The first thing you notice is a screaming tenor saxophone, of the late 1960s avant-garde variety, blasting all over the place. Then there’s a standard jazz rhythm section, playing in what one might recognize as a vaguely hard-bop style – the piano, bass, and drums are playing regular notes and chords in regular rhythm. But what’s really surprising is going on behind all that in the background: a 10-voice choral group. It’s not a vocal jazz ensemble, like Lambert, Hendricks, and Ross, nor one in a gospel groove, like the Pearly Gates Spiritual Singers. It is the kind of choir that might gather to sing madrigals or motets.

In the abstract, there is absolutely no reason why this should work. Yet work it does. Andrew Hill is a master of taking seemingly incongruous ingredients and joining them together in new ways. Indeed, many of the ideas in his classic series of Blue Note albums of the 1960s – the music described above is from “Lift Every Voice,” recorded in 1969 – still seem novel today.

Mr. Hill is leading what he calls his “Retro” Quartet through Sunday at Birdland, featuring tenor saxophonist Greg Tardy, John Hebert on bass, and Leroi Jones on drums. This weekend the group features a special guest, the veteran trumpeter Charles Tolliver, who appeared with Mr. Hill on his 1968 “Dance With Death,” as well as two sessions included in a unique new box set that Mosaic Records recently released of unissued material by Mr. Hill.

Mr. Hill’s melodies are memorable, but they are never about catchy hooks. In his compositions, it’s the combination of sounds that’s most important. Free improvisation is a key part of his composed music. He is that rare composer who can incorporate anarchy into his music, even control it. You hear a tenor playing “outside” – shrieking his fool head off – yet in Mr. Hill’s music it is clearly part of a larger picture. Even the most abstract, atonal solo in a Hill piece seems to be going somewhere; Mr. Hill’s purpose is to give it a destination.

Andrew Hill was born in Chicago in 1937. A child piano prodigy, he studied orchestration and classical music with William Russo, one of the progenitors of the jazz-classical third-stream music, as well as the renowned symphonic composer Paul Hindemith. He recorded his first album for Warwick at 17 or 18 and worked as accompanist for crooner Johnny Hartman. He entered jazz’s major leagues by working with multi-instrumentalist Roland Kirk, who encouraged him to study jazz’s tradition as he had that of European music.

In 1963, Mr. Hill began his eight year association with Blue Note Records, the most productive years in his career. Mr. Hill first appeared as a sideman to the equally precocious young tenor saxophonist Joe Henderson and veteran bop trumpeter Kenny Dorham on “Our Thing.” But before the year was out, he recorded “Black Fire,” a quartet album done under his own name with Henderson this time serving as his own sideman.

The following year, he recorded “Point of Departure,” the album that is, if not his masterpiece, certainly his best known and most widely loved project. At a time when veteran players like Miles Davis and John Coltrane were addressing the changes to the jazz world wrought by Ornette Coleman, Mr. Hill came up with a compositional style that incorporated all the jazz approaches of the era – bop, modal, free – but really was something unique unto itself.

“Spectrum,” on “Point of Departure,” pivots on a simple eight-bar figure and is played by the three horns in unison, then by Hill, then by the ensemble alternating with the soloists. Virtually every episode is in a different time signature, all held in the groove by the outstanding bassist Richard Davis and then-18-year-old drum phenom Tony Williams. The front line was Henderson, Dorham, and a ringer: avant-garde avatar Eric Dolphy.

“Point of Departure” is not people’s favorite Andrew Hill album, but it’s my single favorite recording of Dolphy, who solos twice on “Spectrum” and is heard on all three of his main horns: alto saxophone, bass clarinet, and flute. No other leader employed Dolphy’s talents so skillfully, not even Charles Mingus, Coltrane, or Dolphy himself.

Blue Note major domo Alfred Lion regarded Mr. Hill as one of his three most important discoveries, along with Thelonious Monk and the lesser-known pianist Herbie Nichols. As Lion later told reissue producer Michael Cuscuna, he knew Mr. Hill’s music was so important that he gave the pianist carte blanche to record what he wanted. Issuing the results, however, was another matter. His best-known albums were released in the 1963-66 period, but as the jazz market began to dry up, much of his 1967-70 Blue Note material went unreleased and could not be heard until decades after the fact.

Two years ago, Mr. Cuscuna and Mr. Hill prepared the release of “Passing Ships,” an outstanding nonet session from 1969. It was one of the better jazz albums of recent years, although the music was 34 years old at the time. Encouraged by its success, Mr. Cuscuna and Mr. Hill have now released “Mosaic Select: Andrew Hill” (www.mosaicrecords.com), which contains all of Mr. Hill’s remaining unissued sessions for Blue Note – of the set’s three discs and 31 tracks, only six could have been heard previously, and then only on a rare two-LP set from the 1970s.

The box is essentially five different projects: two sextets, from 1967 and 1970; an exciting trio date with bassist Ron Carter and drummer Teddy Robinson and Mr. Hill tripling on piano, organ, and soprano saxophone; a septet from 1967; and, wildest of all, a meeting of jazz quartet and string quartet from 1969. Although I deconstructed the box down into its five sub-groupings on my iPod, I prefer to think of it as one mega-album, which shows off every aspect of Mr. Hill’s music-making ability.

The 1967 sextet session with two saxophonists, Sam Rivers and Robin Kenyatta, shows Mr. Hill’s wild, extroverted side. Mr. Rivers is that most thoughtful of avant-garde screamers, a perfect foil for Mr. Hill, who also plays a lot of organ here. The seven string tracks are an uncompromised delight, and a good companion to the choral album.

There’s less in the way of free playing here, but no one would call it conservative. Mr. Hill’s long, sinewy melodies, such as “Monkash,” are even more engaging and challenging than usual. Mr. Cuscuna’s notes explain that this tune was named for a friend, Dr. Jeff Monkash, but the melody is, perhaps not coincidentally, Monk-ish in its catchiness. It brings out the best in tenor saxophonist Carlos Garnett.

Once again, Mr. Hill has defied every kind of pigeonholing by juxtaposing the most free of free jazz with the most formal kind of chamber music. From now on, whenever I hear a string quartet playing Mozart, I will be disappointed if there’s not a screaming tenor saxophone simultaneously playing “outside” improvisations. And vice versa.

Until May 21 (315 W. 44th Street, between Eighth and Ninth Avenues, 212-581-3080).