It is also the first without Graham Coxon, the talented and tormented guitarist who left the band last year, reputedly storming out last year after problems with drink and depression and a series of stints in psychiatric hospitals.
There are few signs of dementia tonight, however. Dave Rowntree - the drummer captured in 1993 on the notorious video for the song Star Shaped diving semi-naked and sloshed into a freezing cold river - stops to discuss the relative merits of European cities, before making his excuses and heading back to the hotel for an early night.
Alex James, who recently described himself as "a dandyish, elegantly wasted genius" while admitting he "blew a million quid on champagne", sits quietly in the corner of the room, ignoring the bottles of beer and discussing his imminent marriage to video director Claire Neate. He is particularly thrilled with the wedding outfit he's had designed for the occasion. "It's got a secret pocket," says the bassist, breathlessly.
Only Damon Albarn is drinking tonight, which is perhaps why he is loudly holding forth to French journalists and berating the road crew for not having made available the right sort of pizza. Even with alcohol inside him, and buzzing from a rapturously received performance of new material and greatest hits, it is clear the singer has mellowed.
Fatherhood ("The whole focus of your life changes from the moment your first child is born," he says), a secure relationship with painter Suzi Winstanley, plus the 5m sales of his Gorillaz sideproject with artist Jamie Hewlett - all of these have helped to bring out aspects of Albarn's character that were previously missing presumed nonexistent. Such as humility.
Albarn is less competitive now. The sort of manic compulsion to beat all-comers depicted in John Harris's forthcoming book, The Last Party: Britpop, Blair and the Demise of English Rock, has been replaced by a new magnanimity. The Strokes or the White Stripes? Bring 'em on. "I want to let people through," he says with a dramatic sweep of his arm.
But what about being number one? "I don't get up in the morning and sign a pact with the devil and hate the world and want to topple everything, you know?" And then he, too, calls it a night, stumbling towards the band's tour bus and a date with his bed.
"Yeah, Damon still drinks a little, but that's as far as it goes, really," says Rowntree the following week, back in England, reflecting on the changed perspectives of a band approaching their 15th anniversary.
Blur have been through many phases over the years. They began as shuffling indie-dance merchants during 1990 and 1991, became Britpop sensations in the mid-90s before retreating left-field with a hybrid of grunge, krautrock and electronica in the late 1990s. On Think Tank, they purvey a kind of punk world music. But every one of their metamorphoses has happened in public. "We've grown up a lot," he admits. "We were like kids being given the keys to the sweet shop. The trouble was, none of us knew that if we ate too many sweets we would be sick."
Rowntree is talking, and eating, in a Turkish cafe next to the offices of his animation company, Nanomation, in central London. It is there that he makes title sequences for TV - that is, when he's not playing drums.
Four years older than Albarn and James (he will be 39 in May), Rowntree has long been the sensible band member, the one responsible for their finances and contracts. And although he has experienced Blur's career highs and lows first-hand, he is still able to observe it all with droll detachment. For the unlikely star-musician, fame is more amusing than anything else.
"It never comes gradually enough for you to inch your way into it," he says of celebrity, which Blur acquired as soon as their second single, There's No Other Way, crashed into the Top 10 in April 1991. "It always comes suddenly and ferociously. One day you're no one, the next you're on the front of all the newspapers. It's not something that human beings are designed to deal with."
It was the Britpop years - when the Parklife album was ubiquitous and the Blur vs Oasis rivalry made the News at Ten - that hit them hardest, according to Rowntree.
"If you're going to be a pop star, they should let you know in advance so you can prepare for it. But it happened overnight. The morning after we won four Brits, we were pop stars and the paparazzi were chasing us round. The weirdest thing was the cartoon strip, The Blur Story, in the Daily Star. What an extraordinary time that was."
Rowntree's response to the pressure of instant renown was similar to his response to the prospect of Blur being dropped by their record label, Food, for poor sales before their second album, 1993's Modern Life is Rubbish, came along and rescued them in the nick of time: to drink as copiously as possible.
