The character David X. Machina is a play on the phrase "Deus Ex Machina", a plot device whereby a seemingly unsolvable problem in a story is suddenly and abruptly resolved by an unexpected and unlikely occurrence, which seems to happen frequently on "The Man From U.N..C.L.E."
When Ilya begins to play the horn during rehearsal, the first 7 notes are beginning of the "Man From U.N.C.L.E." theme.