It’s notoriously difficult to capture pure collective improvisation on a jazz studio album, but cornetist Kirk Knuffke’s Arms & Hands (The Royal Potato Family) achieves the combustible energy of a live set without any audience interaction. The album was recorded in one day at Acoustic Recording in Prospect Heights, Brooklyn, with drummer Bill Goodwin and bassist Mark Helias forming the core trio and tenor and soprano saxophonist Jeff Lederer, alto saxophonist Daniel Carter and trombonist Brian Drye adding texture. Knuffke’s 15th album as a leader or co-leader harks back to an earlier era, when studio sessions required alacrity and risk-taking trumped finesse. Two years after Knuffke, Goodwin and Helias took their maiden voyage as a band, they still haven’t rehearsed.

“I’m a big fan of very few takes. There isn’t a song on that record that we did more than two takes of,” says the 35-year-old cornetist, sitting down recently at his cozy walk-up in midtown Manhattan, where he lives with his wife. “Even though there were only two at the most, half of them were first takes.” Longtime collaborator Drye was accustomed to Knuffke’s penchant for spontaneity and compositional vagaries. “I showed up and sight-read the tunes in one take and left. He writes in a way that allows for that to happen,” Drye says. “Still, I didn’t know what to expect.”

Knuffke facilitates the unexpected, starting with the collage-style handwritten sheet music. “Atessa” takes up two staves; the rest of the page consists of a black-and-white print of a cantilever bridge. “Next” fills three staves, juxtaposed by an orange dental diagram. “Tuesday,” marked “through composed,” is the freest, consisting entirely of retro emoji; Kirk is “blissful,” Mark “suspicious” and Bill “surprised.” The atonal result somehow swings. “Every time we play is like the first time,” Goodwin says. To Goodwin, Knuffke’s sound is as forward-thinking as they come, but steeped in a long tradition of melodic improvisation. “Kirk reminds me of a pure melody player, like Bobby Hackett or Ruby Braff, Warren Vaché perhaps-beautiful sound, tone production and a melodic gift,” he says. “It’s straight from the heart.”

Arms & Hands evades the standard conventions of a tribute album, offering a more oblique hat-tip, and not to his cornet forebears. There is only one cover: country crooner Ernest Tubb’s twangy “Thanks a Lot,” a stylish reinvention consistent with the Sonny Rollins school of repertoire selection. Yet the album pays subtle homage to Knuffke’s many saxophone totems: “Bright Light” for Daniel Carter; “Root” for Cecil Taylor saxophonist Jimmy Lyons; “Pepper” for Jim Pepper; “Use” for Art Pepper; “Chirp” for Steve Lacy, the inspiration for baritone saxophonist Josh Sinton’s Ideal Bread, of which Knuffke is a member. “Safety Shoes,” “Elevator” and “Bonderizer” spring from the pages of Studs Terkel’s 1974 oral history of industrial life, Working, interspersing the album with a mechanistic pulse.

However, its clearest debt is to Ornette Coleman, notably absent from the potpourri, but whose polytonal approach to what he called harmolodics suffuses every track. Knuffke plays in Goodwin’s Orntette, a tribute band, with saxophonist Adam Niewood and bassist Chris Higgins. Despite his reputation as a straight-ahead player with Phil Woods and Art Pepper, Goodwin’s range extends far beyond the parameters of bebop. Helias collaborated with former Coleman bandmates Ed Blackwell, Dewey Redman and Don Cherry. Knuffke, for his part, never performed publicly with Coleman, but got to know the free-jazz progenitor well in his final years.

Several years ago, Knuffke attended what turned out to be Dewey Redman’s final show, and after the set he got up the courage to approach Coleman, who was also in the audience. Coleman invited him to Central Park and then to his apartment. “The very first time I went over, he said, ‘Come on in and have a seat.’ Then he said, ‘I’ve got to go to the bank.’ And he just left,” Knuffke says. “As he was going out the door, he said, ‘If you need to take a nap, take a nap.'” While Coleman was out, Charlie Haden called and left a message. When he returned, Coleman showed Knuffke his inner sanctum, a practice room with leopard-skin carpet, original album art from his back catalog adorning the walls, the ideal backdrop for an intimate duo jam. When they finished, Ornette gave Knuffke a standing invitation to come over. For several years, he went at least once a month.

“Just playing with him changed who I am and how I play. There was something about being so close to his sound that was really informative,” Knuffke recalls. He absorbed a lot by osmosis, grazing through the sheaf of papers on Coleman’s desk, myriad lists of melodic lines, mostly diatonic. Some began, unconventionally, on B sharp; one such list was eventually bequeathed to Knuffke. Surprisingly, though, none of Coleman’s lead sheets had any rhythm specified. Melody was chief, rhythm had to be felt out. “He’d say, ‘I don’t remember how it goes until I play it a few times,'” Knuffke says. “Even though it was super rhythmic music, it was all about the line.”

Knuffke grew up in Fort Collins, Colo., and began playing the trumpet at 12; by 14, he knew he wanted to be a professional musician. Largely self-taught, he learned by listening to Al Hirt, Lester Bowie, Bill Dixon, Graham Haynes and Chet Baker. After high school, Knuffke completed one year at the University of Northern Colorado, then moved to Denver, where he cut his teeth in blues and funk bands. He had two prominent Colorado-based mentors: pianist Art Lande, who educated him in the fundamentals of jazz theory and improvisation, and trumpeter Ron Miles. When he was 25, Knuffke moved to New York.

He has since become one of the most in-demand sidemen, performing with the Matt Wilson Quartet, Helias’ quartet with saxophonist Tim Berne and drummer Mark Ferber, Allison Miller’s Boom Tic Boom and groups led by Butch Morris, Michael Formanek and Uri Caine, appearing on more than 60 albums.

Soon after joining the Wilson quartet in 2009, Knuffke began taking annual trips to Denver on tour. During his first homecoming, he offhandedly floated the idea of switching to cornet to Ron Miles. Knuffke had given up the more subdued horn after high school in favor of the trumpet, which he considered more marketable. Miles invited him to his house to try out his cornet collection, including the Monette 900 Series horn that Miles had played on Bill Frisell’s Blues Dream. “Ron said, ‘All right. Dave [Monette] says take care of it.’ And he just gave it to me,” Knuffke says. Specific instructions from Monette himself followed by e-mail several days later. Knuffke has played it ever since.

“When I was a kid, and would get together and play free improvisations with this drummer friend of mine, I would play cornet. I just felt freer on it for some reason,” Knuffke says. “When I moved to New York, it was pretty obvious that I wasn’t getting any work because I was a trumpet player. I was only getting work because I was me.”