There is a piece of classical music playing in the background of
episodes 6 & 15 of Azumanga Daioh, just as the relay races starts,
that isn't on any of the soundtracks. Just wondering what the name of
it is, and who is the composer?

For many questions concerning movies and TV www.imdb.com is an
excellent resource. The only information I could find refering to
soundtracks for the episodes is in the link below. You may want to
check back often to see if it gets updated.

http://www.imdb.com/title/tt0339955/soundtrack

Yeah, I ended up finding out the answer from a friend, who found the
song on another anime soundtrack. Apparently, It's 'Comedian's Gallop'
by Dimitri Kabalevsky, and it was on one of the Fooly Cooly
soundtracks. Just lucky, I suppose.