Finck was born in Rochester, New York. He studied under Sam Goradetzer and Michael Shahan of the Philadelphia Orchestra, and graduated from Eastman School of Music in 1980. He played with Woody Herman in 1980-81 and then moved to New York City, where he played with Joe Williams, Annie Ross, Mel Lewis, Al Cohn, Ernestine Anderson, Rosemary Clooney, Tom Harrell, Jerry Dodgion, Phil Woods, Clark Terry, and Al Grey in the 1980s. He worked with Paquito D'Rivera and Steve Kuhn in the 1990s, as well as Freddie Hubbard, Makoto Ozone, and Eddie Daniels.
Finck's debut release as a leader, Future Day, was released in 2008 on Soundbrush Records. The album features Joe Locke, Tom Ranier, and Joe LaBarbera, as well as guest appearances from Jeremy Pelt and Bob Sheppard.
- Future Day (Soundbrush, 2008)
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With Jon Benjamin
- Well, I Should Have…* (Sub Pop, 2015)
With Steve Kuhn
With Tisziji Munoz
- Incomprehensibly Gone (Anami Music, 2010)
- Beauty As Beauty (Anami Music, 2013)
- Realization of Paradox: Melting the Mind of Logic (Anami Music, 2014)
- Songs of Soundlessness (Anami Music, 2015)