Definition and background:
The third part in a Medieval polyphonic composition. (1) the third voice to be composed. In organum and the early motet, it may share the range of the duplum/motetus, but usually has a higher tessitura (being the top note at cadences, for instance). In the fourteenth century, however, the triplum is often a countermelody which is found above the primary cantus line at the top of the texture regardless of the total number of voices. (2) in organum, a piece which has a third "triplum" line can be called an "organum triplum" or "triplum" for short. Perotin wrote a number of tripla.
For more information on the music term "Triplum" check out these other resources:
Wikipedia - Glossary of Musical Terminology
Virginia Tech Multimedia Music Dictionary
ORB -- Medieval Music Glossary