NEW YORK (NYT NEWS SERVICE ).- Almost 15 years ago, the men of the Danish String Quartet — they were in their 20s at the time and still called themselves the “Young” Danish — said in an interview that they would need to become more mature before daring to play Beethoven’s late string quartets in public.

It didn’t take that long for these prodigiously gifted musicians to get over their youthful reticence. In 2014, for their first Chamber Music Society of Lincoln Center performance at Alice Tully Hall, they ended a program with an urgent yet cogent, spacious and mystical account of the Quartet in A minor (Op. 132). And they were back there this month for a series of six concerts, again presented by the society, devoted to all 16 of Beethoven’s quartets performed in chronological order.

Over the years, the Danish quartet has played these pieces individually. But this is only the second time they attempted a daunting complete cycle — about nine hours of music. Tully was packed for all six programs, which opened Feb. 7 and concluded Tuesday. This year, the 250th anniversary of Beethoven’s birth, is going to offer lots of complete cycles. It’s always difficult to bring fresh takes to pervasively familiar repertory, but the standard set by this quartet will be especially hard to top.

As a rule, I’m not keen on complete Beethoven cycles, precisely because these scores remain such fixtures of our musical lives. I prefer programs that take on Beethoven by juxtaposing his pieces with works by other composers. That’s the approach the Danish quartet took in two recent recordings on the ECM label, “Prism I” and “Prism II,” which place two late Beethoven quartets in context with Bach fugues and 20th-century quartets by Shostakovich and Schnittke.

But perhaps Beethoven’s string quartets could benefit from being heard complete if they’re in order, spanning the composer’s career. And how many opportunities do we get to hear, say, the “Serioso” Quartet?

Or, for that matter, the six early quartets published together as Op. 18. For all the command these probing musicians demonstrated in their riveting accounts of the late quartets, their excellence came through most in bracing performances of the early works.

Beethoven modeled those Op. 18 quartets on Haydn, his teacher, who essentially invented the genre of string quartet. Here Beethoven was honoring him by writing works within the protocols of form and style that Haydn had developed. But there is an element of one-upmanship at play. Beethoven is taking on the master, showing what, in a sense, a “Haydn” quartet could really be.

There was an impish quality in the Danish quartet’s playing, despite its elegance and brilliance. From the first phrase of the opening work, the Quartet in D (Op. 18, No. 3), the blissfully lyrical violin melody, soaring over seemingly supportive harmonies in the other strings, sounded cozy and alluring but just a touch sly. Sure enough, before long, an episode of spiraling passages in triples seemed like a warning for us to listen below the sunny surface.

It’s difficult to explain what makes the Danish String Quartet’s playing so special. Other ensembles arguably match these players in technical excellence and interpretive insight. To say that their performances represent a marvelous balancing of qualities suggests that they occupy some place in the middle of the road. The results are anything but: There is a winning mix of studied concentration and willful freedom in their playing. “All Scandinavians feel like they have a bit of an anarchist inside them,” Asbjorn Norgaard, the group’s violist, said in a 2016 interview. That came through during this entire series.

Their technical command resulted in precise execution. Yet they played with enough leeway to allow instinctive responses to take over in the moment. You might assume that musicians in their 30s would bring youthful energy to bear, but I was struck by how often they opted for a raptly restrained tempo. Rhythms were dispatched with clarity and exactitude without a trace of rigidity.

They have a shared sensibility and richly blended sound. But that doesn’t stop their individual musical characters from continuously shining through. (The members, besides Norgaard, are Frederik Oland and Rune Tonsgaard Sorensen, who trade playing first and second violin parts, and Fredrik Schoyen Sjolin, a Norwegian cellist who has been with the ensemble since 2008.)

The big endurance test of the cycle was the third program, in which they played the three “Razumovsky” Quartets (Op. 59), from 1806. Here is the towering Beethoven, the composer as revolutionary, striding across the pages of these scores — brash, adventurous and ingenious. I found the performance here of the middle one, in E minor, especially distinguished. The Danish quartet brought out both the brooding weightiness and near-crazed intensity of the music. The ebullient third, in C, ends with a quasi-fugue finale, a breathlessly fast tour de force with streams of rapid-fire notes. For an encore, they repeated the final large section of that movement. And, with nothing to prove, they played with an extra dose of daring.

It’s hard to single out movements, or even moments, from the ensemble’s accounts of the late quartets. I loved how they began the Quartet No. 12 in E-flat (Op. 127), which opens with what seems a fanfare, in thick chords, that soon spins off into a genial exploration of a winding theme. The Danish quartet underlined this passage with grit and urgency that sent a signal: A gate to a new path had been opened.

They ended the Quartet No. 13 in B-flat (Op. 130) with the original finale, the Grosse Fuge, not the benign substitute Beethoven replaced it with at the urging of his publisher. That section still comes across like the fugue to end all fugues, with outbursts of sputtering rhythms, obsessively hammered attacks and tangles of wayward counterpoint. As played here, it sounded audacious, extreme and, finally, exhilarating.

The players showed imagination in closing programs with short encores, including a harmonically juiced-up arrangement of Beethoven’s popular piano piece “Für Elise” and, on Valentine’s Day, “My Funny Valentine.” But after the series ended Tuesday with the Quartet in F (Op. 135), they returned to the stage without their instruments, joined arms shoulder-to-shoulder and smiled to the cheering audience. They gave Beethoven the last word. Which was right.

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