Mastodon’s Brann Dailor can cross Primus off his bucket list of musical heroes to jam with.

The Atlanta, Georgia, metal quartet has of course been touring with Les Claypool’s progressive-funk trio since early May; the show will hit Les Schwab Amphitheater on Saturday. But Mastodon’s drummer and occasional vocalist has gotten extra stage time with Primus on a cover of Peter Gabriel’s “Intruder” (per setlist.fm) at recent shows during the run.

“First and foremost, I’m a fan and have been since the early ’90s when I first heard (Primus’ 1990 debut album) ‘Frizzle Fry.’ I remember standing in my living room just like, ‘Wow, this is just kind of different,’” Dailor said from a tour stop in Lincoln, Nebraska. “It just made sense to me and it was right in the wheelhouse of the kind of music that I was getting into at that point in time with bands like Mr. Bungle — you know, discovering that whole world of John Zorn-style music and Bill Laswell and Buckethead and all those kind of things. It kind of fit into what I was really getting into at that point in time, so all these years later, it’s so cool to be able to just become friends with those guys.”

Dailor and the rest of Mastodon — lead guitarist/vocalist Brent Hinds, rhythm guitarist Bill Kelliher and bassist/vocalist Troy Sanders — have made lots of musical friends over the last nearly two decades of hard touring. He rattled off a list of heavy metal’s who’s who: Black Sabbath, Iron Maiden, Metallica, Judas Priest, Slayer, Alice in Chains and Deftones, among others.

But while Primus gets heavy at times, calling it a metal band is a stretch. Same with alt-rock titans Dinosaur Jr., with whom Mastodon will tour in the fall.

“I think obviously we err on the heavier side of things, but we’re also a band that sort of can float in between different genres and stuff,” Dailor said. “We can play with Mars Volta one day, play with Queens of the Stone Age, play with Slayer and play with pretty much whoever if it makes sense in the rock world. Throughout our seven albums, we’ve peppered in some weird tracks here and there. … I think that a lot of our fan base is a little more open-minded and maybe into lots of different versions of heavier music.”

That open-mindedness has helped Mastodon claw its way to the top of the modern metal heap. Since streamlining its sound for its fifth studio album, 2011’s “The Hunter,” the band has racked up four Grammy nominations on top of a nod in 2007. And “Sultan’s Curse,” the opening track from last year’s “Emperor of Sand” album, finally netted the group its first Grammy Award for Best Metal Performance in January.

“I figured we were gonna be going home empty-handed again, which would have been totally fine because it doesn’t really affect anything,” Dailor said. “And really to be honest, the stress of sitting there waiting to be let down is like, man — you let some of this hopeful feeling seep in and it’s dangerous; you shouldn’t do that. … But when they called our name, my wife just looked over and she goes, ‘Holy s—, that’s you!’ I didn’t even recognize that it was (us), because they said the name of the song first, and they didn’t say our band name. They said, ‘The winner is “Sultan’s Curse,’ and I was like, ‘God, that sounds really familiar to me, like I swear I’ve heard that before.’”

“Emperor of Sand” found the quartet returning to the concept-album format that marked earlier albums such as 2004’s “Leviathan” (often considered the band’s defining release) and 2006’s “Blood Mountain.” The album’s story of a man sentenced to death via wandering in the desert reflects the band members’ experiences caring for loved ones who have cancer (Kelliher’s mother died of brain cancer during the album’s recording).

“My mom was always sick when I was a kid, and she’s still sick now, and it has been since I was a kid, I’ve been a caregiver in that respect,” Dailor said. “We definitely wanted to put it into this analogy of desert and kind of being lost out there, as the person who is handed the disease is wandering by themselves. Having a disease is a very isolating thing. As many people want to gather around you and try to prop you up and hold you up and be there for you, you’re still alone with your disease.”

As usual, the band tackled music first before its three vocalists added their parts. The songs, while still heavy and progressive, continue in the more melodic vein of the band’s more recent releases, with acoustic guitar and synthesizer popping up in the mix. The group stretched even further in this direction with the four-song EP “Cold Dark Place,” also released last year and featuring outtakes from “Emperor of Sand” and its predecessor, 2014’s “Once More ’Round the Sun.”

“I never considered myself a singer or a lyricist or anything,” Dailor said. “I just like drums and riffs, and I get inspired by that. But the riffs really create the cinema that happens in my mind. When I’m listening to riffs, I can see this story play out.”