An appreciative audience welcomed the Turtle Island String Quartet, fresh from its Grammy for Best Classical Crossover Album, to the Lied Center Friday evening. The album, “4 + Four,” featured the Turtle Island/Ying Quartet collaboration originally commissioned by the Lied Center four years ago.

But Friday it was all Turtle Island, with each member a talented instrumentalist, composer and arranger. The group members’ mood varied from jubilant to ecstatic, having received their Grammy notification just two days before. Their happiness lit up their music – and the audience.

Their always exceptional communication was heightened by frequent exchanges of smiles and nods, and their musicianship was spot-on.

The first set opened with Michael Brecker’s lively “Straphangin” and closed with Dizzy Gillespie’s “A Night in Tunisia.” Jazz fare included Oliver Nelson’s “Yearning,” the Billie Holiday song “You’ve Changed,” and original compositions by Mads Tolling (viola) and Mark Summer (cello).

The group’s founder and first violinist, David Balakrishnan, received Grammy nominations for his arrangements of the Gillespie and Holiday numbers, and rightly so. The genius of this group is its ability to sound not like a string quartet playing jazz, but like a jazz band, period.

It was soon easy to forget that those harmonies were not a brass chorus, that the violin was not a sax, that the cello was not a bass and a drum and a marimba. After listening to the brilliant founding member Mark Summer, one wondered why any band ever bothered to use anything but a cello for a jazz bass line and rhythm.

Solos were passed smoothly from one instrument to another, and the conclusion of “A Night in Tunisia” featured an extended violin conversation sprinkled with quotes from unexpected sources that had the audience chuckling appreciatively. The number ended in an impressive frenzy of presto bowing.

The evening’s second set, “A Love Supreme,” was a tribute to John Coltrane, and an effort to represent the inspirational nature of that jazz work. Here the illusion of traditional jazz instruments yielded to the awareness that a string quartet was playing jazz ensemble music very well. Balakrishnan, Summer and violinist Evan Price contributed arrangements and created the feeling that if Coltrane would have written for strings like this. The third number, “Countdown,” was notable for a fine, blindingly fast solo by Price strongly reminiscent of Claude “Fiddler” Williams.

The four-movement finale ended with a prayerful and solemn “Psalm” movement, and the evening concluded with Stanley Clarke’s “Song for John,” arranged by Tolling. As the stage lights dimmed to a pale blue-white, the group played this number with great sensitivity, at times lyrical, always elegiac, even when animated. The audience abandoned a rote standing-ovation attempt for enthusiastic and prolonged applause, and Turtle Island obliged with a hypnotic Miles Davis encore.