The Collaborative International Dictionary of English v.0.48:

Triforium \Tri*fo"ri*um\, n. [LL., fr. L. tri- (see Tri-) +
   foris, pl. fores, a door.] (Arch.)
   The gallery or open space between the vaulting and the roof
   of the aisles of a church, often forming a rich arcade in the
   interior of the church, above the nave arches and below the
   clearstory windows.
   [1913 Webster]