUDIO: Project: Twilight)
The composition played in 2007 on John Ellis' 1950s radio; John smiled at the tune. (TV: Out of Time)
Behind the scenes
Neither story identifies the name of the piece, but the famous symphony is clearly recognisable in both.