Had Sigur Ros’ new album “Takk …” been released in 1999, it would be considered a modern classic, a clear link between the shoe-gazing glory of My Bloody Valentine’s “Loveless” and the atmospheric indie pop of Broken Social Scene or The Arcade Fire. But way back then, the Icelandic Sigur Ros was just releasing its first full-length stateside album, the less realized but critically acclaimed “Agaetis Byrjun.” “Takk …” outshines both “Byrjun” and Sigur Ros’ untitled follow-up: It is less obscure, more exciting, dynamic and instantly wondrous than anything the band — and most others — have ever done.

“Takk …” opens with the title track, a 1:57 introduction that matches the stoned ambience of Brian Eno’s ground-breaking early work. But there is an almost pulsing urgency in the quivering of the strings, and immediately the album separates itself from both the noise-rock that influenced “Agaetis Byrjun” and the follow-up’s lullabies.

When the next track — “Glosoli” — begins, Sigur Ros roots itself in its prior work by demonstrating how much the group has developed. The song opens with a guitar slowly playing chords over twinkles of a music box and high-pitched harmonies — sung, as always, in singer JT Birgisson’s made-up hybrid of Icelandic and English. But then the percussion kicks in, ascending into a fury that somehow channels the transcendence of late Beethoven.

The album moves into “Hoppipolla,” another standout track. It is powerfully driven by a string arrangement that alone upstages every like-minded album this year. The relatively short track rises and falls effortlessly but not tiresomely, demonstrating a newfound and welcome conciseness.

The rest of the album returns to the sometimes ponderous boundlessness of Sigur Ros’s earlier work. “Milano” slowly climbs to a crescendo and then falls, only to rise to an all-too-similar peak at seven and a half minutes, only to slow-fade for another three. Sigur Ros may have mastered the art of the ebb-and-flow, but this 10-minute track, on first listen, gives the listener the strong but ultimately inaccurate feeling that the group’s bag of tricks is fairly shallow.

Thankfully, those moments of disappointment are rare. “Gong” achieves a much moodier tone, anchored by mournful strings and percussion straight from Radiohead’s “OK Computer” (the two bands recently collaborated with the Merce Cunningham Dance Company). The drumming falls away to reveal an initially strong guitar which, as its tone softens, carries the album into a more ambient section. “Saeglopur” brilliantly channels “OK Computer” as well (in this case “Climbing Up the Walls”). And then the distortion pulls back, laying bare a simple and moving string arrangement.

Each of the emotions that course through the album, in fact, seem to deeply resonate with the band until, nearing the end, they can barely hold back tears. The climax of the second-to-last track, “Svo Hljott,” isn’t the highest peak in the album, but it certainly feels like the most powerful. One cannot help but sit back as the album warmly bids farewell.

Perhaps fans of Sigur Ros’ more obscure work may be disappointed at the headway “Takk …” makes into more conventional song structures and melodies. But the transition is not only wise but also incredibly natural; even at the album’s least impressive points, the music never sounds derivative, forced or thoughtless. Praiseworthy peers — even the folksy likes of Sufjan Stevens or the easy-going Architecure in Helsinki — often try hard to make their music seem effortless and rough. They would do well to learn from Sigur Ros that precision can be powerful and emotive.

As a side note, I used to think that a person only needed one of Sigur Ros’ albums — whichever one they heard first. “Takk …” changed my mind. It builds upon the youthful roar of “Byrjun” and the airy beauty of the untitled work, achieving a unique and winning combination of the two, a combination that is honest and organic. Each crescendo affirms the band’s vitality, poignancy and relevance, pulling back the curtain on Sigur Ros to reveal them as the most important band making this kind of music today.