It seemed like a good idea at the time. The Golden Tower is a tall structure not unlike a stunted Big Ben with a row of seats strung around it. It looks like a lot less of a drop than you think even when you’re stranded several hundred feet up, legs dangling, mouth dry, gazing out over the flat expanses of Denmark’s capital city. It almost seems fun. Then, just when you’re getting to appreciate the vista, you are dropped several hundred feet. For half a second you’re completely weightless, a guinea pig in some oversized physics experiment. Moments ago the three members of Biffy Clyro were joking, laughing and egging each other on over who could handle such a trip. Now they are being mocked openly by a group of Danish teens waiting next in the queue who take great delight in highlighting their collected ashen complexions.

The Danes are smart people, embracing progress for the greater good. This is manifested not only in the huge number of wind turbines that line the edges of the Oresund Straights which cut through Copenhagen but because they built a theme park in the middle of their city. Tivoli Gardens is part Alton Towers, part Chelsea Flower Show, with a bandstand, aquarium and restaurant thrown in. If Edinburgh Castle had a year-round fun fair up on the hill, there would be a lot more satisfied customers.

Biffy Clyro have just arrived in Copenhagen from Glasgow after appearing at the Scottish leg of the Give It A Name Festival where they shared the bill (somewhat mismatched) with the great and not so good of American emo (American bands: never happy. Even the rich ones, confides a member of the Biffy crew later).

It is an indication of the kind of people they are that they choose to spend an afternoon goading a journalist onto rollercoasters rather than sleep off hangovers in their bus. The Golden Tower (or Golden Shower as bassist James Johnston christens it given the fear that he could do himself a urinary mischief mid-ride) is among the last stops before the band - James Johnston, along with drummer/vocalist and twin brother Ben Johnston and guitarist/vocalist Simon Neil - reconvene for their latest performance as tour guests to Bloc Party who are somewhat immodestly steamrollering mainland Europe with their new album.

While Bloc Party’s time in the sun is very much now, you get the feeling that Biffy Clyro’s might just be coming up. Some records are important, some records are ahead of their time, some spawn a million imitators, but the best ones are just great. Puzzle, Biffy Clyro’s fourth album, is more than great, it is truly special. It is the most coherent, accomplished record the trio have made, after years of being perceived as a little too tricksy for mass consumption. Indeed, single Living is a Problem Because Everything Dies predicted to go top 20 in the UK at the time of going to press.

As Gordon Moakes of Bloc Party and head Biffy cheerleader emphatically puts it: ‘They’re one of our favourite bands full stop. These guys couldn’t have a sunnier, more enthusiastic attitude towards this life and of course there’s the guns-cocked majesty of their songs to boot.’

This is not the first time The List has encountered Biffy Clyro. Back in 2002 we travelled to Kilmarnock to meet the boys at their rehearsal space: an almost derelict YMCA. The room was so small that equipment had to be lifted out of the room so anyone other than the band could fit in. The List was treated to a brief flash of the band running through their set; bass, drums, guitar and three harmonious vocals fused together like nothing we’d ever heard before. We were blown away, seated among the overflowing ashtrays, admiring FHM posters on the walls. Five years on, standing at the side of the stage at the Vega, the beautifully dog-eared, Barrowland-esque venue for tonight’s show, watching Biffy surge through their 40 minutes of kinetic, frenetic, majestic rock shapes, it becomes apparent that time, money and hard earned experience has made them bigger, better and more organised, but the beating heart of the band, the chemistry, remains. They rock, they pop, they thrash, they swell, they swoop. Even some of the fanatical Bloc Party faithful seem moved.

Post-show, standing in the band’s scarlet dressing room - which in truth is more a dressing cupboard - we hear Kele Okereke from Bloc Party doing his operatic vocal warm up exercises. Simon Neil, meanwhile, has lost his voice after his ferocious performance tonight and is sworn to silence by his tour manager for 30 minutes. He stands just inside the door of his cupboard, a sheet of A4 in his hand with an explanation for his being mute, dripping sweat quietly onto the red painted floor. Elsewhere, the post-gig comedown ritual for the brothers Johnston involves regaling us with tales of meeting Watford punks and true Biffy fans Gallows (‘they want to tattoo us when we go on tour together!’), watching footage of Rage Against the Machine live at Coachella Festival on YouTube (‘they’ve still got it!’) and taking turns riding around on Bloc Party’s backstage folding Ikea bicycle.

