For the bulk of his early career Jenkins was known as a jazz and jazz-rock musician, playing baritone and soprano saxophones, keyboards and oboe, an unusual instrument in a jazz context. He joined jazz composer Graham Collier’s group and later co-founded the jazz-rock group Nucleus, which won first prize at the Montreux Jazz Festival in 1970. He joined the Canterbury progressive rock band Soft Machine in 1972 and co-led their very last performances in 1984. The group defied categorisation and played venues as diverse as The Proms, Carnegie Hall, and the Newport Jazz Festival. The album on which Jenkins first played with Soft Machine, Six, won the Melody Maker British Jazz Album of the Year award in 1973. Jenkins also won the miscellaneous musical instrument section (as he did the following year). Soft Machine was voted best small group in the Melody Maker jazz poll of 1974. After Mike Ratledge left the band in 1976 Soft Machine did not include any of its founding members, but kept recording on a project basis with line-ups revolving around Jenkins and drummer John Marshall. Balanced against Melody Maker’s positive view of the Soft Machine of 1973 and 1974, Hugh Hopper, involved with the group since replacing bassist Kevin Ayers in 1968, cites Jenkins’s “third rate” musical involvement in his own decision to leave the band, and the band of the late 1970s has been described by band member John Etheridge as wasting its potential. In November 1973, Jenkins and Ratledge participated in a live-in-the-studio performance of Mike Oldfield’s Tubular Bells for the BBC. It is available on Oldfield’s Elements DVD.
Jenkins has created a good deal of advertising music, twice winning the industry prize in that field. Perhaps his most-heard piece of music is the classical theme used by De Beers diamond merchants for their television advertising campaign focusing on jewellery worn by people otherwise seen only in silhouette. Jenkins later included this as the title track in a compilation called Diamond Music, and eventually created Palladio, using it as the theme of the first movement.