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Haun got his start in the music business through his childhood friend Michael Sherwood and toured with Michael's father’s (Bobby Sherwood) jazz band at 15 years old playing jazz standards and hits of the day. Too young to go into the lounges they toured, the boys were forced to sit backstage between shows and work up their own brand of music. Out of this came their first band Lodgic and later Michael's younger brother Billy Sherwood would take over the role as bass player. They released an album in 1985 on A&M Records produced by David Paich and Steve Porcaro of the group Toto. Some touring followed by opening for Supertramp in the mid-'80s. Thanks to Steve Lukather, Haun got some gigs working as a studio musician playing on numerous recordings. Around this time, Haun met Michael Ruff and did some west coast touring where he then met Sheryl Crow who was singing backup for Ruff and joined one of her first bands along with Greg Phillinganes, Amand Grimaldi and Jerry Watts on bass. Haun went on to play much of the guitar on Yes's Union album (1991; produced by Jonathan Elias), before work in The Key with Billy Sherwood and The Chris Squire Experiment with Sherwood and Chris Squire. He also did session work for Air Supply, Kitaro and Jon & Vangelis during this period.
Haun got involved with Yes producer Jonathan Elias and started composing music for TV and film. In 1996, Haun composed the Mandalay pictures logo. Haun worked extensively with Elias, including his American River album (2004) but also on music for commercials. Haun created the original yodel logo for Yahoo in 1996 as well as the voice for Expedia.com.
Haun re-joined Sherwood in the Yes spin-off supergroup Circa, formed in 2006, also with Tony Kaye (ex-Yes) and initially Alan White (currently of Yes). Sherwood, Kaye and Haun were also in Yoso with singer Bobby Kimball. Yoso subsequently disbanded and Haun left Circa to focus on commercial work.
In September 2012, Haun announced that he was working on a Jon Anderson solo album with Elias and M. Sherwood, but the project subsequently stalled before completion.