John Zorn doesn’t look 60. He looks 35, or whatever age it is where you’re still just about young enough to get away with wearing baggy camouflage pants. “And that’s how it’s done!” he yells through his mic at the end of the first half, over huge applause and cheering from this passionate Barbican audience. The first half has consisted of vocal music written by Zorn, spanning a few light, Latin-style numbers drawn from film soundtracks, a gorgeous composition for five female voices about the 12th-century nun Hildegard of Bingen (The Holy Visions), and – just before the break – some heady, intense music for band that is essentially heavy metal.

And that is how it’s done, whatever it happens to be. What’s astonishing is the complete sincerity of the music in whichever style he’s written it. His heavy metal is real heavy metal; his delicate a cappella vocal writing is real choral music, and his second-half string quartet The Alchemist is as classically composed as they come, too. It’s all sincere, and it’s all excellent; it’s proof, overall, that quality transcends style – though it’s compelling enough, moment by moment, that all you think about in performance is the music.

We are given over three hours of material, and Zorn presents it all, bursting with enthusiasm in true MC style. He only gets his sax out briefly in the second half, but he involves himself in the performances throughout, usually nestled right in among his band and conducting through all manner of bizarre, unorthodox, sudden gestures – a flap of his hands here, a tap on the guitarist’s arm there, lots of pointing. He’s obviously shaping the whole thing, but the band are so tight that it seems unnecessary; he comes across as an awkward cross between a control freak and an air guitarist. He gets in even deeper for the heavy metal numbers, donning a hoodie, raising the hood, and blurring the boundaries between conducting, moshing and performance art.

For the more classical numbers, he doesn’t conduct, but nor does he fully leave the stage: he perches ponderously on the steps at the side, where we can see him listening. He may not join in, but he doesn’t quite let go of any of his music all evening.

He has no cause to worry about it in these hands: the players are all superb. Steve Gosling gives an (apparently) meticulous realisation of the dizzying, nuts piano piece Illuminations (performed here with improvised bass and drums from Trevor Dunn and Kenny Wollesen); a string quartet of Jennifer Choi, Jesse Mills, David Fulmer and Jay Campbell do The Alchemist proud; Mike Patton, Jesse Harris and Sofia Rei all excel vocally, in their extremely different ways. Most impressive of all are the band, billed as Electric Masada (Marc Ribot, guitar; Jamie Saft, keyboards; Trevor Dunn, bass; Joey Baron and Kenny Wollesen, drums; Cyro Baptista, percussion; Ikue Mori, electronics); their final set is searing, brilliant, with everything from epic Guns n’ Roses-style guitar solos to blurty Ornette Coleman-style outbursts of free jazz. Zorn himself joins in on his alto saxophone only for the most free, wildest numbers, and when he does so it sounds like a donkey giving birth to the Antichrist. It’s great.

Obviously, you can’t categorise John Zorn. All the music is of the highest quality, and all is uniquely his, there’s no question – he even curates the improvised sections, through his conducting – but as a whole it can’t be grasped. And when his music is presented as haphazardly as it is tonight, with little clue about even what sort of music is going to be next up, the whole thing becomes even more thrilling than usual. This concert proves that Zorn is a great musical performer in more or less the same way as any other: he presents his music with a conviction that is compelling to watch, whether you truly understand it or not. The music is inexplicable, but only in the same way that it’s impossible to properly account for any music, ever. What we listen to is how it’s done.