The Skye Boat Song

The Second Doctor played a slow tune on his recorder near the conclusion of the London Event. It was in part a diversion to prevent Colonel Lethbridge-Stewart from overhearing his discussion with Jamie McCrimmon on the Piccadilly Circus tube station platform. (TV: The Web of Fear)

After getting access to the TARDIS years later, the produced the Doctor's recorder and proceeded to play this song. (TV: The Power of the Doctor)

Behind the scenes
The tune played by the Second Doctor in The Web of Fear, and the in The Power of the Doctor, is "The Skye Boat Song", a nineteenth century Scottish folk song. However, the song is never referred to by name in either story. By the time the tune made a re-appearance in The Power of the Doctor, it had become known as the theme song from the television series Outlander.