The most famous photograph of Fletcher Henderson, jazz’s first great big-band leader, shows the dapper, coffee-colored man looking as cryptic as the Sphinx. His eyes are half-closed: You can’t tell if he’s supposed to be looking at you or around you. And his lips are curled into something too ambiguous to be called a smile. The difficulty the viewer has in defining that expression parallels the problems jazz fans and historians have had agreeing on Henderson’s legacy.

Henderson’s one-time producer and unofficial rabbi, John Hammond, credited Henderson with almost single-handedly inventing big-band swing, only to see his ideas usurped for greater profits by white bandleaders (in particular Benny Goodman, Hammond’s own brother-in-law). Yet lately an opposing point of view has become more prevalent, as expressed by historian Dan Morgenstern, who refers to Henderson as “perhaps the most overrated figure in the history of jazz.”

The truth, as author Jeffrey Magee shows us in his excellent new book, “The Uncrowned King of Swing: Fletcher Henderson and Big Band Jazz” (Oxford University Press, 310 pages, $30), is somewhere in between.

As befits a man whose band would nurture the development of the jazz soloist, Henderson (1897-1952) had an essentially passive nature – one at odds with the stomping, passionate music that his orchestras created. From the beginning he went the way the wind blew him: The Georgia-born son of an early giant of African-American education, he came to New York to study chemistry at Columbia but found himself drawn instead to popular music.

In his early days, Henderson assembled an astonishing collective of great musicians, including virtually every black star of the 1920s and 1930s – Louis Armstrong (for 13 glorious months), Coleman Hawkins (for a whole glorious decade), and Benny Carter, who was equally skilled as a instrumentalist and an arranger. Yet he had little involvement in what they played, concentrating instead on keeping the band together and taking care of business.

For this reason Mr. Morgenstern and others have claimed that Henderson himself doesn’t deserve much credit. But that’s not the whole story. By the late 1930s, Henderson was arranging much of the band’s music himself (the irony is that he had set aside the financial and other responsibilities, and nobody was minding the store).

Henderson did more than any other to anticipate the swing era. The pioneering orchestrations of his chief arrangers, especially Don Redman and later Benny Carter, started with source materials from the earliest jazz – the Original Dixieland Jazz Band, King Oliver, Jelly Roll Morton, Bix Beiderbecke, and Frank Trumbauer (note that there are as many white sources as black) – and transformed them into modern, swinging large-ensemble jazz. Essentially, they took the excitement of the best small-group jazz and translated it to a larger canvas.

The primary way they did this was through contrast, making a melody more exciting by dividing it up between different instruments, different sections of the band, and different soloists. Jelly Roll Morton’s composition “King Porter Stomp” may be the archetypal Henderson production. It opens with a blasting trumpet that signaled a clarion call to several generations of dancers, includes several soloists alternating with the full band, and climaxes in a wild battle with the saxes and the trumpets shouting back and forth at each other.

Henderson was forced to break up his most celebrated band in 1934, which – as it turned out – was just in time to help Benny Goodman “build the kingdom of swing,” as Mr. Magee puts it. Henderson wrote more than 200 charts for the clarinetist, in a sense making him the central musical director for the entire swing era. Mavericks like Duke Ellington, Count Basie, and Woody Herman would later take the jazz big band in all kinds of new directions, but all subsequent orchestral jazz would be measured against what Henderson and Goodman created in the mid- to late 1930s.

In the wake of Goodman’s triumph, both Henderson and his very talented younger brother, Horace, were able to start new bands. He continued to lead bands off and on, and also to play for one of his original employers, diva Ethel Waters, until he suffered a debilitating stroke in 1950. But though he went on to discover more stars (Roy Eldridge, Chu Berry), none of Henderson’s later bands had any lasting success.

