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 has three children. Henry (b. 1995), Olivia (b. 1997) both adopted from Brazil, and biological daughter Vivian (b. 2001)
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 Steve Kuhn