The Mid-Ocean Amphitheatre at Fairmont Southampton Resort was about three-quarters full. So about 600 people had braved the wind, horizontal rain and the vagaries of parking to attend Saturday’s show by the Eliis Marsalis Jazz Quintet — an enthusiastic, affable and knowledgeable crowd. There was definite buzz in the air.

Enter quintet: pianist Ellis Marsalis on the left, bass and drums on the right, sax and trumpet centre.

They were introduced by Marsalis in his clear and unhurried voice: Derek Douget, tenor saxophone; Ashlin Parker, trumpet; Jason Stewart, string bass; and Stephen Gordon, drums. After a few short chords and arpeggios from Marsalis to test the sound system (which was perfect: crisp, not too loud, not too digital, a nice punchy analogue warmth), the musicians launched into Delilah, a piece by Clifford Brown from the late bebop era.

Building on the insistent piano riff, which sounds rather early Herbie Hancock, the soloists took over and gave the tune a swinging, modal Middle Eastern flavour. In some ways Delilah is firmly in the lineage of Gillespie’s A Night in Tunisia, though less percussive, yet at the same time looks forward to modal cool and beyond.

Marsalis expressed his desire to pay tribute to the music of Horace Silver (1928-2014) with the concert. Both he and Silver had initially been tenor saxophone players, and both had switched to piano early on in their careers.

Accordingly, the quintet played Nica’s Dream and Ecaroh. The Nica of the song title refers to Baroness Pannonica de Koenigswarter (1913-1988), an English aristocrat who was a jazz enthusiast and, despite her family’s intense disapproval, befriended and helped (among others) Thelonious Monk, Charlie Parker and Horace Silver.

It’s a pure, “hard” bop tune in the Silver style with tender lyrical interludes. Ecaroh shows Silver in playful mood, the name of the composition being his name backwards. It starts with a Middle Eastern/African-style riff on the piano and after saxophone and trumpet, the piano re-enters with a dazzling and scintillating solo with reversals, inversions and key changes, all beautifully executed by Marsalis.

Next, a change of tempo with Marsalis’s own composition, Homecoming. The theme is redolent of nostalgia, of course, but digs deeper with Douget’s saxophone solo showing conflicting and tumbling varieties of emotion.

The first set ended with an up-tempo 12-bar blues by Silver, which showed off the superb high legato of Parker’s trumpet. While it is a technique that goes back to the birth of jazz blues, Parker uses it to naturally punctuate the end of phrases.

The second half kicked off with another up-tempo swing blues boogie with a driving, almost rock ‘n’ roll, saxophone solo. Stewart and Gordon also enjoyed a turn in the spotlight with great bass and drum solos.

Marsalis’s masterful piano quoted from other compositions continually, but changed before your reviewer had a chance to identify them. But in the middle of this number I did identify “When the red, red robin comes bob, bob, bobbin’ along”, which was entirely in keeping with the tune.

Horace Silver’s Peace followed, a lyrical ballad written at the height of the Cold War when the doctrine of nuclear MAD (Mutually Assured Destruction) prevailed and we were reading doomsday books. The music is like a lullaby and does convey a deep sense of peace and comfort.

Next, Wayne Shorter’s 1963 One by One, which featured a brilliant vibrato trumpet solo by Parker. Unconventionally, Marsalis stopped playing altogether during the trumpet and saxophone solos. Once again, drums and bass had their solo turns.

The set concluded with two untitled pieces, a lovely arpeggiated ballad and a rousing 12-bar blues played at a cracking pace.

By now there was some serious head-bopping among the audience, who were on their feet yelling for more as the quintet departed.

Marsalis obliged with a beautiful solo which I did not recognise but was identified by Ron Lightbourne as a highly stretched version of The Party’s Over. Which, alas, it was.