Jazz improvisation is often misconstrued as merely a blank-slate scrawl, its basic substance involving something clever made up on the fly. The reality has more to do with shorthand strategies, half-tested hypotheses, a resourcefulness in the realm of what-if. Only in rare instances does a group manage coherence without prescriptions, speaking freely and spontaneously but in whole paragraphs, with proper syntax and a persuasive argument.

You’d have a good chance of catching one such instance this week at the Jazz Standard, courtesy of the alto saxophonist Lee Konitz and several inventive but exacting sidemen: the pianist Danilo Pérez, the bassist Rufus Reid and the drummer Matt Wilson. Their second set on Tuesday night was an astonishment of collective attention and unmannered epiphany.

Mr. Konitz has worked in this mode often over the course of his career. At 80, he’s a certifiable jazz legend, with peerless high-modernist credentials and an integrity equal to his talent. His aversion to cliché can make him seem stubborn, sometimes yielding music that’s more about the searching than the finding. But that wasn’t the case here.

The chemistry of the group had a lot to do with it. While Mr. Konitz has some history with Mr. Reid and Mr. Wilson, he was forging a new alliance with Mr. Pérez, who opened the set with a scrap of suggestion. Mr. Reid responded with a pendulumlike line, and Mr. Wilson began tapping his toms and cymbals in a halting pattern.

Mr. Konitz found his way into the conversation with borrowed melody, interpolating phrases from a songbook standard, an old spiritual and a Charlie Parker tune. Where some jazz musicians, in the spirit of Parker, use quotations to signal wit, Mr. Konitz seemed to enlist them as a means of orienting himself in uncertain territory. When he invoked Miles Davis’s “Solar,” it stuck: the band lunged into the tune. By the time it was over, nearly half an hour had slipped by.

The set went on to include two lustrous ballads. “You Stepped Out of a Dream” was artfully tentative, with Mr. Konitz leading what felt like a search party under cover of darkness. “If You Could See Me Now” became a feature for Mr. Reid and Mr. Pérez, who both produced patient, knowing extractions of the melody.

Then came “Cherokee,” a tune once held as the ultimate test of mettle for jazz improvisers. Mr. Konitz sought to make it so again, imbuing his solo with harmonically restless phrases, and finishing with a step up into a new key. Without missing a beat, Mr. Pérez and Mr. Reid shifted with him, resetting the tonality of the song.

Mr. Pérez took this premise further in his solo, nudging each new chorus up a smidge, until the song had made its way through all 12 keys. At that point Mr. Konitz rejoined the action, as if stepping onto a moving Ferris wheel.

Performances continue through Sunday at the Jazz Standard, 116 East 27th Street, Manhattan; (212) 576-2232, jazzstandard.net.