The jazz of pianist and composer Abdullah Ibrahim is unlike any the world has ever heard. Throughout his 70-minute performance at Jazz Standard Tuesday night, Mr. Ibrahim seemed to be telling us to rid ourselves of the archaic, Western notion that music — or any artistic statement that exists in time — has to have a beginning, a middle, and an end, and in that order. Mr. Ibrahim played a single, uninterrupted piece with no obvious starting point or finishing line, just a lot of middle — and there were middles within middles and interior monologues galore.

Mr. Ibrahim, a native of Cape Town, South Africa, who was formerly known as Dollar Brand and who turns 73 this fall, has long been the most celebrated of all African jazz musicians. This week at Jazz Standard, he played solo piano Tuesday through last night, and will play with his trio (with Beldon Bullock on bass and George Gray on drums) tonight through Sunday.

There are composers in the classical and jazz avant-garde who work without any traditional linear form; Karlheinz Stockhausen and Cecil Taylor come to mind. Mr. Ibrahim did go through a phase when his music was more chaotic and discordant and had more of an element of free jazz to it. But the remarkable thing about what he’s currently playing is that it is so traditionally melodic, harmonic, highly rhythmic, and completely accessible. The only thing that’s avant-garde about it is his rather unique concept of form.

The traditional narrative shape of music preconditions the listener to expect certain things — in a sense, as soon as a piece begins, you’re waiting for it to get to the next step, to see how it transitions through these agreed upon points, and you try to follow the “plot” of the music as if it were a movie. Mr. Ibrahim’s set-length piece was very much like a long, scenic journey, in which you’re not sure where you’re heading or where you began. You’re not necessarily sure of where you happen to be right at the moment, but wherever it is, it’s someplace mighty attractive. You may not be in any particular hurry, but there’s an unmistakable feeling of going somewhere. And you’re thinking less about making your way to the end than simply enjoying where you happen to be at the present time. (It’s all rather Zen.)

Another convention of modern music says that it should start quietly and get noisier and more exciting as it progresses. Mr. Ibrahim’s piece, by contrast, starts quiet and gets ever quieter at different points, so quiet that you can hear a knuckle crack. At Jazz Standard, there were a few passages in which everything was so motionless that even my glass of club soda had more activity.

At these moments, the music is truly minimalist — there’s so little going on that when anything actually does happen, you can’t possibly miss it. So much contemporary jazz is about young players showcasing their chops and their ability to reel off barrages of notes over complex chord patterns at lightning speed, jazz’s equivalent of a car chase in a summer movie that would be advertised as a “thrill-packed roller coaster ride.” Mr. Ibrahim’s music, on the other hand, is pure tranquility — eerily hypnotic and completely compelling, jazz that’s based on ideas rather than technique.

At various points, the tempo picks up and gathers momentum, like a rock rolling down a hill. For the most part, the melodies and harmonies are completely transparent, but there are other sequences in which the tune becomes more intricate and the chords more complex. In certain spots, Mr. Ibrahim seems to be playing a long vamp; at others, it’s a set of rhythmically charged riffs that play off one another.

Here, there’s a sequence of Ellington-like romantic chromaticisms; there, it’s a bunch of Monkish dissonances. Eventually, we travel past something that sounds vaguely like the blues, and then in the distance we sight an object that could be a waltz; at certain times you could swear that you’re listening to a love song or a ballad, and at many times, it all becomes more spiritual, like a hymn. There’s not much in this particular piece that makes me jump to my feet and dance, but there is a lot to make me get on my knees and pray.

On Tuesday, Mr. Ibrahim continued to play with his audience’s expectations to the very end: At one point, an hour or so in, he started to hit the keyboard with greater force and fury, producing dark rumblings from the bass end. He sounded like he was building toward a conventional conclusion, but no: The piece went on for another few minutes, and when it ended, it just stopped, with no forewarning or fanfare. Abdullah Ibrahim rose from the piano bench and placed his hands together and bowed slightly, in deference to the crowd.

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Mr. Ibrahim and his wife, the singer Sathima Bea Benjamin, left Cape Town in 1960 to escape apartheid. While Mr. Ibrahim returned to his native land at the end of that oppressive regime, Ms. Benjamin continues to reside in New York at the Chelsea Hotel, her home for 30 years. Last fall, not long after her 70th birthday, Ms. Benjamin released “SongSpirit” (Ekapa) a CD anthology of her favorite tracks from her eight previous albums.

Ms. Benjamin sings mostly American jazz standards, including a lot of Ellington, with a distinctively African timbre — a cool, even hypnotic tone and beguiling pitch that often seems just slightly under the note. She also uses a slight catch in her throat and a style of bending notes that make show tunes seem like an incantation.

For a singer who has been indirectly allied to a movement of liberation, Ms. Benjamin also sings quite a few standards that are tied to old “imperialist” European operetta, such as selections by Franz Lehár and Noël Coward. The highlight of the current collection is a previously unissued reading of “It Never Entered My Mind,” done as a piano-voice duo between husband and wife, in which they show that even Rodgers & Hart can benefit from their cool, Cape Town sonorities. Whatever Ms. Benjamin sings, she is a vocalist of daring and imagination, and should be heard.

wfriedwald@nysun.com