"I suppose I'm more able to draw a line between serious activity and frivolous activity these days," he says. Rowntree, who reckons Blur "still behave like fucking arseholes when we want to", is a qualified pilot who flew several of the band's auxiliary musicians to Marrakech for some of the Think Tank recording sessions.
"But with drink I had no 'off' switch. If I had a drink I wouldn't be able to stop until the drink had run out or I was in a police cell somewhere." How many times did that happen? "It's in double figures, certainly. They say God looks after little children and drunks, and it really is true. How I managed to get through that period relatively unscathed I have no idea. By and large the police are tolerant, good, forgiving people. Thank God for that."
At the root of his drinking, Rowntree says, was insecurity. Blur's reputation, at the height of the Britpop madness, for being, as Rowntree puts it, "fucking awkward" to the point where they would reduce journalists to tears, was founded on good old-fashioned fear and self-loathing. "I was incredibly insecure and shy of meeting people and handling social situations," he says. "I passionately cared whether people liked me or hated me. That was the root of all my insecurities. I was suffering, and that's why I drank."
They've been calling Alex James rakish and urbane since he was a teenager. But he's not just the witty boulevardier whose model looks have made him the thinking female student's crumpet for years. He also has his demons. He is, as he will readily admit, only "the third most important person in the studio" when Blur go in to record. It is perhaps this low self-esteem, although wry in its estimation, that has accounted for both his drinking problems in the past and the withering one-liners that have made him a sort of latterday Peter Cook.
James has been described variously as "charming" and "a cunt". "I'm a charming cunt, that's me," he offers breezily, in the coffee shop beneath a Covent Garden hotel, not far from the five-storey house where he has lived for 12 years. "Am I insecure? Reasonably. Sometimes." A complex figure, James is, like Rowntree, a qualified pilot. His collaborators include Sophie Ellis Bextor, Marianne Faithfull and, in the art-thug concept called Fat Les, Keith Allen and Damien Hirst. Fat Les's single Vindaloo was the second biggest selling of 1998.
Meanwhile, a serious fascination with science led him to team up with Professor Colin Pillinger of the Beagle 2 space exploration unit. A British unmanned spacecraft called Beagle 2 will land on Mars in November this year, at which point a Blur track, also called Beagle 2, will be broadcast. "Science is pretty rock'n'roll," he says.
As for rock'n'roll itself, it seems to be losing its allure, especially the extracurricular trappings. "Celebrity is the elixir of marketing," he decides. "It's the most precious currency in the world."
James no longer dreads being a piece of product, even though, since their last album, 13, released in 1999, the band have been making some of their most personal, confessional music. "The media turn things that are delicate and precious into a commodity, which I can totally cope with now," he says. "We sell the way we feel." Besides, to sell or not to sell is no longer the dilemma. The issue is whether or not what you are selling, and being praised for, has merit. For James there is a world of difference between "being famous" and "being good".
"They're two completely different things," he says. "It's like being rich. When you're young, you think all the good people are rich. But there is absolutely no correlation between being a good, nice person and being a rich person. In fact, you've got to be a ruthless cunt to be rich." You're pretty rich, aren't you? "Yeah," he says. "I'd actually rather be rich than famous any day. Imagine being on Big Brother." He digresses, "There is something delicious about crap telly. But it's exactly the feeling you get after a wank: really bad guilt."
He compares his level of fame with the kind enjoyed, or rather endured, by Damon Albarn. "I'm not famous. But I've hung around with people who are famous and it's an amazing drain on your resources, walking into a room and everyone knowing that you're there."
There was, he remembers, a time when Blur "couldn't walk through any European city without bringing traffic to a standstill". But for Blur teen worship was only ever incidental. "We never really played our looks, did we?" That's all over now, anyway. "We're all grown ups," he says. "Have I changed? I think so. You spend your 30s renouncing everything you loved in your 20s. It's quite positive. The road of excess leads to the palace of wisdom, and all that. Close scrapes? If you don't have those you're not doing it right. We've hung on for dear life at times."
