Mick and Keith are back in your town, baby! That’s right—the Stones are on tour again, tramping across America with their multiple vans and background singers in tow, and you’re turning out to see them one more time because (after all) they are the greatest rock ‘n’ roll band of all time.

Or,... yawn.

Which is it? The Stones are or at the very minimum were a very great rock band—a band that it’s become easy to worship or take for granted, depending maybe on your age. They’ve even got a new album out (and some say it’s their best in [insert large number here] years), but you know you really want to hear “Satisfaction”. Which, I think, is cool with the Stones. They want to play it for you.

There’s no longer any need to resolve the once-defining Stones v. Beatles argument. The Beatles long ago ascended to myth while the Stones have mopped up in the here and now. And so it is that covering Beatles songs is either standard practice (Phish’s Halloween concert covering the entirety of The White Album, jazz pianist Brad Mehldau’s brilliant variation on “Blackbird”) or a frightful mistake (Sinatra’s “Something”), whereas the Stones just aren’t covered much. There’s not much need to cover them—they’re covering themselves.

And maybe the least likely genre for a Stones cover is jazz. The Beatles were all about fancy harmonies and shifting genres, but the Stones were always a rock band, man. How is a jazz cat going to cover “Yeeeahhh, yeeahhh, yeahhhh—whooooo!” from “Brown Sugar”? I mean… c’mon!

Or… wrong.

Saxophonist Tim Ries, who has toured with the Rolling Stones since 1999, has done the improbable on The Rolling Stones Project—he has recorded a legitimate album of Stones songs played in a variety of jazz styles. And, with an exception or two, it works. This is a group of mostly instrumental Jagger/Richards classics, flavored by the sound of straight-up modern jazz and played by an all-star line-up off great jazz players. Look, I know you want to look askance at a project like this—Lord knows, I do—but here are the folks making this album good: Jon Scofield, Bill Frisell, Bill Charlap, Brian Blade, Luciana Souza, John Patitucci, Norah Jones, Clarence Penn, Jeff Ballard, and Larry Goldings. Add to that Sheryl Crow, Lisa Fischer, and three guys named Keith, Charlie, and Ronnie—you still think it’s gonna stink?

Too bad it opens on a dud. Ries’s boogaloo arrangement of “Satisfaction” isn’t awful, what with Sco playing some blues and the reharmonization of the tune being perfectly hip. But the impulse on this track (and on “Slippin’ Away” and ““Waiting on a Friend”) to engage background singers in snatches of the lyrics, adds a hint of smoove jazz to the proceedings. It’s deceptive. This is decidedly not a Kenny G kind of thing. Even on “Satisfaction”, the improvising is tough and exciting—particularly Scofield’s ripping solo over Clarence Penn’s funky blend of Billy Higgins and Dennis Chambers. But the hint of cheese never quite fades here.

Much better is the organ trio version of “Honky Tonk Woman”, which matches Ries’ tenor with Larry Goldings on B3 and none other than Mr. Watts swinging the drums. It’s the kind of performance you might hear in a little bar and not even catch as a Stones cover at first—the bluesy groove of the tune sounds perfected suited to this instrumentation. No background singers necessary. It was bold of Ries, then, to include a later version of the same tune, with Keith Richards lending the original guitar riff, this time with Goldings on acoustic piano, Stones bassist Darryl Jones on the bottom, and Lisa Fischer wordlessly singing about the edges and on the chorus. This track is an instrumental rock version and, as good as Keith’s presence is, mostly out of place among the jazz covers.

The only performance that delivers a proper and complete vocal is the Norah Jones collaboration with Bill Frisell on “Wild Horses”. Whatever your feelings about Norah, this is an indisputably original approach to a great song. Norah is extraordinarily relaxed here, indulging every word, then letting the tempo creep up on her for the chorus. Frisell does his thing—barely there, yet coloring each bar like a master pastel painter. Ries plays obbligato on soprano sax to lovely effect. The duet between Ries’ soprano and Frisell on “Ruby Tuesday” is similarly delicate and lean.

The most original and striking transformation is on “Street Fighting Man”. Ries has recruited Brazilian-American singer Luciana Souza, pianist Ed Simon, and percussionist Mauro Refosco to turn this into a compelling Baiao groove that gurgles with polyrhythm.

Two of the best tracks come from Ries’ demos for the project. “Paint it Black” and “Gimme Shelter” feature Brian Blade on drums, Ben Monder’s guitar, Charlap and Patitucci. Playing Stones covers with Wayne Shorter’s rhythm section? That’s a cute trick, and it utterly works. Monder is pensive and distorted by turns, and Blade kicks up enough of a storm to make Mr. Watts feel OK about it.

So, who is this guy Tim Ries anyway? He’s made a hot and cold album, but one that is simply way better than you had to fear it would be. He ends the disc with an original tune, “Belleli”, perhaps in an attempt to define himself. He shares the reaching melody with Mr. Frisell, and the track is chamber-jazz of a very high order—something you wouldn’t be surprised to hear on an ECM disc, maybe something by John Abercrombie or Jack DeJohnette. You try to imagine this kind of talent being embalmed in the horn section for the Stones on a world tour, and you just realize how much incredible talent there is in jazz, too little of it getting paid anything much.

Still, the Stones are the Stones, and maybe you’re willing to part with a few hundred bucks for tickets to see them live, one more time. Tim Ries will be in the band, if that makes a difference. But maybe you should pick up his album, this one or what I hope will be a new one eventually, a disc of all his own songs, mostly unadorned, that might earn him a thousandth of his boss’s acclaim. Which would be huge—and well deserved.