San Francisco's Turtle Island Quartet looks like any other well-scrubbed chamber ensemble, with one crucial difference: Onstage, the group can improvise like seasoned Blue Note veterans. Since forming in 1985, TIQ — violinist/founder David Bala­krishnan, violinist Mads Tolling, cellist Mark Summer and violist Jeremy Kittle — have cross-pollinated the classical repertoire with everything from bluegrass and bebop to Indian and Latin American folk music. Their 2005 CD 4 + Four (Telarc) — a collaboration with Chinese-American siblings The Ying Quartet, who join them Tuesday — is a wonderful introduction to the TIQ's far-ranging milieu, weaving the bluesy swing of "Yearnin'" and an arrangement of John Lennon's "Because" around works by French modernist Darius Milhaud and Balakrishnan's own "Mara's Garden of False Delights" (based on the three states of being in Hindu philosophy) and even taking a swipe at Prince's "Delirious" in "Variations on an Unoriginal Theme." 4 + Four netted TIQ a 2006 Grammy for Best Classical Crossover, and last year's A Love Supreme, their tribute to John Coltrane's iconic 1964 suite that throws in jazz standards like Thelonious Monk's "'Round Midnight" and Miles Davis's "So What," seems certain to follow suit next month.