At the height of David Murray’s New York period, which lasted from the mid-1970s until the mid-’90s, the incredibly prodigious, incredibly prolific saxophonist-composer was frequently described as one of those rare musicians in whose playing one could hear the entire history of jazz. From the cry of the blues up through the dance rhythms of New Orleans, the Jazz Age and the swing era, the complexities of bebop and, the avant cry of free music — not to mention such detours as funk and world music — it’s all there in Mr. Murray’s horn.

One of the key elements that made Mr. Murray such a cutting edge player was, perversely, a sense of the jazz tradition above and be yond the call of duty. On Wednes day, for his opening set of a week long stand at Birdland, Mr. Murray began with two numbers that showed a unique take on three of the canonical colossi of the tenor — John Coltrane, Sonny Rollins, and Stan Getz — all of whom were driv en by rhythm.

Coltrane virtually invented the modal waltz, taking an antique time signature associated with the most old-fashioned European music and suddenly making 3/4 time the hottest and hippest way to play the music; in “Waltz Again” (which Mr. Murray recorded with a full string section), Mr. Murray deployed his signature circular breathing technique — in which he virtually never pauses for air — as a complement to the circular nature of the tune, playing fast but amazingly legato.

Introduced by Mr. Murray as a new composition, “Kiama” is a samba (in the same vein as his 1988 “Ming’s Samba”) that seemed to owe equal props to the jazz calypso, as pioneered by Mr. Rollins, and the jazz bossa nova, as perfected by Getz. All manner of Pan-American rhythms were referenced, as pianist Lafayette Gilchrist laid down a dense, polyrhythmic pattern reminiscent of Horace Silver’s “Cape Verdean Blues.” In both numbers, Mr. Murray stuck resolutely to the rhythmic parameters that he established: He played every beat of the measure on both the waltz and the samba without ever cheating or cutting corners.

In almost every number, Mr. Murray started “inside,” sticking to the bop tradition of using the notes from the chord changes, but gradually got further and further out, spicing his solos with upperregister shrieks from the free jazz vocabulary. And yet even when getting “free” tonally, he still stuck to the rhythmic trajectory he laid down for himself. This was especially effective on the samba, which, when played amid these “outside” blasts, had the effect of a conga line of dancers strutting around the edge of a volcano.

Although he has celebrated past jazz masters from Sidney Bechet to Albert Ayler to Paul Gonsalves, Mr. Murray has rarely dwelt on his own past. However, he calls his current group, which also features Jaribu Shahid on bass and Andrew Cyrille on drums, “The Black Saint Quartet,” in reference to the Milan-based label for which he recorded between 1978 and 1993. Just like music can be jazz and Latin at the same time, the jazz of the Black Saint Quartet is at once bebop and avant-garde. It fills all the requirements of both, playing off the chord changes in a fast, boppish tempo, but also mixing in outside elements for additional flavor, as well as free notes and elements of discord not merely for shock value, but as legitimates parts of the music. An atmospheric piece called “Vanished” spotlighted Mr. Shahid, whose arco bass solo was complemented by dark low notes from Mr. Gilchrist and the leader on bass clarinet. Even more exotic was an untitled piece which, Mr. Murray said, recently came to him in a dream, but which suggested the brooding jungle-isms of Duke Ellington’s later African-styled pieces; Mr. Cyrille used mallets to give the piece a more mystical and classical feeling.

The opening set also included a beautifully moving ballad, “In the Spirit,” by the late pianist Don Pullen, with whom Mr. Murray recorded the tune on his 1991 “Shakill’s Warrior,” as well as a funk number, which he said helped him celebrate Thanksgiving in Paris. Only 52, Mr. Murray has been in the vanguard of jazz for more than 30 years, and there’s still no one better at utilizing an inside approach to outside music — or is it the other way around?

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Mr. Murray’s tune “Vanished,” he explained, comes from the soundtrack of a forthcoming documentary on the subject of racism in America. In “Black and White Baby,” Bobby Short’s memoir of his childhood experiences in vaudeville, the late pianist-singer talks about the unbearable bigotry he encountered in all walks of life in the 1930s. Rather than letting it defeat him, Short vowed that he would use music to bring people together.

These days, it’s taken for granted that black and white musicians can work together, but the fields of jazz and cabaret — both of which are dependent on interrelated foundational elements — don’t share a stage nearly as often as they should. Thus it was fitting that, in Short’s memory, two of the industry leaders in these respective fields — the pianist-singer Michael Feinstein and the trumepter Wynton Marsalis — should collaborate in his honor. At the climax of Mr. Feinstein’s show on Tuesday (the start of the second week of his two-week run of “Celebrating Bobby Short” at Feinstein’s at the Regency), the two paid homage to one of Short’s friends and idols, Duke Ellington.

The up-tempo “It Don’t Mean a Thing (If It Ain’t Got That Swing)” showcased Mr. Marsalis’s ability to leap into a tricky arrangement without any forewarning. But it was the ballad “I Got It Bad (And That Ain’t Good)” that showed how good jazz and cabaret can be when they get together. Mr. Feinstein was simple and spare, never deviating from the essence of the melody or the core of the lyric; Mr. Marsalis, surprisingly, played a lot (both his obligato and solo were extremely busy), yet he never exceeded his boundaries or stepped on his collaborator’s toes. These guys should get together more often, maybe even on a record; it’s rather like watching Steve Jobs play Pac-Man with Bill Gates.

wfriedwald@nysun.com