Sat with her back to the audience, Madison Cunningham is performing an eerie, unreleased ballad, her white dress mirrored in a glossy black upright piano. The theatre is pin-drop silent as the California singer-songwriter’s astonishing voice, classic and characterful, pours into a heavily reverbed microphone.

“That was a funny way to say hello,” she grins afterwards. The “pitch” for this one-off show, part of Celtic Connections festival, she tells us, is that it will comprise almost entirely new songs, from a record not yet announced. It’s a bold gambit – usually fans come to hear the songs they know – but Cunningham has a winning confidence in her rich, complex new material, and how it develops her sound. One song, with a restless piano melody that echoes Joni Mitchell, slips organically into a verse from Cunningham’s Grammy-winning folk album Revealer, the lyrics altered to address the wildfires in her home town: “You’re all I’ve ever known, Los Angeles,” she mourns.

Earlier, opener Louis Abbott (of Glasgow’s Admiral Fallow) asked any fellow musicians in the crowd to raise a hand. “Yeah, I thought it would be that kind of gig,” he laughed, after an enthusiastic response. His point – that Cunningham is a musician’s musician – is evident in her technicalities and unusual tunings, but also in her intuitive, unexpected melodies and knack for a twist. Like Revealer, these new songs bridge folk, alt-rock and Americana, but with a far heavier atmosphere. When her band join her, and Cunningham picks up her electric guitar with its uncanny, underwater tone, one rock song builds into a rolling boil, her voice thickening into a growl.

Finishing on a handful of familiar singles, Cunningham and band radiate ease. Hospital has a wonky, wry riff, and Life According to Raechel, a love song about her late grandmother, starts so vividly, just her and a guitar, that you scarcely notice her bandmates slipping in.

It’s obvious Cunningham could have taken the easy road tonight – playing through Revealer would have worked just fine. But it’s her pursuit of the new, the challenging and even the uncomfortable that makes her so distinctive. She is a musician’s musician, after all.