So did you really put away a bottle of champagne every day for 18 months? "No," he says. "It was three bottles. No, that's not true." He smiles. "Well, it was probably nearly true." Did he drink alone, or did he have help? "God," he says stagily, "if I could only have got a minute to myself, darling!"
Newly slim and into yoga, these days he is more likely to be found either "wandering around in the sunshine" or at home, "making soup and thinking about architecture", than he is propping up the bar at the Groucho or playing, as he once did, with rock pigs Zodiac Mindwarp. "I'm quite inward-looking," he says.
James once described Damon Albarn as the most famous man in the country. You wouldn't know it today as he wanders unnoticed from bar to table inside Westbourne Studios, a busy, cavernous recording complex with its own cafe-restaurant in west London near the home he shares with his partner and their two-and-a-half-year-old daughter Missy.
Barely an eyebrow is raised as he shambles over, looking more scruffy grunge-slacker than media-savvy pop star, which is ironic considering Albarn virtually single-handedly ended Britain's infatuation with dowdy US rock (Nirvana, Pearl Jam et al) with his music and neo-mod image circa Modern Life Is Rubbish and Parklife.
He is at least wearing a jacket. He started wearing it after finding himself in the Jubilee Rooms at Parliament. This was a week before the Stop the War march. Albarn had attached his name to the campaign as the bombing of Baghdad seemed, to his disgust, increasingly likely. "I realised I had to smarten up," he says. "I was looking like a sort of mad mutoid cross between American trailerpark and Britpop. It was confusing me as much as it was everybody else."
It must be equally confusing to be one day a 2D cartoon character out of Gorillaz, the next an anonymous tributary for something like the Mali Music project (put together in 2002 with various indigenous musicians), the next the leader of Blur, the next a fierce opponent of Blair and Bush's Gulf War II. "I am a bit like Zelig," he says, referring to the Woody Allen character who assumes radically different guises according to the situation. He likes Tom Hanks's character, Forrest Gump, for the same reasons. But he denies juggling careers. "I'm just doing what I do, which is write songs."
After more than 13 years in the game, Albarn knows how to avoid scrutiny. It is two days after the revelations about his relationship with Justine Frischmann of Elastica appeared in the Guardian, and he is not impressed. "That's not what I got up to behind closed doors," he snaps. "That's someone else's opinion." He is perturbed to have been the subject of newspaper serialisation. "It's no different to serialising Jordan or whoever in the Sun. I appreciate that people want to fill in the gaps, and I know it's annoying, but I just don't want to talk about my private life. I never have done and I never will. I'm consistent in that. I haven't spilt the beans on anybody."
Presumably, then, he won't respond to rumours that the real causes of the great Blur-Oasis farrago of 1995 weren't deep cultural divisions but the fact that Albarn had slept with Liam Gallagher's ex-girlfriend Lisa Moorish behind his back. This time, he laughs. "What can I say? We were all a lot younger and we were having a good time." Britpop, he says, "was nothing exceptional, but it was amazing because it was happening to us for the first time. And anyone who says it wasn't, either it's sour grapes or they are going senile very early." He adds, in case you were wondering: "Liam and I have always got on famously. I've always liked Liam."
Someone calls him on his mobile. When he hangs up, he is conciliatory. "I do want to be as open and as revealing as I can," he says.
But what would he reveal? Albarn is nothing if not the master of disguise. Parklife introduced the cocky Eastender with the penchant for greyhound racing. Since then he has played the art rocker on Blur (1997), the tortured soul on 13 and the cyber pop star on 2001's Gorillaz. So which is the real Albarn? "It'd probably be impossible for anyone to find out," he dodges. "I am all those things quite naturally."
