What do you say to Mick Jagger when he calls up and asks you to join the Rolling Stones? "Let me check my day planner"? "Sorry, Mick, I've got a tapestry-making webinar that week"? (I'm riffing here.) No, when Mick Jagger calls, you're on the next plane out of town. I don't care if he's 103 years old—it's the Rolling Stones. San Diego–based saxophonist Karl Denson got that call from Mick Jagger, or Skype call as it were, and Denson is now the Rolling Stones' newest live member. For 30 years, he's been blowing brawny, golden plumes of sax and flute. He's a true master who's seen time with Lenny Kravitz, the Greyboy Allstars, and Slightly Stoopid. His own band, Karl Denson's Tiny Universe, hurls a multi-horned charge of jazz, funk, rock, and very infectious boogie.

Their latest album, New Ammo, pulls from a love of film scores and a series of tributes the band did honoring Ray Charles, Rick James, and the Beastie Boys. The Seattle show is a tribute to Run-DMC, meaning they've reworked Run-DMC's music and will be re-creating the material with live instrumentation. These are some flat-out talented cats, so do catch it if you can. Denson spoke from Brooklyn, New York. He was walking to meet his daughter for lunch.

You're playing with the Rolling Stones. Man, congrats! Please, tell me everything. How did you get the call to play with them?

I got the call from my connection with Lenny Kravitz. He was having dinner with Mick Jagger and found out they were looking for a sax player. Over that next week, I had a Skype meeting with Mick, and he hired me. A week after that, I got on a plane and went out on the road with the Rolling Stones. Yeah, the fricking Rolling Stones.

Did you know the songs? I saw some footage of you playing "Brown Sugar," and you were throwing down.

They gave me a short list of songs to learn. There's actually not that much stuff with horns in their music. I got in touch with their other sax player, and we decided who would take what solo. I lucked out and got "Brown Sugar."

Is Keith Richards like you thought he would be? Indomitable and wily? How the hell is he still smoking all those cigarettes? Did any of the guys surprise you? Please tell me y'all are playing "Bitch."

"Bitch" does get played, yes. It's been so fun. The cool part is that Mick is all super-business. He's taking care of stuff all the time—you can tell, he's got the busy face on. The other guys are real chill. They're hanging out, but they work really hard. When they rehearse, they get in there and get to work. It's kind of amazing. Mick must have sung for four out of the five hours during the first rehearsal day. Real singing. I was like, "Damn." He works himself into shape. The band does the same thing. You see that and you realize why those guys are so big. They did "She's So Cold" in rehearsal, and I flipped out—that's the quintessential Stones beat for me right there.

Will you continue playing with them?

They have basically invited me on board. I got a welcome-to-the-band dinner and the whole nine yards.

What do the Rolling Stones serve for their welcome-to-the-band dinner?

It was Asian. I had my first Peking duck.

What were some film scores that informed New Ammo?

Our bassist, Chris Stillwell, is a total film-score buff. He's found really great tracks over the years. I'll say, "Chris, let's go into the vault and find some cool stuff." Like, we just added a new tune yesterday called "Running from Danger" by Steve Gray. Chris found it in one of those film-score libraries, where they have tons and tons of songs written by great writers that people can access. He's a thrift-store record buyer as well, so he's always scouring the landscape.

James Brown's score for Black Caesar is the zenith for me.

Oh yeah, for sure. I saw that movie when I was a kid, first run. I think I'd just started playing sax, I was maybe 13 living in Orange County, California. Planning my eventual conquest [laughs]. I also saw Trouble Man. Those are some extremely heavy experiences—hearing that music for the first time as a kid. It's coming off the screen, and you're like, "What in the hell is going on?" Marvin Gaye wrecked my world with that.

I can't stop listening to your song "New Ammo." Where did you all record?

We've been doing our work out at a studio called Sound Design in San Diego. It's a friend of mine's studio. I decided I was going to stay local, stay small, and develop our own sound out of that place.

The album comes off the heels of your Beastie Boys tribute. How do you all work these tributes? Does it ever get too daunting to honor the song and the artist? Do you ever stress about it?

I don't stress too much. The band stresses, though [laughs]. We've been doing these tributes for the last few years, starting with the Rolling Stones. It's evolved from there. We did our version of Beastie Boys' "Sure Shot" and we were like, "Okay, we're going to keep that one." I believe we know enough about music and our instruments. And if we take the time to learn the song, then we're going to figure out what it says to us and what we should do with it. So I just try to go at it really organically. We'll learn it and then decide what our approach is going to be. With Run-DMC for instance, their music is so minimal—'80s drum beats and small samples. We looked at it and thought, "Let's do a remix."

Is there a tribute that was trickier than the others?

I'd say the Rick James one. Just because there's a lot of orchestration in his stuff. Organizing the singers and the horns, it was a big production.

Now it's embouchure time. Most people don't know what embouchure is—the use of the facial muscles and the shaping of the mouth and lips that goes into playing brass and woodwinds to get sound.

That's all technical, silly stuff that you don't really concern yourself with, because the practice that goes into building your embouchure is very boring. And probably boring to talk about! It's like holding your guitar. You put the time in, become proficient at the instrument, and then you can go about trying to make music. We're far enough along that we get to go about making music, and that's the beauty of it.

Seems like fewer musicians are putting the time in on their instruments these days. Too many DJs hitting the space bar.

I don't know, I think there are tons of fine musicians out there who are just starting up and are ready to put their stamp on the world. DJ culture is like anything—you have to put the time in to get proficient at it. There are kids playing video games making hundreds of thousands of dollars a year playing and winning tournaments. And guess what, those kids are putting in thousands of hours. The video-game controller is their instrument. Mankind grows and changes. I look at it like we're ants. There are worker ants, musician ants, medical ants, all different kind of ants. Some are smarter. Some are different. Some are more fun. It's like, you get in your car, and you have no idea how that car works. There are guys in the lab right now, coming up with things that are going to change the future.