Bergsma attended Stanford University for two years (1938–40) before moving to the Eastman School of Music, where he earned his bachelor's and master's degrees; his most significant teachers there were Howard Hanson and Bernard Rogers.
Students of Bergsma include composers Jack Behrens, Philip Glass, Karl Korte, Robert Parris, and Steve Reich.
He eschewed the avant-garde—his obituary in the Seattle Post-Intelligencer describes him as having "never deserted tonality" and seeing "dozens of his former avant-garde colleagues returning to the fold"—though he did embrace aleatoric techniques later in his career.
The first is a somber tale of a 16th-century French peasant's disappearance and return upon which he is suspected to be an impostor; the music is marked by dissonance which emphasizes the tension in the story, particularly in the final courtroom scene.
The partially aleatoric orchestral writing is intended to be the voice of Stalin, and uses quotes from Carmen, La traviata and Don Giovanni for comedic effect.