On a Saturday afternoon in August, while the Red Sox were trouncing the Yankees in the Bronx on the way to a series sweep, members of the Baseball Project, a supergroup that sings about baseball, were settling into the greenroom of the Sinclair, a small venue in Cambridge, Massachusetts, not too far from Fenway Park. The night before, they’d performed in Brooklyn—Yankees territory, more or less. “They were yelling for ‘The Yankee Flipper,’ ” Scott McCaughey, one of the band’s guitarists and singers, said. He was referring to a ballad he wrote about the time Jack (Black Jack) McDowell, the former Yankees ace, got drunk with members of R.E.M.—including McCaughey and one of his Baseball Project bandmates, Mike Mills. “Sorry,” McCaughey joked in the greenroom. “We don’t know how to play that anymore.” McCaughey is a Giants fan. “You get sick of hearing the Yankees doing so great all the time,” he said.

“You don’t have to worry anymore,” Linda Pitmon, a Yankees fan and the band’s drummer, said.

The Baseball Project also includes another former member of R.E.M., Peter Buck, along with Steve Wynn, the singer and guitarist who leads the Dream Syndicate. The group was born at a party the night before R.E.M. was inducted into the Rock & Roll Hall of Fame, in 2007. Wynn recalled that he and Pitmon, who are married, met McCaughey there, and they started talking about their shared baseball obsession.

“I’d met you before,” McCaughey interrupted.

“In the men’s room in a Seattle club,” Wynn clarified.

“Scott and I, each unbeknownst to the other, had the idea of someday writing a few songs or an entire album about baseball,” Wynn continued. They began to discuss it, and within a week they’d swapped several songs. A few months later, they had a whole album, “Volume One: Frozen Ropes and Dying Quails.” That was fifteen years ago. In the meantime, R.E.M. broke up.

The Baseball Project has just come out with its fourth album, “Grand Salami Time!,” produced by R.E.M.’s old producer Mitch Easter and recorded at his compound in North Carolina. They play mostly on small stages, although they often throw out the first pitch or sing “The Star-Spangled Banner” before ballgames. When the group had a night off in Boston a few years ago, Josh Kantor, the organist at Fenway, who has toured with the band, took them behind the scenes of the Red Sox’ ancient bandbox. Pitmon rolled around in the warning-track dirt in front of the Green Monster.

“I was, like, ‘Oh, man, what’s it like to play at, you know, an eighty-thousand-seat football stadium?’ ” Kantor recalled asking Mills. “He’s, like, ‘What’s it like to play at the World Series?’ And I was, like, ‘Oh, right. That actually is really cool.’ ”

Mills once invited Kantor to join their fantasy team—they play in a league with other indie musicians, including Stephen Malkmus, from Pavement, and Ira Kaplan, of Yo La Tengo—but warned him that he would have to take it seriously. Kantor knew that Mills meant it, and declined.

McCaughey usually wins the league, though every member of the band is competitive—except for Buck. “I’ll be in the van back there, and it’s like fantasy whatever—”

“Fantasy baseball,” McCaughey said.

“ ‘Oh, this guy has a groin pull,’ ” Buck went on.

Buck doesn’t follow sports. “He’s our token non-baseball man,” Wynn said.

“I like all these people and I like the songs,” Buck explained. He does appreciate what he calls baseball’s “old characters”: “They were like Wild West gunfighters. They were weird, you know, in a way that’s now impossible.” He went on, “They were like rockabilly stars or rhythm-and-blues stars, except that they were wearing a uniform and hitting a little ball.”

These days, the money is bigger and any weirdness is under wraps. The band doesn’t shy away from the game as it is now—there’s a song about Shohei Ohtani, and a sly take on doctored balls (“Just like Brylcreem, a little dab will do ya / I use that L.A. look to try and fool ya”)—but it can be a struggle to come up with material. “A lot of the current players are kind of boring,” McCaughey said. “They don’t drink and carouse like they used to. I mean, probably for a good reason.”

The big themes persist, in baseball and in music. Wynn was on a solo tour, staying in a run-down hotel outside Buffalo, when Buck sent him the music for the song “Journeyman.”

“It was great music,” Wynn said. “I went, Yeah. A guy goes town to town, just does the best he can, and does something pretty well so he can keep working. And I can write a song like that.” ♦