Jenny Bilfield is president and CEO of Washington Performing Arts Society. (Sarah L. Voisin/The Washington Post)

In a world of old-boy bastions, classical music’s has been particularly entrenched. Even today, conductors are generally male, and female composers are chronically underrepresented on the orchestral stages of the world. The picture is brighter in administration, but even there the glass ceiling is not far behind us; the women who helped break it, like Deborah Borda, now the president and CEO of the Los Angeles Philharmonic, are still active in the field. Catherine French, now a prominent headhunter for nonprofit organizations, remembers the New York Times headline in 1970 when she was picked for a prominent orchestra job: “Girl to Lead American Symphony.”

But in Washington, at least, women have taken over classical music’s traditional male network. In January, Francesca Zambello officially became the Washington National Opera’s artistic director; in April, Jenny Bilfield took over as president of the Washington Performing Arts Society. Add Rita Shapiro of the National Symphony Orchestra and you have the three major classical institutions in the city led by women — not to mention the Washington Chorus, the Choral Arts Society, the Fairfax Symphony and other smaller groups, as well as the presence of Marin Alsop, the country’s highest-ranking female conductor, down the road at the Baltimore Symphony.

In an era in which women are told to “lean in” and recognize the inequities that still exist in the workplace, but in which female leadership is also becoming more common, the question is: Does this matter?

And the consensus from female leaders seems to be: We’re not sure.

The responses, admittedly, run along a generational fault line between those who actually broke through glass ceilings and those who are the beneficiaries of their work.

The National Symphony Orchestra’s concertmaster of the last 12 years, Nurit Bar-Josef, 38, doesn’t feel that being a woman has presented any problems or issues at all: “I have not encountered any conflict or felt that it’s anything different.”

For Borda, 63, one of the first women in classical music to attain a high-profile leadership position, it’s important to look at the bigger picture. “If you look at Fortune 500 companies,” she said in a recent phone interview, “4.2 percent are led by women.” Seen in that light, Washington’s situation may seem more noteworthy. “I think this is a positive statement about the arts,” she said.

Yet “it’s not something that really occurred to me,” said the WPAS’s Bilfield, “until I came here and someone brought it up.”

There are so many other things to worry about in the classical music field that the question of women’s rights tends to be underreported, except when the Vienna Philharmonic comes to town and music critics (predominantly male) get up on their high horses, and rightly so, about the disgracefully slow pace of women’s integration into that traditionally all-male preserve. And classical music is certainly far from the only field to demonstrate gender disparities; they’re still present, alas, in virtually any field. Molly Smith of Arena Stage — another strong Washington woman — has often pointed out that female playwrights are underrepresented in the mainstream theater world. And according to Susie Farr, a Washington power woman who has just retired after 14 years establishing and leading the Clarice Smith Center at the University of Maryland, only four or five of the top 20 university-based presenting organizations are led by women.

“We’re not exactly breaking down the barriers,” Farr said in a recent phone interview, “as fast as I might have hoped when I was younger.”

“I think the biases are still there,” agreed headhunter French. But they’ve abated. She added, “I think it is an evolution. In many ways I see it as 40 years from the major schools going coed. I think that was really the great equalizer. . . . It was no longer strange for the men who were growing up in the ’70s and the ’80s to have women in class with them and clubs with them, and then you see [that reflected] inside the orchestras.”

When the topic of female leadership in classical music comes up, the discussion usually focuses on conductors and composers — the two areas where women are most egregiously underrepresented. Even there, we may be too quick to think of the “woman problem” as solved. In an online op-ed piece in the New York Times this month, the self-styled “composeress” Annie Gosfield argued that female composers have it pretty good — while observing that about 10 percent of an applicant pool for grants and awards she’d juried had been women. When Caroline Shaw was announced as the winner of the 2013 Pulitzer Prize for music, I — like a lot of people — thought it was a striking choice. Shaw was only 30, didn’t even consider herself a composer, and is part of a group of young New York-based musicians with a particular smart indie vibe. It took me two days to register that the fact she is a woman was significant — given that only four of the previous music Pulitzers have gone to women.

Female leadership in administrative positions is less of an issue. The Los Angeles Philharmonic, the Chicago Symphony Orchestra and the Philadelphia Orchestra — three of the leading orchestras in the country — are all run by women. Carnegie Hall was long run by Judith Arron; the Chicago Lyric Opera looks back warmly at the tenure of Ardis Krainik.

“Early in my career all of the leaders of major orchestras were men,” said the NSO’s Shapiro, 58. “It was a different culture and ethos. Deborah Borda broke through that first, at least for my generation of mentors.” At this point, however, she feels that women reaching senior management in any field is no longer much of a story.

But you could argue that the more prominent the exception, the more it proves the rule. The majority of orchestras and music institutions are still run by men. Alsop’s accession to the Baltimore post did not clear the way for a new spate of music directorships for women, though one can name a few (such as the Estonian conductor Anu Tali, who has just taken over the Sarasota Orchestra).

It’s notable that there aren’t more discussions about the issue, and that women themselves seem reluctant to bring it up (“Successful women in particular,” Borda said, “have trouble dealing with it”). It’s also notable that, when women do talk about it, many of them invoke traditional ideas about the differences between female and male leadership — “a sense of collaboration is often missing with male leaders,” Farr said — and the perennial issue of balancing work and personal life. Does it really, still, all come down to who’s taking care of the children? And if so, will social media, which encourage more openness about personal affairs, ultimately help women in the workplace? “Maybe that lack of privacy has seeded a comfortable environment for what people really have to do,” Bilfield said. “It makes you a more interesting person to acknowledge that these other parts of you exist.”

In more concrete terms, having women in leadership positions also has the potential to foster an old-girl network. There’s no guarantee that a female leader will create opportunities for women, but Zambello’s WNO season this year includes both a female conductor (Xian Zhang will conduct Verdi’s “Force of Destiny” in October) and a female composer (the company will give the world premiere of Jeanine Tesori’s “The Lion, the Unicorn and Me,” a family holiday opera, in December).

In any case, when women become the majority in a given field, in a given region, it’s a significant success — perhaps all the more so when it’s not part of any conscious effort or movement or anything more than trying to find the best people for a given job. The fact that it’s happened in Washington may be a pure coincidence — or may say something about the city.

Bilfield is a new arrival after a career spent largely in New York and California. In Washington, she said in a recent e-mail, “I’ve found myself spending a considerable amount of time working with rather extraordinary women. Role models everywhere . . .Ironically — at least from my vantage point and perception — DC seems to be a bit ahead.”