The rise of FKA twigs is one of the most intriguing in modern pop. It has not been an ascent to dizzying commercial success: her releases sell respectably rather than astonishingly. It’s more about the acclaim she’s achieved. Rock and pop stars of any stripe usually struggle to convince the world that they’re actually polymath geniuses; FKA twigs – AKA Tahliah Barnett – seems to have been accepted as one from the start. In the five years since her debut album, she has diversified into everything from arty webzines in which she features as a comic strip superhero called Anomalie to starring in a commercial for Apple directed by Spike Jonze. Live, she frequently spends more time engaged in choreography than actually singing, as if music is a secondary pursuit to abstract tap-dancing and demonstrating her mastery of sword-based kung fu. Her videos aren’t just promotional items but exercises in visual art with mini-documentaries attached, explaining the practice behind them.

The artwork for Magdalene.

Critics could have greeted all this with scepticism, for distracting from the music that drew people to her in the first place, but quite the opposite. The uniformly rhapsodic response to it all has seen Barnett fast-tracked into a very rarefied space, spoken of in the same breath as Kate Bush and Björk, not as a lazy comparison but as their contemporary auteur heir.

The praise is sometimes deserved – but not always. Magdalene’s sound cuts almost all Barnett’s ties with the “alt-R&B” label that attached itself to her debut (and which she has decried as racist). Instead, she heads further into the realms of abstraction: a discombobulating aural scrapbook of pummelling, funeral march percussion, slivers of eerie electronics and white noise and warped samples of voices. As in one of the videos for which Bennett learns pole dancing, martial arts, oncological neurosurgery, hang-gliding or whatever, you can sense the pains that have been staked in its making. However chaotic the musical backdrop appears to get, you know everything has been very carefully assembled, that not a note or a beat or a wisp of feedback is out of place, that nothing has happened at random. It’s strange enough to make the ordinary moments jarring, as when rapper Future appears on Holy Terrain, rhyming over a straightforward beat, Auto-Tune cranked up to full.

Sometimes the results are stunning, as on the beautiful microcosmos of tiny, constantly shifting sounds that fade in and out of Mary Magdalene. Or on Home With You, with its rhythm track made up of bursts of static interference, noises that sound like distorted gunshots, whoops that sound like someone catching their breath mid-sob. Barnett’s voice seems to have shifted slightly from her debut, becoming more refined. A high, perfectly enunciated RP quaver that sounds like a postwar BBC announcer on the verge of tears, it’s definitely an acquired taste but capable of sounding simultaneously wracked and icy – an impressive ruse to pull off.

FKA twigs: Cellophane – video

Sometimes, however, the songs are weirdly stifling. Magdalene is clearly a very personal work: a breakup album that ruminates on the power of female sexuality and the toxic effects of fame and media intrusion on one’s personal life (Barnett became an unlikely tabloid fixture after dating actor Robert Pattinson). “If I walk out the door it wakes a thousand eyes,” protests opener Thousand Eyes. “They want to see us alone, they want to see us apart,” notes closer Cellophane, bitterly. But there are moments where the music seems to work against the listener fully engaging with the emotions of the songs. Fallen Alien features a lot of clever production touches – beats that sound like swords clashing, unexpected explosions of industrial clamour, high-drama bursts of sped-up backing vocals – allied to a meandering, episodic tune. The result is something you admire for its audacity, and for the fastidious technique involved, rather than immerse yourself in.

There are very good songs here that would pack a punch stripped of the accompanying sonic barrage – Bad Day, Mary Magdalene, the exquisitely careworn Mirrored Heart – but occasionally the sound seems like scaffolding to support some quite slender material: the two lines of melody that make up Thousand Eyes; the repetitious Day Bed.

And yet, for all its variable quality, you don’t finish Magdalene mystified as to why FKA twigs has attracted so much praise. Her ambition is lofty. She shows a genuine desire to press forward, to create something new, and that is utterly at odds with huge swathes of current British pop, where trying to fit in and appear as normal as possible is very much the order of the day. Magdalene is not carefully constructed to slot unobtrusively into a curated streaming playlist. Its highlights achieve their ambition in a genuinely impressive style, and if other moments fall short, it’s not for want of trying. At a time when a lot of twigs’s peers aren’t trying to do much more than tick boxes, that’s laudable in itself.

This week Alexis listened to

Cate Le Bon and Bradford Cox: Secretary

Art-rock summit meeting yields hauntingly rickety song: out-of-tune piano, ramshackle percussion, spoken-word interlude from the Deerhunter frontman.