Fans of Icelandic music have had much to pick over at the LA Philharmonic’s Reykjavik Festival, a 17-day extravaganza curated by LA Phil conductor Esa-Pekka Salonen and Icelandic composer and conductor Daniel Bjarnason. There is Björk Digital, as well as an installation by the Icelandic artist Shoplifter, and Xarene Eskander’s Driving At the Speed of the Nordic Sun screened on continuous loop.

But the festival’s jewel was three evenings devoted to the music of Sigur Rós, the Icelandic post-rock three-piece famed for the extraordinary texture and scale of their music, for their used of bowed guitar and for lead singer Jonsi Birgisson’s remarkable falsetto.

Each night culminated in a live show by the band itself – currently on tour in the US, but the early part of the evening saw tracks spanning the length of the band’s career re-imagined by a number of contemporary composers including Pulitzer prize-winning David Lang, also known for his scores for Paolo Sorrentino’s films Youth and The Great Beauty, and Owen Pallett, a Polaris prize winner and frequent collaborator with Arcade Fire.

“What makes it special is something enigmatic about that band,” says Nico Muhly, the American composer known for his work with Björk and Grizzly Bear, as well as for arranging the orchestration for Birgisson’s solo album. “It’s oblique, it creates an atmosphere.”

For the festival, Muhly tackled a new song, Nordor. “For me the older songs exist in such an iconic form and I didn’t want to dismantle them,” he explains. However the commission has given him the opportunity to examine further how the band’s music works. “I love that they’re not scared to make ambient music or afraid to make music that’s very, very quiet or music that’s very, very loud,” he says.

For BBC Scottish Symphony composer-in-residence Anna Meredith, who chose 2008 track Fljotavik, Sigur Rós’s use of space was intriguing. “In a way it was an orchestrational gift,” she says. “It’s beautiful and sparse, and it’s nice to exercise a bit of restraint, to use space, which is something that doesn’t come naturally to me. There were moments, she says, where she was tempted to go “the full Phil Spector” but she managed to resist.

For the Icelandic composer Pall Ragnar Palsson, the band’s music is part of his country’s national identity. “We’ve known each other for a long time, including through the 90s indie rock music scene in Iceland. I have emotional attachment to the music of that time.”

Björk and Arca perform onstage at the 2017 Ceremonia festival in Toluca, Mexico. Photograph: Santiago Felipe/Getty Images

“This whole generation of Icelanders consider this music very dear to them,” he adds. “It somehow just fitted into the zeitgeist – around that time Sigur Rós supported Radiohead on the Kid A tour, and they were paving the way for people to be interested in music that they wouldn’t be interested in before.”

Certainly in the years since, the band’s influence has been tangible: in the soaring structures of Of Monsters and Men and Explosions in the Sky. Iceland, too, has become something of a musical epicentre – Reykjavik’s annual Airwaves festival offers an array of emerging musical talent in venues strung across a cold but walkable city, as well as established names and spectacular one-off productions staged at the stunning Harpur concert hall, right on the waterfront. Alongside the international talent come a diverse selection of local artists such as the hugely successful Asgeir, the pop star Milkywhale, and Halldor Eldjarn, a drummer and programmer who performs with handmade robotic instruments.

[In Iceland] you need to be original, because you don’t want to sound like someone else who lives on the same street Pall Ragnar Palsson

In recent years, Iceland has also become a home for musical newcomers too – the American electronic musician John Grant, for instance, who, after decades of booze and drugs and unhappiness in the US, found a new kind of peace and musical inspiration in the country.

It is this diversity that seems so striking in a country of just 300,000 people. That such a small land could host both a big commercial act such as Of Monsters and Men as well as film composer Jóhann Jóhannsson, folk hero Snorri Helgason, and the Bafta-winning instrumentalist Ólafur Arnalds, as well as punk bands, a symphony orchestra, singer-songwriters and programmers. As Palsson says: “You need to do something original, because you don’t want to sound like someone else who lives on the same street.”

Attempts to describe the particular magic of Sigur Rós’s music often flounder, falling into detailed musical terminology or grasping for the language of architecture or geography. It is testament, perhaps, to the compelling nature of their songs, a quality that still stands today – even in the more aggressive tones of 2013’s Kveikur. As Muhly puts it: “The thing I love about Sigur Rós is it’s music that’s floating and music that’s on the grid. And there’s meaning that’s floating and meaning that’s on the grid.” Their music, its method and its meaning, always seems intriguingly out of reach.