Dave King, the drummer with the maverick trio the Bad Plus, is seated amid the swags, drapes and genteel gloom of the Café de Paris, London, looking a little hurt. “It’s uniquely the British critics who do this. They have this idea of us as ‘piano trio that plays Black Sabbath’. They somehow think we’re not serious. But there’s a complexity and a drama going on that they miss.

“The idea that we’d be out on the road 160 nights a year for a joke is not funny. That’s insulting.” He shrugs and slumps back on the sofa.

Depending on your viewpoint, the Bad Plus are the best thing – or the worst thing – to come out of American jazz in the past five years. The trio from the Mid-west have junked the tired formula of rifling through the Great American Songbook. Instead they have reinvented Nirvana tunes, swung Chariots of Fire, sent up Abba and written originals that veer between romanticism and hyper-active clatter.

This has all won them a healthy international fanbase and the bafflement of some critics. “Please, tell me it’s just a clever dadaist in-joke,” implored The Sunday Times of their Suspicious Activity? album. The Evening Standard thought last year’s Jazz Café show “too clever by even more than half”.

But after six albums, those listeners who think they have the measure of the Bad Plus are about to be wrong-footed. “We decided to do an album in the tradition of a jazz record of standards,” King says. “But some of it is contemporary classical music – some Stravinsky and Ligeti – and some Wilco, too.”

The album, For All I Care, is inspired by the John Coltrane band’s record with the vocalist Johnny Hartman, according to the band’s cerebral pianist Ethan Iverson. Except that Coltrane never mixed up the Bee Gees’ How Deep is Your Love with Wilco’s Radio Cure or Stravinsky’s Variation d’Apollon. A postmodern mess? Actually no. With the discipline of having to back a singer, the band have reined in their more chaotic impulses and delivered one of their most convincing sets.

The vocalist is Wendy Lewis, a friend of King from bands in Minnesota, and she has joined them for this hastily arranged promotional show at the Café de Paris. Like the rest of the band, waiting in the chintzy murk before showtime, she is looking tired from an overnight flight from San Francisco. Lewis is no chick singer brought in for eye candy. “I have two grown-up daughters, I’ve never played outside America before, but they said: ‘Yeah, go.’ They think it’s great.”

Reid Anderson, the bassist, calls the record “a kind of unifying statement that all these diverse musics can live in the same world”. They accept that the jazz police might not get it. “What we’re trying to do is be real. We’re not trying to please everybody but we have to please ourselves. But it’s a misconception to say that we’re antijazz or not serious.”

But isn’t it at least true that the band have played some tunes for laughs – as when they used to beat up Abba’s Knowing Me, Knowing You? “We don’t relate to this idea of irony even though it’s something other people may see in us,” Anderson says. “We don’t start from a place where we say these songs are worthless. We like the way they connect with our life experiences and the life experiences of people in the audience.”

But what about that hyperbolic Black Sabbath cover? “When we met Geezer Butler, who wrote Iron Man, he said it was the best Sabbath cover he had ever heard,” King says. “He came to tell us that. He felt it was a very powerful rendition. He didn’t see it as a joke.”

Iverson dislikes the snobbishness that views rock songs as less valid than jazz tunes. Too many jazz musicians are playing by rote. “Jazz education has turned out a zillion players that play this B-flat moderate jazz and it’s terrible for the music. You can’t tell anyone apart.”

He leans forward and says carefully: “It’s really important for musicians to know a lot about jazz – then choose not to play jazz. Too many people learn just to ‘play some jazz’. If you’re not bringing surrealism, a sense of ‘other’, the creative imagination . . .” he tails off and adds: “I can get quite angry at times.”

Still, the Bad Plus endure. They have been operating for eight years, veering between jazz temples such as the Village Vanguard in New York to sharing a stage with the Pixies – flying the flag for tricky music in the X Factor age.

King says: “What we’re most excited about is seeing the generational quality of the audience – from older jazz fans to young people who have never checked out jazz but just like the improvising. That’s how we were once, the young kids listening to Thelonious Monk. We can be a bridge to checking out the music that excited us. We’re firmly rooted in the idea that this music is available to everyone.”

Later that night, they perform a set that embraces Flaming Lips, Pink Floyd and the sort of intuitive interplay that only comes when three musicians have played long and hard together. Some people might not get it, but the ghosts of Charlie Parker and Miles Davis surely would.

For All I Care is released by Emarcy on October 20 (www.thebadplus.com)

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