Two jazz giants — one recently deceased, and one very much still active — were honored in concerts over the weekend. The late percussionist Ray Barretto was saluted as part of the Highlights in Jazz series on Thursday at the Tribeca Arts Center, and the pianist-composer Randy Weston was the subject of a thoughtful re-imagining Saturday at the Abrons Art Center. Both shows took place on the extreme west and east sides of lower Manhattan, which is perhaps appropriate for two musicians whose work has been distinguished by how it helps to define the relationship of North American jazz to the music of other cultures. Barretto famously denied that there was such a thing as “Latin jazz,” even as he helped to invent the genre; contrastingly, Mr. Weston has insisted that all jazz is fundamentally African.

Barretto (1929–2006) spent his life proclaiming that he was a jazz musician who just happened to work with Latin instruments and rhythms — not that any qualification was necessary. Fittingly, the tribute concert to him on Thursday consisted of fellow Hispanic jazzmen (with one exception) playing classic American jazz standards. Also appropriately, the Highlights concert utilized a Jazz at the Philharmonic-style jam session, in which a rotating lineup of frontline horn players interacted with one another and a core rhythm section consisting of the pianist Arturo O’Farrill, the bassist Junior Terran, and Barretto’s longtime trap drummer, Vincent Cherico. The principal soloists were the trumpeter Ray Vega and flutist Dave Valentin in the first half, and the trombonist Steve Turre and alto saxophonist Sonny Fortune in the second. There also were two conguerros, Bobby Sanabria and Steve Kroon, who, in the spirit of Barretto, took many of the solo honors.

The proceedings were loose, spontaneous, and obviously unrehearsed. But even so, the players called quite a few slightly offbeat tunes: John Coltrane’s blues “Equinox” opened, followed by “Dewey Square,” a lesser-heard Charlie Parker tune played by Mr. Vega as a variation on Bird’s more famous “Yardbird Suite.” Mr. O’Farrill chose Mal Waldron’s lovely ballad “Soul Eyes” as his feature, and the first half of the show concluded with Oliver Nelson’s “Stolen Moments.” Mr. Valentin fit into the proceedings particularly well: In his own shows, his enormous tone on the flute, replete with all manner of sound effects (some of them outrageously cartoonish) can be overwhelming, but he is one of the few flutists who can hold his own sonically against a trumpeter or saxophonist. Regrettably, Mr. Valentin only played the first third of the show before an asthma attack forced him to depart.

The second half spotlighted Messrs. Turre and Fortune, both of whom played well above their heads on “Footprints”; when Mr. Fortune got all mellow-roonie on “Insensatez,” Mr. Turre went romantic with a wah-wah mute and a keenly vocalized sound on the ballad “Body and Soul.” Mr. Turre followed with an “El Manisero”- like parade rumba featuring trombone and his mystical seashells — an amazing thing to experience if you’ve never seen it before (and even if you have, his performances on the shells, an organic instrument if ever there was one, have grown increasingly musical over the last 20 years).

The evening concluded on a terpsichorean note with “Passion Dance,” which seemed like the one pre-rehearsed tune of the evening. This was an adroit orchestration for Messrs. Vega, Turre, and Fortune that re-constructed McCoy Tyner’s 1967 line as a fiery, Afro-Cuban conga and left us fortified to confront the cold outside.

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I’m consistently surprised when the trumpeter Dave Douglas, who has made a point of playing every kind of music from beyond-the-fringe to world music, returns to the jazz tradition, even though he has already masterminded excellent programs of the music of Thelonious Monk and Mary Lou Williams. Randy Weston was and is American jazz’s most outstanding Afro-Centrist, so it seemed inspired for Mr. Douglas, at the Abrons Arts Center on Saturday night, to present his music with a band that included musicians of African, Spanish, and Irish descent.

Where Mr. Weston’s music is essentially driven by rhythm, Mr. Douglas’s reinterpretation dogmatically avoided the use of a conventional rhythm section: The keyboardist Geoff Keezer, playing two electronic keyboards (one that sounded more like an organ, the other like an electric piano, and sitting on two chairs stacked on each other at the same time), supplied few background chords behind soloists, but instead filled in with splotches of color the way a percussionist would. Marcus Rojas played his tuba like a brass instrument rather than a bass fiddle, as one might have expected. The only player supplying rhythm all the time was drummer Nasheet Waits, who seemed to be doing double and triple work.

Still, all the so-called frontliners were playing with more exuberant rhythm than usual; the second tune on Saturday night, “Mystery of Love,” opened with all six players clapping out a Pan-African polyrhythm. Mr. Weston’s own quintets have tended to emphasize soft-sounding trombone and sax, suggesting the mellow qualities of African choirs such as Ladysmith Black Mambazo. Contrastingly, Mr. Douglas’s group was all bright, biting brass, particularly the leader’s dynamic trumpet tone. Saxophonist Donny McCaslin mostly played the sharper-toned soprano instead of tenor.

Mr. Douglas astutely rearranged Mr. Weston’s compositions so that the melodies remained prominent, each featuring a different member of the group and none seeming like an excuse for a string of solos. Mr. Douglas’s treatments emphasized the melodic and rhythmic variety of Mr. Weston’s tunes; even though they’re all at least vaguely African and come in a variety of time signatures, there are love songs (“Mystery”), hymns (“Prayer Blues”), blues (“Blues to Africa”), and party pieces, such as the celebratory “African Cookbook.”

Mr. Douglas saved Mr. Weston’s best-known song, “Hi Fly” (one of several popularized by Lambert, Hendricks, and Ross), for an encore, which the group performed in the most basically boppish rendition of the night. At this point, the trumpeter announced that the composer himself (who turns 82 in April) was in the house, not that it was possible to miss him. I was thinking throughout the show that Mr. McCaslin has got to be one of the taller tenors around — he’s at least 6-foot-5 — but even he is dwarfed by Mr. Weston, who towers closer to 7 feet and at times seems to come out of an African folktale about walking trees. Thank goodness he is a friendly giant.

wfriedwald@nysun.com