But he said that he was proud of the orchestra’s growing flexibility. “I think Alan brought a freshness to his tenure,” said Irene Breslaw, who played viola in the Philharmonic from 1976 until last year. “Not just that he was great in contemporary music, but in his ideas.”

After starting a new-music series, Contact!, Mr. Gilbert took a page from the art world in 2014 and staged an ambitious contemporary festival that he envisioned as classical music’s answer to art shows like the Venice Biennale. He even called it the NY Phil Biennial.

And the Philharmonic continued to push boundaries: Mr. Gilbert got back together with Mr. Fitch in 2013 for a production of Stravinsky’s “Petrushka” that featured puppetry, New York City Ballet stars, and the Philharmonic players clad in Russian hats and jackets. They were willing to stomp their feet; stand up and play; switch chairs; and, in the case of a violist, juggle.

“That’s not what the New York Philharmonic did,” Mr. Gilbert said. “But now they do, and if we’d ask them to do something like that now they wouldn’t bat an eye.”

But there was an increasingly somber drumbeat within the orchestra, and throughout the music field. The Philharmonic, like other ensembles, has been trying to find its way at a time when classical music occupies less bandwidth in the broader culture; the old subscription model of selling a season’s worth of tickets at a time is dying; and high fixed costs make the economics of running an orchestra tougher than ever. Just a couple of years after Mr. Gilbert became the music director in New York, the venerable Philadelphia Orchestra filed for bankruptcy protection.

A debate emerged between those who feared that the Philharmonic could not afford to take risks and those who feared it could not afford not to. It was a theme Mr. Gilbert turned to in a speech in April 2015 while on a tour of London with the Philharmonic during which it played only 20th- and 21st-century works. Invited to deliver the Royal Philharmonic Society Lecture, Mr. Gilbert spoke about what he called the need for “a new paradigm” in classical music.