"Augie's Great Municipal Band", which concludes The Phantom Menace, is very strongly related to "The Sith Theme", as first heard in Return of the Jedi.
"Augie's" contains two primary musical components: the melody (sung by children's chorus) and the fanfare (played by trumpets) - both of which are directly derived from "The Sith Theme". The first measure of both "The Sith Theme" and the "Augie's" melody contain three notes, the first and last of which are the same, the middle of which is comparable. The second measure of both then use different notes, but share intervallic contour (down a minor third, down a major second). The second half of the themes have less in common, although they both do use the tonic note on beats 1 and 3 of the third measure.
The "Augie's" fanfare, then, adopts a similar relationship to "The Sith Theme" for its beginning (first and third notes identical, with second note comparable), followed by the same intervallic pattern (descending minor third, descending major second). Those last two notes (F# and E) are then repeated twice (so heard a total of three times - indicated in brackets at the bottom of the example below) using different fanfare rhythms. The second half of the fanfare uses a copy/paste repetition of the first 13 notes (indicated with the large rectangles in the example below), with the final three tones matching up pitches with the "Augie's" melody.
The use of the modified "Sith Theme" at the conclusion of The Phantom Menace musically illustrates the true (if behind-the-scenes) victor of the film: Senator Palpatine has just been promoted to Supreme Chancellor, and it's only a matter of time before he declares himself emperor and reveals his secret identity as a Sith lord.
The Music of Star Wars
These posts will help focus and develop my analyses of John Williams' film scores.