Vivaldi was the composer of "The Four Seasons", which included "Spring" and "Summer". (PROSE: Outsourcing)
He once offered to teach the Doctor how to play the violin. (AUDIO: The Paradox Planet)
Vivaldi was often playing during the First Doctor's visits to a tavern on Rigel. (PROSE: Byzantium!)
When the Sixth Doctor called Signpost to determine why it was interfering with his TARDIS, the "on-hold" music was "Spring". He listened to it twenty-one times and angrily demanded of Vishesh Abeyesekere that he play something else, even if it was only "Summer". (PROSE: Outsourcing)