Why did the consensus of critics and cognoscenti settle several decades ago on Elliott Carter as the greatest living American composer? Part of the answer is his music’s consistently forward-looking freshness and radical complexity. But the other part of the answer is its essential conservatism, in the proper and best sense.

Carter’s latest string quartet, No. 5, dating from 1995, is a particularly strong case in point. It was the centerpiece of Sunday’s concert by the Pacifica String Quartet, an exquisitely refined California-born, Illinois-bred troupe that has earned raves for its performances elsewhere of the complete Carter string quartet cycle.

The players were violinists Simin Ganatra and Sibbi Bernhardsson, violist Masumi Per Rostad, and cellist Brandon Vamos. The San Antonio Chamber Music Society sponsored the Pacifica’s local debut in Temple Beth-El.

Like all of Carter’s mature music, this quartet is uniquely his own in matters of structure—six short movements of distinct character with introductory “conversations” among the instruments—and rhythm, which moves the voices at several different and constantly changing paces simultaneously. But this music has deep roots—not only in Arnold Schoenberg’s 12-tone method and Charles Ives’ surprising layerings, as you’d expect, but also in 19^th century romanticism.

At least two of the cellular ideas introduced at the beginning sound like fragments of romantic-era rhapsodies; the fourth movement’s long-lined and very beautiful cello melody is positively Brahmsian. The plucked and strummed finale feels strangely dancelike.

Rich chordings, which flow in a way not very different from traditional voice-leading, dominate the second and fifth movements, and even without a formal tonal home base the music very often implies regions of harmonic motion and equilibrium.

The performance was superbly crafted, impeccably timed and alert to both the radicalism and the deep precedent in this music.

The Pacifica opened and closed in familiar territory. The salutatory four chords of Felix Mendelssohn’s Quartet in A Minor announced the troupe’s style of playing, a combination of taut ensemble, detailed planning, finely gauged balances and a richly grained, integrated tone.

If the proceedings seemed too closely planned and the intensity somewhat tethered in Mendelssohn, those traits responded aptly to the composer’s own aesthetic.

Bedrich Smetana’s Quartet No. 1 in E Minor, “From My Life,” elicited freer passions, especially in the tender, ballad-like ending, which alludes to the onset of the composer’s syphilitic deafness.