It’s not surprising that Billie Holiday recorded Fats Waller’s most famous song, “Ain’t Misbehavin’,” in 1956. But given what we know of her life, it is surprising that she sang it with absolute purity, as if she weren’t remotely tempted to misbehave. Backed by an allstar group that included trumpeter Charlie Shavers and tenor saxophonist Budd Johnson, Holiday’s treatment is all the more compelling because she sings with an obvious awareness of the joys of misbehaving – of staying out late and partying – only to reject such pleasures in favor of her true love’s kisses.

That Holiday, whose lifetime of self-destructive misbehavior has been exhaustively documented, could sing so convincingly of sticking to the straight and narrow is a testament to her gifts as a musician, and as an actress.We often think of Holiday as a tormented woman who sang only from her own life experiences, who crooned about errant lovers who cheated on her and beat her only because she knew them personally. But Holiday could do much more: She was not a musical primitive who merely translated her own suffering into song. Rather, like Frank Sinatra or Nat King Cole, Holiday was a thorough professional who could interpret any lyric in the Great American Songbook – from the “Dese ‘n’ Dose” Ebonics of “Porgy and Bess” to the salon formality of Cole Porter or Oscar Hammerstein – and make it seem like an extension of herself.

That point can’t be emphasized strongly enough: In recent years, popular perceptions of Holiday as an icon of suffering have come dangerously close to overshadowing her musical legacy.

A case in point is Julia Blackburn’s “With Billie,” a compilation of interviews published last year that contains virtually nothing about Holiday’s music but plenty of lurid sensationalism detailing her sexual encounters, particularly with members of her own gender, and her experiences as a drug addict and prostitute. (That was the story Holiday, to a degree, told in her own book, but it has to be remembered that her memoir was a kind of performance art: She was interested in selling books and making money.) In place of scholarship or analysis, “With Billie” offers cheap thrills – precisely the opposite of what Holiday’s artistry was all about.

A recently released box set, “The Complete Verve Studio Master Takes”(Verve 8030),produced by Bryan Koniarz, should go a long way toward turning the spotlight back on Holiday’s art.The box is a consolidation of an earlier package, the 10-CD “Complete Billie Holiday on Verve 1945-1959” (517658), a 1993 release that also includes four CDs of live concerts, studio chatter, alternate takes, and informally taped rehearsals.Although the older box is recommended for completists, the current set is the one I will play more often.

The 100 tracks here contain their share of tragic tales and blues – “Stormy Weather,” “My Man,” “Good Morning, Heartache” – but no less a share of happy ones like “Nice Work If You Can Get It,” “Moonglow,” and “Easy To Love.” Holiday is never better than when she’s mixing the two extremes, as on “Travelin’ Light” or even “Lover Man,” songs about sad situations that she renders with more than a hint of optimism, offering every reason to hope that Lover Man will return some lucky night. Far from being a prophet of doom who wiped every smile away, Holiday could sing of happiness just as movingly as she could of despair.

This is especially true in the period covered by the box, 1952-59. In the 1940s, Holiday sang mostly contemporary songs in the pop idiom, including some memorable sessions with orchestrator Gordon Jenkins. In the ’50s, by contrast, she returned to the copasetic smallgroup swing format of her glory years in the late ’30s, working with sympathetic accompanists (generally either Shavers or Sweets Edison on tenor, and a roster of saxophonists that included Ben Webster, Flip Phillips, and Benny Carter).Meanwhile, her producer, Norman Granz, had her concentrate on the Great American Songbook like never before.

On these songs, Holiday wasn’t just singing about her own life – she was singing about the human experience. In “Strange Fruit,” Holiday, who was born and raised in Baltimore, describes the “gallant South” as a living hell for black people. In “Stars Fell on Alabama,” she no less convincingly depicts Dixie as a romantic “fairyland.” And in “When It’s Sleepy Time Down South,” her nod to Louis Armstrong, she perfectly conveys the love-hate relationship that nearly all of us have with the places we come from. Only a great actress can so thoroughly sell you on such completely inapposite points of view on the same topic from one song to the next.

“Sleepy Time” comes from Holiday’s final studio session, an album posthumously released simply as “Billie Holiday” on the MGM label. It’s not produced by Granz and, technically speaking, does not belong in a Verve box, but the package is much richer for its inclusion.This 1959 album is a follow-up to her 1958 “Lady in Satin,” and like it, utilizes a small string orchestra arranged and conducted by Ray Ellis. Holiday’s voice has deteriorated even since the previous year, but she puts everything she’s got into these songs, and the struggle is a glorious one.

The repertoire itself is remarkable in that it consists largely of songs associated with Holiday’s colleagues and heirs – Armstrong’s “Sleepy Time,” Sinatra’s “All the Way” and “I’ll Never Smile Again,” Bing Crosby’s “Just One More Chance,” Ethel Waters’s “There’ll Be Some Changes Made,” and even a recent Johnny Mathis hit,”It’s Not for Me To Say.” In singing their songs, Holiday seems to acknowledge that she’s handing them the keys to the kingdom as she prepares to leave the world behind.Yet no one could take her place.

wfriedwald@nysun.com