A few years ago, an electronics company claimed in its ads that its products were “slightly ahead of our time.” That’s a perfect description of the way the music now known as Miles Davis’s “The Birth of the Cool” was originally received. It wasn’t until about five or six years after the tracks were recorded that the music began to have a profound influence on the development of jazz from the mid- and late-1950s onward.

Now a new recording of that music, expanded into suite form by the renowned composer and conductor Gunther Schuller, who played on the original 1949–50 sessions, as well as a new album by the saxophonist Joe Lovano, shows that perhaps the time for this remarkable body of music has only arrived today.

Five years ago, Mr. Lovano, who is playing two shows a night through Saturday at Birdland with his group, Shades Of Jazz, was asked by the Monterey Jazz Festival to assemble a program for his nine-piece band (which itself was originally inspired by the Birth of the Cool band) on the occasion of the 75th anniversary of Davis’s birth. Mr. Lovano turned to Mr. Schuller, and together they began work on the piece now called “The Birth of the Cool Suite,” which forms the center of Mr. Lovano’s new album, “Streams Of Expression” (Blue Note 41092), and which has just been released in Davis’s 80th birthday year.

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The original band began to come together in 1947, thanks to the 21-year-old Davis, who was working nightly at the time with Charlie Parker, and Gil Evans, in whose basement apartment they convened. At the time, Evans was a veteran arranger best known for his many years with the innovative orchestra led by the pianist Claude Thornhill. Together, Evans and Davis marshaled an impressive body of young musicians and composers, all of whom had all been part of the bebop movement and/or worked with Thornhill — including Gerry Mulligan, John Lewis, Max Roach, J.J. Johnson, Lee Konitz, Kai Winding, Al Haig, and the academically trained Mr. Schuller (who also conducted some of the selections) on French horn.

Throughout the 1940s, Evans’s arrangements for Thornhill’s band had brought a unique classical sensibility to swing band music; with this new group, Evans wanted to push that idea a step further by incorporating ideas related to the concurrent musical revolution of the decade. The band, announced as the Miles Davis Organization on one of the few radio broadcasts it made, became the second wave of modern jazz — the counter-revolution to bebop.

The music breathed a tranquility, a contemplative character then unknown in jazz, as if a whole ensemble of various horns were all taking after Lester Young in his most laconic moments. This was composer-driven music, distinguished by rich harmonies and unusual instrumental combinations (including the first appearance of the tuba in post-Dixieland jazz), and it proposed a middle ground between the concert hall and the jam session.

The band never quite made it out of the basement, playing only one major paying gig, at the Royal Roost in September 1948, where it shared a bill with Count Basie and played to an indifferent crowd. Thankfully, the astute arrangerproducer Pete Rugolo was supervising a series of modern jazz dates for Capitol Records at the time, and he was able to document 12 pieces during three sessions with the group in 1949 and 1950 — after the players had stopped working together. But a few years later, just as Davis’s career was getting ready to kick into high gear, the music was released on LP, eventually as “The Birth of Cool,” and almost instantly it became some of the best-known and widely influential recordings in all of jazz.

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More than half a century later, Mr. Schuller’s long-form adaptation of the “Birth of the Cool” material includes just three selections, of which the first and most important is Evans’s thoughtful revision of “Moon Dreams.” The song was originally written for the Glenn Miller Orchestra by Miller’s pianist, Chummy MacGregor, and illustrates the transition from the big band swing of the 1940s to the orchestrated modern jazz of later generations. Mr. Lovano’s presence here underscores that the one thing the original “Birth”band missed was a great tenor player, a void he ably fills.

The first movement of the new suite begins with a prelude, using prominent flute (played by Michael Parloff), an instrument not in the original band, before transitioning into the original Evans arrangement of “Moon Dreams.” Mr. Lovano takes charge only eight bars into that theme, but after it has been faithfully stated by both the soloist and the ensemble, both Mr. Lovano and Mr. Schuller extend the piece with new variations, both pre-written and improvised, that are perfectly within the character of the 1950 recording. The other two augmented movements are perhaps even more impressive in that they juxtapose various tempos, with new, slow, or rubato introductions before moving into the bouncy boppers “Move” (by drummer Denzil Best) and “Boplicity.”

The 20-minute “Birth of the Cool Suite” is only the highlight of a typically brilliant offering from Mr. Lovano, who wrote the five-part suite, “Streams Of Expression,” which occupies most of the album. Not coincidentally, the second movement of “Streams” is titled “Cool,”and is also inspired by Davis and Evans, specifically their three masterpiece albums of 1957–60. Of special interest are several pieces featuring the saxist on a new instrument called the “Aulochrome” (invented by a French tinkerer named Francois Louis), which looks like a Siamese twin set of two B flat soprano saxes welded together with a new fingering system. As heard on “Big Ben,” a blues dedicated to the late Ben Webster, the Aulochrome sounds like two soprano saxists playing together in distinct voices but in perfect sync.

Still, the centerpiece of “Steams” is the revision of the “Birth” material. In his notes to the album, Mr. Lovano informs us that he was set to perform the suite live with his nonet at Monterey in September 2001, but the events of 9/11 prevented it, and that this recording represents the first time the work has been played. Let’s hope Mr. Lovano brings the full nine-piece ensemble and the suite to a New York venue soon.”The Cool,” having been born nearly 60 years ago, is ready to be reincarnated.