Which type of music features a vocal quartet in which each voice has its own role and the bass singer handles the repeated foundation part of often in the form of a short phrase of nonsense syllables?

The type of music you’re referring to is likely "barbershop" music. In this style, a vocal quartet typically consists of a lead, tenor, baritone, and bass, with each voice fulfilling a specific harmonic role. The bass singer often provides a foundational line, sometimes using nonsense syllables as part of the arrangement. This creates a rich, close-harmony sound characteristic of the barbershop genre.