When alto saxophonist Rudresh Mahanthappa recorded his Charlie “Bird’’ Parker tribute album “Bird Calls’’ two summers ago, he didn’t include a single cover from the bebop legend’s own songbook. Instead, Mahanthappa whipped his crack new quintet with trumpet phenom Adam O’Farrill, pianist Matt Mitchell, bassist François Moutin, and drummer Rudy Royston through new compositions he created from nuggets he had lovingly culled from songs in the Parker canon during long nights in a rented studio space near his Montclair, N.J., home.

Hints at the source material are contained in the new tunes’ titles. “Both Hands’’ is derived from “Dexterity,’’ “Maybe Later’’ from “Now’s the Time,’’ “Talin Is Thinking’’ (Talin being the name of Mahanthappa’s young son) from “Parker’s Mood,’’ “Sure Why Not?’’ from “Confirmation,’’ “On the DL’’ from “Donna Lee,’’ and so forth. The clues help: Even Mahanthappa’s band needed prompting to recognize from where the new music they were recording had been conjured.

The album, released in February 2015, won Mahanthappa album of the year in the DownBeat Critics Poll and tied for top honors in the NPR Music Jazz Critics Poll.

“I don’t know if surprised is the right word,’’ says Mahanthappa, 44, who will perform material from “Bird Calls’’ Wednesday in a sold-out show at the Institute of Contemporary Art (and be interviewed onstage beforehand by a sometime collaborator, guitarist and Berklee professor David Gilmore). “I was definitely happy. I had a sense that it was really timely and refreshing for people to hear me going back to an acoustic format, which I hadn’t done since ‘Codebook,’ and a more traditional lineup. And this orientation with one of the most prominent figures in the history of this music — maybe it took them by surprise! For me, it’s all part of the continuum of what I’m doing, but maybe it’s kind of the right time at the right place as far as my discography goes.’’

His next album will be with his Indo-Pak Coalition partners Rez Abbasi (guitar) and Dan Weiss (drums, tabla). They’ll record in August for release early next year. He hopes to eventually record another album with the “Bird Calls’’ band as well.

In the meantime, he’ll continue touring “Bird Calls.’’ Sometimes this requires finding substitutes for his busy sidemen, but Mahanthappa’s bench is deep enough that the music doesn’t suffer. “I’m not just hiring anybody,’’ he notes. “I’m looking for people where the music will very quickly become second nature to them. Part of it is me being lucky enough to have the bandmates that I have, the subs that I have, and keep the company that I keep.’’

At the ICA, Thomson Kneeland and Jordan Perlson will sub for Moutin and Royston, respectively. But Mitchell, who has two critically acclaimed albums of his own behind him (and, like Royston, is also a member of Dave Douglas’s formidable quintet), and O’Farrill, whose debut album as a leader is out next month, will both be on hand.

Mahanthappa’s relationship with O’Farrill, 21, is especially noteworthy. O’Farrill’s trumpet makes him a Dizzy Gillespie-like foil to Mahanthappa’s Parker, particularly on the irresistible track “Chillin’?’’ (Mahanthappa’s reimagining of Parker’s “Relaxin’ at Camarillo’’). His recording “Bird Calls’’ at age 19, three months before placing third in the 2014 Monk Institute trumpet competition, calls to mind Miles Davis recording with Parker at a similar age. And O’Farrill and Mahanthappa both appear on “Cuba: The Conversation Continues,’’ which earned O’Farrill’s father, Arturo O’Farrill, best Latin jazz album at last month’s Grammy Awards.

Playing with Mahanthappa has been invaluable for O’Farrill. Mahanthappa mostly teaches by example. “Every night he was just delivering it full throttle,’’ O’Farrill recalls of a recent tour. “The thing that really makes Rudresh stand out is how he’s able to bring such a high level of intensity and action to everything he does. And emotion, too. And soul. That’s what’s been amazing to take from the experience.’’

Sometimes the lessons are more explicit. “We have this tag at the end of a tune where we’re kind of playing off of each other,’’ says O’Farrill, describing one such lesson. “I kind of kept imitating him too much, without creating my own ideas for him to play off of. Onstage, he said, ‘Play something else!’ He didn’t yell it aggressively. It was kind of funny, but it was also good. I think it’s important for those things to happen. It’s the only way you really learn and grow.’’

What Mahanthappa gets in return from sidemen O’Farrill’s age is being reenergized, onstage and off. He recalls having a couple of days off in Paris with O’Farrill and substitute pianist Joshua White, both on their first tour of Europe. “Adam and Joshua were up at 8 a.m. trying to go to all the museums, and I was like, ‘Wait, I’ll do that too.’ There was a time where I did all that stuff. As I’ve started touring more, unless I’m really in a place I’ve never been, I have a million things to do and don’t necessarily have time to do that. I wind up doing a lot of work. That’s all to say that, being in some of these places, I kind of felt like I was there for the first time, seeing it through their eyes.’’

Consider Wednesday’s concert taking place at a museum a fitting coincidence, then. And know that Mahanthappa’s artful tribute to Bird is every bit as cutting-edge as it ought to be for any institute of contemporary art.

Bill Beuttler can be reached at bill@billbeuttler.com.