Gunther Schuller is credited with coining the term Third Stream, meaning compositions where twentieth-century art music forms exist simultaneously with jazz. Furthermore, Schuller specifically states in the liner notes to the debut recording of Eine Kleine Posaunenmusik "The work is not a Third Stream piece." Yet the concerto alludes to jazz through a multitude of slide glissandi and plunger mute effects, Solotone mute passages, specific references to the jazz trombone styles of Tommy Dorsey and Lawrence Brown, musical quoting or indirect reference, and the use of a walking bass line in Movement V, Finale. What makes one piece Third Stream and another simply a modern composition with jazz implications? Is Third Stream primarily a compositional designation or a performance practice stipulation? How does a celebrated trombone soloist inspire and collaborate with a distinguished composer in the creation of a major work? The somewhat conspicuous title, Eine Kleine Posaunenmusik, seems to point towards Mozart's famous string serenade Eine Kleine Nachtmusik. What connection to Mozart, if any, does Schuller's title suggest? All of these questions are elucidated in this study through careful investigation and research of Gunther Schuller's Eine Kleine Posaunenmusik. New interviews with John Swallow and Gunther Schuller are included.