For some bands, the feeling of having made it comes not with number-one singles and sports-arena fireworks, but a sit-down set in a major cultural ­institution.

“Play the hits!” demanded one wag among The National’s notably fervent fanbase. “Dude, these are the hits,” replied none-more-dry singer Matt Berninger, knowing that his band didn’t need chart-toppers for the evening to feel significant.

Headlining here between numerous festival appearances, this was the first chance for a UK audience to hear how the Brooklyn-based quintet will follow their near-universally acclaimed fourth album from 2007, Boxer. It cleaned up their sound to a condition of doomed grandeur, characterised by ringing guitar, stately piano and Berninger’s bruised, low tones.

His vocal weariness inevitably leads to comparisons with Leonard Cohen, but you wouldn’t catch Len barging deep into the crowd, fending off embraces, screaming like he’s just caught his hand in the cutlery drawer. What began politely seated, with the frontman mainly singing to the floor, ended up as something far less controlled.

Older tracks Abel and Mr November reached roaring heights, while new song Blood Buzz, Ohio built up a similar wall of guitar over racing drums.

Runaway and Vanderlylle Cry Baby were more restrained ballads but still promised great things for next year’s new album. Continued cult status is assured even if The National don’t go global.