THE WHITE HOUSE became a world-class performing arts center during the Kennedy administration when Pablo Casals was invited to perform in the East Room. Recording engineers from RCA Victor were on hand to preserve the occasion for posterity.

Unfortunately, they also recorded the fact that the event lacked a world-class audience. The Washington elite applauded at the wrong time -- after the first movement of a four-movement Mendelssohn trio.

Fortunately, even before the Kennedy administration, in places like the Library of Congress, the National Gallery and the Phillips Collection, there were Washington audiences who knew when to applaud for chamber music. The Library of Congress audience, in fact, is one of the most sophisticated -- and best served -- in the world. And from here (as from the Phillips Collection and the National Gallery) regular, live radio broadcasts spread Washington culture far outside of Washington. The city's problems in becoming a cultural center have focused largely on those who were elected to come here -- people like President Grant, whose musical knowledge was once summed up in a terse statement: "I know two tunes; one of them is 'Yankee Doodle' and the other one isn't."

Happily for Washington, those with political power are not the arbiters of Washington culture. For that kind of power, you look to people like Roger Stevens at the Kennedy Center, Zelda Fichandler at the Arena Stage, David Lloyd Kreeger (the city's busiest patron of the arts), and the decision-makers at the Cafritz Foundation and the National Endowment for the Arts. Douglas Wheeler, successor to Patrick Hayes at the Washington Performing Arts Society (WPAS), makes the most imortant decisions about what the city will see and hear in the performing arts, along with Mstislav Rostropovich at the National Symphony and Martin Feinstein at the Washington Opera. On the whole, these decisions are in good hands.

Washington has had some claims to cultural life since the pioneering Friday Morning Music Club was founded a century ago, and it has undergone a cultural renaissance since the WPAS came on the scene in 1965.

Particularly in the last generation, culture has been slipping into Washington whether the movers and shakers know it or not. In this, Washington is a microcosm: the arts are sprouting all over the United States. Part of the reason is increased leisure, and part of it is probably the long-term effect of the G.I. Bill of Rights, which after World War II brought a college education within the reach of many, many Americans.

But Washington is a special case, because the people who elected to come here have been hiring highly educated assistants to help them cope with the growing complexities of legislation and administration in the late 20th century. The city's other major industries tend to deal with words and thought processes rather than tangible, manufactured products; we have lobbyists, national associations, public relations firms, law offices and communications media of all kinds -- from specialized newsletters to network television. These thought-and-word people take the seats next to yours at the Arena Stage or the Eisenhower Theater.

Then there are the local institutions that are also national or even international institutions: for example, the National Institutes of Health, the National Academy of Sciences, half a dozen major universities and a few more, like George Mason, that are coming up fast. All are magnets for bright, talented people. You don't see these bright people at Cabinet meetings, though they prepare some of the position papers. But you can see them at the free lunch-hour concerts that are sometimes given in the new auditorium of the World Bank or at the "Midday Muse" programs in the Folger Shakespeare Library on Capitol Hill: serious, alert audiences of people eager to spend their lunch hours listening to a concert of chamber music or a poetry reading. The special qualities of the Washington population may account for the fact that this city has been building a major league symphony orchestra and opera company while it lacked a major league baseball team. Washington may not be the only American city where more tickets are sold for performing arts events than for sporting events, but it is certainly one of them.

Embassies are still among the brightest cultural spots in the Washington scene. Most embassies will engage in some discreet promotion and offer a gala reception when one of their native stars is appearing in the Washington area. Quite a few embassies have also given substantial aid to Washington museums or performing arts organizations that call attention to the art of their countries -- not necessarily financial aid, but mailing lists, refreshments and an elegant room for a reception, sometimes publicity and sometimes performing space.

In the last few months, a new element has been added on the embassy circuit: La Maison Francaise, a small theater with superb professional equipment, excellent acoustics and handy parking facilities; it is part of the new French Embassy on Reservoir Road NW. The embassy has generously put this theater at the disposal of several Washington organizations -- a unique example of international cooperation aiding purely local artistic enterprises.

