Growing up in the San Francisco Bay Area, Alessi eventually found freelance jobs with various chamber ensembles as well as work with the San Francisco Symphony Orchestra. During the mid-'80s Alessi attended the California Institute for the Arts where he studied with bassist Charlie Haden and performed with various jazz luminaries, including saxophonist Bill Perkins and pianist Frank Strazzeri as well as Haden's Liberation Music Orchestra. In 1990, armed with a B.F.A. degree in jazz trumpet performance and an M.F.A. degree in jazz bass performance, Alessi moved to New York City. Since then he has performed with such longtime collaborators as saxophonist Ravi Coltrane, pianist Uri Caine, and saxophonist Steve Coleman.

between 2001 and 2013, Alessi recorded ten albums as a leader for a variety of labels including RKM Music, Clean Feed, Between The Lines, CAM Jazz, and Whirlwind. Later that year he signed to Manfred Eicher's ECM label. Baida, his debut there, won wide acclaim for its imaginative compositions. He co-led a date for Babel entitled Sugar Blade with Steve Davis and Kris Davis in 2015, and delivered Quiver, his sophomore ECM offering with his quartet--Nasheet Waits, Drew Gress and Gary Versace in early 2016. ~ Matt Collar