Mr. Magee has given us a marvelous read, a worthy companion to the late Walter C. Allen’s “Hendersonia: The Music of Fletcher Henderson And His Musicians.” That 1973 “biodiscography” compiled every imaginable fact about Henderson, his recordings, and his stellar sidemen. Jazz fans have waited 30 years for a trained musicologist such as Mr. Magee to use Allen’s data to create a book that evaluates Henderson’s strengths and weaknesses and attempts to place him in the history of American music.

“The Uncrowned King of Swing,” which draws extensively on Mr. Magee’s own original research, will be most enjoyed by those with some technical knowledge of music. Mr. Magee is generous with musical samples and transcriptions, and the book is not laid out like a conventional biography. Rather, it is a series of articles with subheadlines, somewhat like a textbook. Mr. Magee analyzes and discusses dozens of classic Henderson recordings, placing them in both a historical and musical context, and his conclusions are free of any agenda. Though he doesn’t neglect the contributions of the soloists, Mr. Magee’s main emphasis is on the orchestrations, most by Redman, Carter, and Henderson himself.

Perhaps his most surprising finding is that one of the hoariest cliches about the early days of the band simply isn’t true. The first thing you generally read about Henderson is how his band sounded like some kind of boring parlor salon orchestra (or, even worse, a white band) until Armstrong stepped up to the plate and transformed it into a great jazz ensemble. Yet even a cursory listen to Henderson’s many 1923-4 recordings shows the band getting hotter and hotter, almost as if Henderson and Redman were preparing for Armstrong’s arrival.

Between October 1924 and November 1925, Armstrong would invent and perfect the art of big-band soloing, but he couldn’t have done so if Redman hadn’t concurrently perfected the art of writing hot, swinging orchestrations for him to shine in and if Henderson hadn’t given him the stage to do so.

THE ESSENTIAL FLETCHER HENDERSON RECORDINGS

A STUDY IN FRUSTRATION (Sony Legacy 57596) Although the sound on this three-CD set is somewhat worse than on some of the English productions (and the title may now seem inaccurate), this is as comprehensive a retrospective as we are likely to get of Henderson’s entire band leading career. It samples all of his groups, from the earliest acoustic era recordings up through the breakthrough with Louis Armstrong, the great bands of the late 1920s, the Connie’s Inn Orchestra of the early 1930s, and even the later ensembles he led in the wake of his success with Goodman. The notes and photos provided by historian Frank Driggs are an added bonus.

BENNY GOODMAN PLAYS FLETCHER HENDERSON VOLUMES ONE & TWO (Hep 1058 & 1059) An appendix in Mr. Magee’s book lists more 200 charts by Henderson for Goodman. Obviously, they can’t all fit on two CDs, but this pairing from the Scottish Hep label features 50 of the absolute greatest in superb sound quality. Perhaps now that Columbia and Victor are part of the same corporation, the new mega-label will release a complete, multi-disc Goodman-Henderson package.

THE CLASSICS SERIES The 16 volumes of Hendersonia released by this Belgian label have the same pros and cons as the rest of the Classics releases. The remastering is inconsistent, but the series is complete, except for alternate takes (which are on two volumes from an Austrian label called Neatwork). It’s also highly unlikely that anyone else will ever attempt such a thorough accounting of Henderson’s entire output, especially the pre-Armstrong acoustic period.

LOUIS WITH FLETCHER HENDERSON 1924-25 (Forte F3 8001-3) Another import, this three-disc project contains all the recordings Armstrong and Henderson made in each other’s company, including alternate takes and accompaniments to blues singers. The material is in the best possible sound, as prepared by the late audio restoration savant John R.T. Davies. Would that Henderson’s entire legacy – and for that matter, Armstrong’s – were this well preserved.

FLETCHER HENDERSON AND HIS ORCHESTRA ‘LIVE’ AT THE GRAND TERRACE, CHICAGO 1938 (Jazz Unlimited JUCD 2053)

Live air checks by Henderson’s own bands were exceedingly rare until this Danish firm unearthed these two exciting half-hour broadcasts. The only question remains why they put “live” in quotes, as these performances are most assuredly live and swinging and much recommended.