WHAT WASHINGTON most lacks in its arts life is an ingredient abundant in such cities as Cleveland and Pittsburgh: smokestacks. The local, word-and-image-oriented industries supply good audience material but not much in the way of capital contributions. In a factory-based economy, there are the families of factory owners -- corporate families sometimes touched with a sense of community responsibility that makes them arts patrons. True, Andrew Mellon gave the city the National Gallery, and Joseph Hirshhorn gave it the museum across the Mall that bears his name. But most of the endowments for the arts in Washington have been relatively modest; the one outstanding beneficiary has been chamber music, which is the least expensive kind to produce. Washington may be the world's leading city for chamber music, period. It is almost certainly the leader in free, high- quality chamber music.

Free museums, too. Under the terms established by Andrew Mellon, the National Gallery cannot charge admission; and those terms are still respected in an age when most major American museums have begun to sell tickets. Not only does this mean that Washingtonians get free admission to blockbuster touring exhibits that require paid admissions in New York, Chicago or Los Angeles; it also means that other notable museums in town such as the Corcoran and the in town, such as the Corcoran and the Phillips Collection, cannot afford to charge. How could any museum selling tickets compete with a free collection like the National Gallery?

The situation is wonderful for museum fans but problematic for museums, which have enormous costs for maintaining their collections, as well as constant temptations to go out and buy new material. A similar situation prevails in chamber music. How can the Manchester String Quartet, made up of National Symphony members, sell tickets when the Juilliard is performing free (25 cents a ticket) at the Library of Congress? Still, the Manchester plays on, as do the Emerson, the Tokyo, the Cleveland and the Guarneri, all of which perform here frequently, superbly and to enthusiastic audiences.

One answer is endowments; other chamber musicians are competing with programs at the Library of Congress, the Phillips Collection and the National Gallery that are supported by endowment funds. The Kennedy Center's Terrace Theater -- a superb location for chamber music, intimate opera and less pretentious forms of theater -- is economically problematic for the same reason that it is artistically outstanding: it seats only about 500 people, and even with a capacity audience, income from ticket sales can't come near meeting expenses for most performances. Marta Istomin, artistic director of the Kennedy Center, has begun raising money for an endowment fund named in honor of the late Abe Fortas. Its purpose is to keep ticket prices low (if not free) for chamber music in the Terrace. In a factory town, the local industrial leaders might be asked to subsidize this kind of activity. In Washington, the leading local industry is Congress; and when it has money to spend on the arts, it prefers to send most of that money back home where the voters are.

Still, somehow, the arts continue to grow. When the Kennedy Center was established in the early 1970s, there was some fear that it might overshadow and snuff out performing arts in other Washington spaces. But the exact opposite has happened: activity has accelerated enormously. In the shadow of the Eisenhower Theater, the National has been renovated and made the scene of one touring Broadway production after another. The Arena has continued to build its audiences for home-grown productions; Ford's Theatre works out imaginative solutions to its problems of small stage and small auditorium; and dinner theaters have blossomed all over the suburbs. In Rockville, the Round House has offered some ingenious productions, while hardy, adventurous young companies downtown (the Studio and the Source) have been pioneering a new small- theater district along 14th Street NW.

Only a few blocks away from the Kennedy Center, where the big visiting companies perform, the Washington Ballet has built a solid constituency at the Lisner Auditorium. And smaller companies offer distinctive productions all over town.

The old Pension Building and the Old Post Office Building tell the story of the arts in Washington as well as anything. Both are monuments of American Victorian architecture at its most spacious and pretentious -- tall buildings for the era when they were built, and still fairly tall for Washington, where buildings do not grow very high. Today these shrines of 19th-century bureaucracy have become shrines of 20th-century culture. The Pension Building is now the National Building Museum, while the Old Post Office has become the Pavilion -- headquarters of the National Endowment for the Arts.

