Imagine, for a moment, that you are in one of pop's most cherished alt-rock bands. After 15 years together, during which you have dealt with all manner of upheaval, from record company fall-outs and line-up changes to debilitating drug addiction, your image is that of a serious band that tackles serious subjects. Now, with your latest album, you want to cheer things up a little, offer some light relief to your listeners. So what do you do?

"Simple," grins Wilco's Jeff Tweedy. "You hire a camel and throw it a birthday party."

As Wilco devotees – and they are increasingly many – will already know, Tweedy is referring to the cover image of the band's self- titled seventh album, released to glowing reviews earlier this year, which depicts a camel standing in a garden terrace in a party hat. In the foreground is a table with a birthday cake on it and six empty chairs.

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"Originally, we thought of having a really old man having a birthday party but once we found the camel it became painfully obvious that the party should be for him," smiles Tweedy. "I wanted someone to look at the image and go, 'What the hell is that? How did that happen, and why wasn't I invited?'"

It had the desired effect. When the album was released, online forums started debating the cover's meaning. While a few serious souls read it as a statement on the Middle East peace process and man's futile pursuit of happiness, the majority saw it for what it was: Wilco letting their hair down.

Wilco (the album) is a work that radiates confidence and contentment, recalling the simplicity of the band's early LPs A.M. and Being There. The first track, simply titled "Wilco (the song)", finds Tweedy paying tribute to their fans against chugging Velvet Underground guitars, and pledging that no matter how tough things get, "Wilco will love you, baby."

Elsewhere, he cheerfully grapples with matters that he has come to accept are beyond his control. Pondering existential mysteries in "Deeper Down", he states, "I adore the meaninglessness of the 'this' we can't express," while on "Solitaire" he concludes, "Once I thought without a doubt, I had it all figured out... It took too long to see I was wrong to believe in me only."

It's tempting to look at Wilco's upbeat demeanour, together with their decision to name their seventh album after themselves in the manner of a new band releasing its debut, and conclude that they are drawing a line under the turmoil that has come to define the group, that they are somehow wiping the slate clean.

"I guess there's an element of that," says Tweedy cautiously. "After 40 years Glen Campbell put out a record called Meet Glen Campbell, which is kind of cool. But we struggled with a lot of other titles that felt more exemplary of what the music was. But nothing else felt quite so succinct."

If the album has an overriding theme, it is the acceptance of life's uncertainties. "I think there's a liberating nature to that concept," says Tweedy. "It allows for a playfulness and an engagement in life that is more enjoyable than the alternative. I've aspired to convey some of those things for a long time now, maybe not so clearly before because it hasn't been so clear to me. But I do believe that the greater ability you have to tolerate ambiguity, the more successfully you can steer your life. The alternative point of view – the complete dismissal of ambiguity, trying to rationalise irrationality – can be very destructive."

Tweedy is known for his philosophical bent, and it has served he and his music well. If there's a band that has learned to negotiate and overcome acts of randomness and misfortune, it's Wilco. First, there have been the frequent line-up changes – only Tweedy and the bassist, John Stirratt, remain from the original group, who formed in 1994 from the ashes of the alt-country outfit Uncle Tupelo. Since then, a succession of musicians have come and gone, including Jay Bennett, who left in 2001 after relations broke down with Tweedy and who died unexpectedly earlier this year.

More unsettling still has been Wilco's relationship with the industry in which they operate. In 2001, their record company Reprise, an offshoot of Warners, rejected their album, Yankee Hotel Foxtrot, on the grounds that it lacked any discernible singles, and dropped them from the label. The band responded by buying back the master tapes and streaming them for free on the internet. Half a million people downloaded the album and, as word got around that they were back on the market, Wilco found themselves in the unlikely position of being courted by 30 different labels. After much deliberation they re-signed to another Warners imprint, Nonesuch, meaning they were effectively paid twice for the album by the same record company.

In 2003, the band endured further turmoil when, during the recording of A Ghost Is Born, Tweedy checked himself into rehab for an addiction to painkillers, taken to suppress the migraines and panic attacks that had afflicted him since he was a child. The ferocity of these attacks were grimly captured in Sam Jones's Wilco documentary, I Am Trying to Break Your Heart, which showed Tweedy bent over and vomiting from the pain.

Wilco's unwillingness to wallow in difficulty is as admirable as their ability to confront it. Tweedy's hackles rise slightly when I ask to what extent personal and professional drama has fed his art. "I didn't aspire to have drama surrounding records and I didn't welcome chaos as some sort of gateway to creativity," he says. "These things are just our experience. Aside from the consolation that music can provide in itself, I think there's a consolation in seeing that suffering doesn't have to go to waste. The unfortunate flipside of that is that people romanticise it to the point where they think that it can't exist without the suffering. The cliché of the tortured artist is something I really detest. The thing that I find most repulsive is that, more often than not, what is perceived as suffering by followers of any particular artist is really an unwillingness to suffer. Drug addiction in particular grows out of an unwillingness to suffer when suffering is part of the human condition."

Tweedy certainly knows a thing or two about this, although with his painkiller addiction under control, he has fewer headaches than he used to. Where once they were almost constant, they are now reduced to one or two a month. "There is a relationship between migraines and panic disorder, and once I had managed my panic disorder, I think that eliminated a fair amount of headaches that escalated through panic into a migraine," he says. "One thing I'm certain about is that when you're active in an addiction to painkillers, your body really wants to have pain, especially if you're wanting a rationale to be provided for something you know is wrong."

When Tweedy was growing up in Chicago his heroes were not rock'n'roll peacocks, but outsiders and misfits. "The Minutemen, Hüsker Dü, The Replacements, I liked the anti-hero kind of thing," he says. "I tended to be pretty suspicious of people that were comfortable enough in their skin to act like rock stars. I'm just not that guy. I'm not David Lee Roth, as much as I'd love to be. I mean, that would be a handy set of skills to just get up there and shake it about for a couple of hours."

Along with their finely tuned survival instincts, the secret of Wilco's longevity, I suggest, has surely been the enduring modesty of their ambitions.

"Well, I gave up on the idea of being a sensation a long, long, long time ago," Tweedy grins. "That made appeals to my ambition a lot less attractive. Record companies always make appeals to your ambition in the same way that credit card companies appeal to people's desire to have status or things. We couldn't afford those things – tour buses, videos, for instance – so we didn't have them. That's always been the mindset."

Even so, Wilco's fan base has broadened incrementally over the last 15 years, with each album selling more than the last. Wilco (the Album) peaked at number four in the US album charts, a career best for the band. Tweedy is delighted but at the same time he isn't one to dwell on chart positions and units shifted. These days, he says, he sees a calmer future for Wilco in the years ahead. Is that just wishful thinking?

"Put it this way, I would think that it will be easier to survive the next 15 years than it has been to survive the last 15. We're more fit now in almost every way for whatever is coming down the pike. I think we'll be around for a while yet."

Wilco (the Album) is out now on Nonesuch

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