Perhaps Albarn is just on the defensive after numerous critical maulings. He is used, he says, to "people trying to get a rise out of me". And nothing gets him going like a debate about the authenticity or otherwise of the character he assumed on Parklife. The suggestion that he is some arty posh boy winds him up. "I know I shouldn't worry about this, but I didn't go to public school, I went to shit state comprehensive schools," he says, sucking on a cigarette. "I've got four or five O-levels. I got a B for English and the rest were Cs, and I've got a D for A-level English and an E for History."
Nor did he, like Coxon and James, attend Goldsmith's College, part of the University of London. "No, I didn't go to Goldsmith's." He raises his voice. "I went to drama school for a year, dropped out and, because everyone else was at Goldsmith's, got a part-time, twice-a-week course on some spurious music thing so that I could get into the student union and hang out with everyone."
Has he succeeded over the years partly driven by this lust to prove his credentials? "Lust? Well, that's your word. Am I 'for real'? I don't care if I'm for real or not for real. I'd just like it to be understood that I'm not what people think I am." Which is? "Well, fuck knows." You are different things to different people? "Obviously."
Albarn does acknowledge his unusual upbringing. His father managed the Soft Machine and wrote books on Arabic design while his mother worked for Joan Littlewood. The family, originally from Lincolnshire, moved to Leytonstone, and then to Essex. "They just wanted us to have a bit of countryside in our lives," he says. His phone rings again. He tells whoever is on the line: "I'm just trying to convince the Guardian that I'm authentic."
Never mind Zelig, is there something of the machiavellian dilettante about Albarn? "I'm determined," he says, calming down. Have there been casualties along the way? "Well, Justine seems to have perceived herself as a casualty of all types of wrong-doing," he says, referring to Harris's book.
"That is really sad, you know? I feel in some way that I am responsible for that. I don't feel good about that. I loved Justine and when we first met we were incredibly in love and, just by the nature of being really into music, we ruined our relationship. Both of us did. That wasn't machiavellian, it's just really sad that a relationship that started off so nicely ended so badly."
He repeats that he doesn't "feel the need to expose every nuance of my personal life in the press". Later, he will say: "I am a quite overemotional person. And it gets me into trouble all the time. But it also helps me write the songs that I write. On this record, I got really emotional."
James says "13 was a record about love gone wrong", while Think Tank is about "love gone right". The new album's closing track, Battery in Your Leg, a poignant elegy to good times and old friends, is the only one to feature Coxon. "It's about all of us," corrects Albarn, explaining what it was like going back in the studio a man down. "It was horrible! We got there at 11 and it was like, 'He's a bit late.' And then it was, 'Maybe he's forgotten that it's the first day.' And then by the end of the day we found out that he wasn't coming. That was a shock to the system."
It's unclear why the guitarist left. Was it good, old-fashioned "musical differences"? Was he sacked? Did his history of mental turbulence become too hard to handle? Coxon turned down several requests for an interview, although speaking to Teletext, he declared that, after hearing Think Tank, he was "very glad" not to be in Blur any more.
Is it important to Blur that Coxon likes Think Tank? "Of course I'd prefer him to like it," says Albarn. "I hope he likes the track he's on, at least. Graham is as close to a brother as I've ever had in my life. He used to live round my house when he was younger, basically."
Both Rowntree and James say that Think Tank is Blur's finest hour and consider Albarn, and not the guitar whiz, to be Blur's resident genius. They miss him as a friend but not his musical input. "We've proved we don't need Graham," says James. "We've made a different record without him. Not better or worse, just different."
For Albarn, Coxon's departure is further confirmation that Blur are a sort of "dysfunctional family" who "only see each other at weddings and funerals". Will he ever come back? "I still don't feel that, if he said, 'Let's just make some music together and forget all of that stuff,' I would hope that I wouldn't say, 'Piss off.' I would hope that I would say, 'Yeah, definitely.' Like I say, if you consider someone family you never close the door on them, do you?"
So have Blur lost their Paul McCartney? "No," says the singer, flatly. "We've lost our Graham Coxon."
Think Tank is released on EMI on May 5.
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