When Osmo Vänska became music director of the Minneapolis Orchestra in 2003, he gave his players a tough order: Become America’s top orchestra. It was a brash objective, especially given the Finnish conductor’s soft-spoken manner. “It doesn’t mean we’re against anyone,” he quickly pointed out. “But we would like to be the best, and this gives us a goal.”

New Yorkers will have a chance to make their own assessment of Mr. Vänska’s progress toward that goal when the orchestra plays in Avery Fisher Hall on Sunday. Many who have never heard the orchestra under Mr. Vänska will already know the potency of their partnership thanks to their award-winning recordings of Beethoven symphonies on the Swedish label BIS. As Mr. Vänska explained in a phone call from Leipzig, where he was guest-conducting the storied Gewandhaus Orchestra, the Beethoven project was an important early step in their pursuit of excellence, yet the idea arose from BIS, not from him.

“Why did we record the Beethoven symphonies when there were already more than 100 recorded cycles? That’s a good question. When my job in Minneapolis was announced, BIS said it wanted a Beethoven cycle.” Mr. Vänska had already recorded more than 70 discs for the label, but the Beethoven project prompted some soul-searching. “They thought we could do something new, something fresh with the symphonies, but I needed to be convinced.”

Reviews have praised the spontaneity and lucidity of the performances, which still have an element of traditional weight. By thrusting conductor and orchestra into an enterprise that demanded something approaching perfection, the project constituted “a great challenge,” Mr. Vänska said. “But it also brought us closer together. The BIS producer, Robert Suff, understands musical ideas, and the recording sessions were master classes for everyone.” Next on the recording agenda are the Beethoven piano concertos, with Yevgeny Sudbin as soloist.

Beethoven, however, though featured in the orchestra’s last New York appearances in 2007, is absent from Sunday’s program. The concert leads off with “The Dryad” by Sibelius, the composer with whom Mr. Vänska is most closely identified. This short piece “is not deep, dark Sibelius, but ought to make people smile.” Also on the program is Mahler’s First Symphony (“Titan”), for which Mr. Vänska will disregard Mahler’s final expressed preference and include the original Andante second movement, known as “Blumine.” In addition, Lars Vogt will join the orchestra for the Schumann Piano Concerto. Might Mahler be the subject of the next cycle? “He is on the list, and Bruckner is, too,” Mr. Vänska said, but it is too early to say.

Mr. Vänska came to prominence at the helm of a regional orchestra in the southern Finland town of Lahti. He built the Lahti Symphony Orchestra into a first-rate ensemble known for its distinctive performances of Sibelius. His results in Lahti helped win him the post of chief conductor of the BBC Scottish Symphony Orchestra in Glasgow (1997–2002), where he won the admiration and respect of British critics.

Mr. Vänska received his principal training at the Sibelius Academy in Helsinki as a student of the legendary Jorma Panula, who taught nearly all of today’s prominent Finnish conductors, including Esa-Pekka Salonen, Mikko Franck, Sakari Oramo, and Jukka-Pekka Saraste.

Since going to Minneapolis, Mr. Vänska has returned to a former pursuit: playing the clarinet. Before he turned to conducting, he was co-principal chair of the Helsinki Philharmonic. In February, he played the Brahms Clarinet Quintet with members of the Minneapolis Orchestra and last summer performed in a Mozart wind serenade at the Mostly Mozart Festival here. “I went three years not playing the clarinet and started to miss it very much,” he said. “The clarinet gives me a direct connection to playing music. A conductor needs the orchestra; with the clarinet, it’s myself.” No doubt his clarinet performances add something to his conducting as well. But, fortunately, conducting will continue to come first.