Showered, dressed and hungry, we set off in search of sustenance, only to end up enjoying an accidental extended gad about Copenhagen’s back streets, unable to locate an area ‘just round the corner’ packed with places to eat. Some time later, we stumble upon an Italian trattoria. Here, the band finally find time to reflect on the making of their album.

‘There’s a feeling from some corners that things are falling into place on this record,’ Ben explains. ‘Which is great but the only pressure we feel is from ourselves. The pressure we put ourselves under to make it.’

‘You learn as much from the bad mistakes you make and the bad bands around as you do from the good ones,’ adds James. ‘How not to do it is as important as how to do it.’

The brothers share that connection that only twin siblings do. Ben is excitable and emphatic, while James is more reflective and analytical, digesting questions behind his spectacular mane of copper hair and beard. Simon Neil slots in perfectly between; his soft voice, piercing eyes and mischievous smile camouflaged behind his black whiskers. That’s not to say he’s being evasive, the band are cheeringly honest about their past achivements and future ambitions. They have learned how to be confident without being cocky. They are genial hosts, unpretentious artists and fevered, insightful musicians.

The last decade has been eventful for them. A self-financed debut single was followed by a release on Stow College’s Electric Honey label, the first after Belle and Sebastian’s Tigermilk. This led to a deal with Beggars Banquet. While their debut, 57, was a creditable post-Nevermind pop record they remained modestly underground in the post-grunge, nu metal wastelands. Singles from their last two albums, 2002’s The Vertigo of Bliss and 2004’s Infinityland battled their way into the top 30 and there was some sense that they were building in increments.

‘We might not have exploded but we’ve had the chance to get gradually better,’ James says. ‘Not enough bands are afforded the opportunity to develop like we have. We’re eternally grateful to Beggars for that. I don’t think we’ve been ready to make a big, major label album until now anyway.’

The band have steadily built a fanatical following over the years, aptly displayed at tonight’s show when Danish teens are singing along to songs from an as-yet unreleased album.

‘We were asked what our ambitions would be for our next record and we realised they were different from our label’s,’ says Simon.

They parted ways with Beggars Banquet but went through protracted wrangles getting out of their contract.

‘In a way, having that time was good for us,’ explains James. ‘Normally we’d go into the studio with a dozen or so songs to record, this time we had the opportunity to write and rewrite and ended up with about 40.’

While that might be viewed as an upheaval, nothing could compare to what happened next: Simon Neil’s mother died.

The band were literally struck down and it is an understatement to say that Simon struggled. While he’s keen to avoid detailing the litany of bad feeling he’s been through of late, suffice to say, copious drink, drugs - prescription and recreational - and depression have all snarled him in recent times. The product of exorcising such beasts from his system has been Puzzle. The death of Eleanor Neil is at the broken heart of the album. Lyrically, the songs meditate heavily on his loss. The centrepiece is a moving tribute entitled ‘Folding Stars’, which sounds like someone really not wanting to say goodbye.

‘There are songs I’m not sure we’re going to be able to play live,’ Simon admits. ‘Not yet anyway. That’s one of them.’

That’s not to say this is a maudlin record, it is rightfully peppered with moments of fury and despair but also hope.

‘We’ve been through a lot. But it’s what has made this record what it is,’ says Simon. ‘Surviving.’

If there was ever a time when a band like Biffy Clyro were equipped musically to become a more common concern it would be now. They hold the same emotive hooks that make Foo Fighters and Queens of the Stone Age so appealing and there is a sense that they deserve it.

Puzzle is a complex, complicated album that is imbued with the redemptive spirit that all great rock music should have. Nirvana did it, the Pixies did it, Weezer do it, Biffy Clyro do it too. They scare with blusterous, unpredictable noise but kiss it better with melodies from their bittersweet pop heart.

It’s a little after 1.30am as The List returns wearily to our modest digs after bidding the band a safe onwards trip to Sweden for the second of another ton of dates before they return for their own UK headline shows. The hope is that when they next return to this fine city that they’ll be the ones packing out the Vega.

Great music makes us feel good to be alive. Like being dropped off the edge of a huge metal tower with gravity as our only combatant. Great music is a genuine, undiluted thrill. Biffy Clyro’s music reminds us why it’s good to be alive.

Biffy Clyo play Potterrow, Edinburgh, Thu 31 May; Barrowland, Glasgow, Fri 1 Jun. Puzzle is out Mon 4 Jun on 14th Floor Records. See review.