There’s a 20th-century tradition of composers finding audiences in museums and galleries. Washington’s museums present concerts in more conventional spaces than the art galleries where Steve Reich and Philip Glass found listeners in the 1960s, but they’re doing a lot to support new music in a city not always attuned to it. The Phillips Collection spotlights living European composers; the estimable Verge Ensemble and the 21st Century Consort are based at the Corcoran and the Smithsonian American Art Museum, respectively; the National Gallery of Art has a resident new music ensemble.

And the Smithsonian’s museums of Asian art, the Freer and Sackler Galleries, have a strong concert program with an East-meets-West emphasis. On Thursday, the Lark Quartet joined the koto player Yumi Kurosawa to offer the Washington premiere of “Genji,” a concerto for koto and string quartet that Daron Aric Hagen wrote for the group in 2010.

Encounters between traditional Asian music and Western music have come to occupy their own niche on the concert stage: Too frequent to be a gimmick, they retain an undeniable curiosity factor, particularly when they involve a composer who, like Hagen, had never written for the koto before. The instrument is a kind of dulcimer, a board laced with plucked strings tuned with pyramidal blocks, set at intervals under each string, that give the curving surface the look of a mountainscape diorama.

The sound is slightly twangy, robustly banjo-like, with a range and even volume beyond a banjo’s. Hagen presented the piece as a dramatic dialogue in five sections between the Western and Eastern voices: Now the koto sang a solo over suspended string lines; now the two groups exchanged folk-like melodies, the Westerners’ contribution sounding like a sea shanty; now the solo strings traded off romantic solos over the koto’s plucking.

The result was a vivid and appealing piece that even sounded idiomatic. Just how idiomatic only became clear, at least to Western ears, when Kurosawa followed the concerto with two solo pieces — “Disorder,” by the 17th-century composer Kengyo Yatsuhashi, and “GreenPt,” written by herself — that showcased the instrument’s somewhat timeless voice.

Bookending these Eastern offerings were two purely Western works. The evening opened with a group of little-known quartet movements by Mendelssohn, including an elfin Scherzo in A Minor that was the last scherzo he ever wrote, which the quartet played with a scrubby alacrity. The closing piece, Janacek’s “Intimate Letters,” was an even better fit for this group’s dark, graceful toughness; they gave a vital and robust reading that carried over the folk-song flavor of “Genji” without lapsing into emotional excess or losing sight of the work’s innate grace.

For its next East-West musical encounter, Nov. 3, the Freer Gallery will present the ensembles Music from China and Music From Copland House in world premieres by the composers Derek Bermel and Wang Guo Wei, as well as works by Bright Sheng, Chen Yi, Lu Pei, and Yang Yong.