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McCartney was speaking to Howard Stern last week when he agreed with the radio host's assertion that The Beatles were the better of the two groups.
The 77-year-old said: "You know you're going to persuade me to agree with that one.
"They are rooted in the blues. When they are writing stuff, it has to do with the blues. We had a little more influences... There's a lot of differences, and I love the Stones, but I'm with you. The Beatles were better."
He continued: "We started to notice that whatever we did, the Stones sort of did it shortly thereafter.
"We went to America and we had huge success. Then the Stones went to America. We did Sgt. Pepper, the Stones did a psychedelic album. There's a lot of that.
"We were great friends, still are kind of. We admire each other... The Stones are a fantastic group. I go see them every time they're out. They're a great, great band."
Now The Rolling Stones frontman has responded with some interesting points of his own, while very much retaining the polite and flattering tone; this is not a Stormzy vs Wiley style beef by any stretch of the imagination.
When told about McCartney's comments by Zane Lowe in an Apple Music interview, the 76-year-old said: "That's so funny. He's a sweetheart. There's obviously no competition.
"The big difference, though, is and sort of slightly seriously, is that The Rolling Stones is a big concert band in other decades and other areas when The Beatles never even did an arena tour, Madison Square Garden with a decent sound system. They broke up before that business started, the touring business for real."
He continued: "So that business started in 1969 and the Beatles never experienced that. They did a great gig, and I was there, at Shea stadium. They did that stadium gig. But the Stones went on, we started doing stadium gigs in the '70s and [are] still doing them now.
"That's the real big difference between these two bands. One band is unbelievably luckily still playing in stadiums and then the other band doesn't exist."
But what do you think, which was the best of the best?
Jagger and McCartney probably won't mind which you choose, after all, you can't always get what you want, sometimes, you've just got to let it be.
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