How many opera compa- nies are there in Washington? One of them, the Washington Opera, is internationally known, but there are at least a half-dozen others (and also an ad hoc company assembled each summer at Wolf Trap) -- all giving occasional performances worth the attention of hard-core fans. They perform in a variety of places outside the Kennedy Center: school auditoriums, community centers and several churches. The Prince George's Civic Opera, under music director David Abell, has done some remarkable work for a small company. The Summer Opera Theatre, which performs in the Hartke Auditorium at Catholic University, has established a substantial following on the strength of only two productions each summer -- but productions that meet high professional standards.

One of the most promising developments of the current season was the discovery of a hall for opera on the Mall at 10th and G streets. It is an old auditorium, small and well-preserved, of late 19th- century vintage, and once part of a school complex. It was discovered by Opera Southwest, which had become cramped in its former performing space, a church near the Arena Stage. Now, Opera SW has become Opera DC, and this small but inventive company may be on the brink of a breakthrough.

Even in its former situation, of course, this company must have been the envy of yet smaller ones, such as Capitol Opera and Potomac Valley Opera, which perform with piano rather than orchestra. At the other end of the spectrum, the Washington Opera has been producing work tht meets high international standards and doing well at the box office -- but still runing into financial problems. Its 1985-86 season was reduced from seven productions to five for lack of funds, but the Cafritz Foundation has given it a grant that should allow a full schedule next season.

In the shadow of the National Symphony, there are more than a dozen other or- chestras (some of them excellent) performing regularly in the Washington area. Recent growth has been most spectacular in the suburbs, where both theater and music seem to be booming. It is now possible to have a full, rich musical life without ever coming inside the city limits, with the Community Concerts presented at the University of Maryland and the excellent chamber music series at the Wolf Trap Barns, the National Institutes of Health, and the Jewish Community Center. Excellent orchestras available outside the Beltway include the Fairfax Sym phony and the Montgomery Chamber Orchestra. Besides Wolf Trap, opera worth seeing can be had from the Opera Theatre of Northern Virginia and the Prince George's Civic Opera.

Still, the center of gravity in Washington's cultural life remains downtown. Its most spectacular manifestations are on and around the Mall. In the long, narrow strip of territory that extends from the Folger Shakepeare Library across the Library of Congress and the Capitol, through the National Gallery and the rest of the Smithsonian complex, there is probably a greater concentration of literary, historic and artistic treasures than can be found in any similar-sized space anywhere else in the world.

If Washington has so much to make it one of the world's great cultural capitals, what does it still need? Money would come first on the list of anyone involved in the city's cultural life, but the other needs are strangely varied.

*The current D.C. public library system may be adequate in the size and location of its physical facilities, and in the number and training of its staff, but it could use a substantial upgrading of its collections -- not only books, but also audio-visual materials.

*It's probably too late to do anything about it, but it might be helpful if the Kennedy Center could be relocated -- perhaps to Dupont Circle. The Kennedy Center now is buffered from the rest of the world by the Potomac River on one side, and by the Watergate, several apartment complexes and the State Department on the other side.

And one does not have to be a part of many Kennedy Center audiences to notice that it draws almost all of its customers from one-third of the city's population. The missing two-thirds are black people, some of whom turn out in good numbers for occasional events specially targeted to them, but only a small fraction have been inte- grated into the Kennedy Center's mainstream audience.

*The funds voted by Congress year after year for support of the Washington Civic Opera were arbitrarily taken away several years ago by the city's Commission on the Arts and the Humanities, and the Civic Opera died. The funds should be restored and that institution should be revived. That unique company reached audiences otherwise unreached; it had a level of integration in its audiences and performers that the Kennedy Center might envy; and it gave Washington musicians excellent opportunities to display and develop their talents.

*Somebody should be planning now for the building of a performing arts center and the development of performing arts programs in Anacostia, which will be Washington's most spectacular growth area in the next generation. Such a center and such programs, in the nature of things, should be very different from what now exists downtown and at the Kennedy Center. But they should rise above commercialism and be based on the unique characteristics and resources of the Anacostia neighborhood.