You can’t tell a story until it’s finished. So now that Jazz at Lincoln Center is telling the story of jazz, encapsulating it in what amounts to a museum of the music, does that mean jazz is finished?

Believe it or not, the opening of Jazz at Lincoln Center’s Rose Hall – the first performance complex in the world completely dedicated to jazz – would have been seen as a step in the wrong direction by many who supported the music during the past century. When Paul Whiteman staged the first notable jazz concert of modern times, in 1924 at New York’s Aeolian Hall, he said he was going to “make a lady out of jazz” – an unfortunate phrase that suggested that jazz was a low-class guttersnipe before he arrived. The implication that jazz needed to be made “respectable” is made people wary of moving the music from club to concert hall.

As Jazz at Lincoln Center artistic director Wynton Marsalis put it in an interview last year, “I think people are too quick to confuse formality – or dignity, let’s say – with institutionalization,” he said. “There’s a Hollywood myth that says jazz has to be played downstairs in a club, where there’s a low ceiling and a cloud of smoke. Plenty of great jazz has been created in settings like that – but it’s also a fact that Benny Goodman and Duke Ellington played regularly in Carnegie Hall.”

That ain’t the half of it. Even before jazz proper had been heard anywhere outside of the Deep South, major African-American roots musicians were making a plea for wider attention. Ragtime master Scott Joplin wrote several operas. James Reese Europe, a ragtime-era conductor whose dance band in many ways anticipated the swing era, led several highly successful concerts of African-American popular music at Carnegie Hall as early as 1912.

Whiteman – who also appeared at Carnegie and other leading concert halls across the country – set an important precedent. In commissioning and introducing George Gershwin’s “A Rhapsody in Blue,” he established that the chief reason to bring jazz into the bastions of symphonic music would be not to play jazz as it was heard in dance halls and theaters but to introduce extended compositions that blended jazz elements with classical structure.

One notable exception was Benny Goodman’s groundbreaking appearance at Carnegie Hall in January 1938 – the most famous jazz concert of all time. The Goodman band played essentially what they played in ballrooms; in essence, Goodman was making a considered statement that jazz, or swing, as they played it every night, was as much of an art music as that normally heard in Carnegie. (The one concession to the concert format was a segment called “The History of Jazz,” which gave samples of key points in the music from 1917 to 1938.)

That same year, Duke Ellington’s music was heard at Carnegie, as commissioned and performed by Paul Whiteman. Ellington and his Orchestra made their own Carnegie debut in January 1943, with the most ambitious jazz work yet attempted. “Black, Brown and Beige:

A Tonal Parallel to the American Negro” was three times as long as the 15-minute “Rhapsody In Blue,” but not nearly as well-received. Ellington continued to appear annually at Carnegie until 1948 and frequently after that, generally using these appearances to introduce extended works. Woody Herman presented both “Summer Sequence” and Igor Stravinsky’s “Ebony Concerto” at Carnegie in 1946. Dizzy Gillespie played in a small group (at least once with Charlie Parker) and also with his orchestra on several occasions.

In 1944, a radical new kind of jazz concert had been introduced in Los Angeles. Jam sessions were after-hours affairs before Norman Granz began presenting them in a formal concert setting. Where Ellington and Whiteman championed extended composition, Granz featured the informal ensembles and encouraged players to take long solos and “battle” one another. That he called his concert series “Jazz at the Philharmonic” shows just how novel the idea of jazz musicians appearing in concert halls still seemed.

When George Wein held the first Newport Jazz Festival in July 1954, it was instantly a commercial and cultural event. He, too, combined star soloists and full-scale working bands, and folded in everything from Gospel to pop under the rubric of jazz. Performers like Miles Davis played festivals like Newport while continuing to make appearances in concert halls (Davis’s famous 1961 Carnegie appearance included his working quintet leading a full-scale big band) throughout the next two decades.

Having perfected the jazz festival, Mr. Wein went on to develop what has since become known as the Jazz Repertory Movement. In 1974, the New York Jazz Repertory Company became the first jazz band that operated like a symphony orchestra: It played the compositions of classic composers like Ellington and Jelly Roll Morton and honored legendary soloists like Louis Armstrong. Though it only existed for a few years, the NYJRO provided the blueprint for the American Jazz Orchestra (whose artistic directors were Gary Giddins and John Lewis of the Modern Jazz Quartet), launched in 1986; the Smithsonian Repertory Orchestra; and, more recently, the Smithsonian Jazz Masterworks Orchestra.

Not to mention Jazz at Lincoln Center. Ever since Mr. Marsalis’s first jazz repertory style-concert at Lincoln Center (also in 1986), it’s been clear that his goal is to bring the music a level of recognition it hasn’t had before. That meant forming an orchestra to perform the repertory. That meant building halls designed for the unique acoustical needs of its musicians. That means placing the music before as large a live audience as possible and (as in the case of tonight’s opening night concert) a national television audience.

Jazz doesn’t need to be made respectable. But it does need to be heard.