For what the Thimble Island Saxophone Quartet thought was going to be its last number, leader and composer Frank Vasi picked a medley of Cole Porter tunes that he had arranged into a single piece — a “tip of the hat” to one of his (and many people’s) favorite songwriters.

However, he added, “unbeknownst to me, I put in a tune that wasn’t his.” The other quartet members had enjoyed pointing that out, “gently,” said fellow quartet member Will Cleary. “We called the piece ‘Some Cole Porter,’ Vasi said, “which is actually correct.”

Vasi’s knowledgeable, self-deprecating banter, combined with the obvious musical and personal camaraderie among the four members of the New Haven-based Thimble Island Saxophone Quartet — Tim Moran on soprano, Will Cleary on alto, Frank Vasi on tenor, and Skyler Hagner on baritone — made for a concert at Best Video on Saturday that was energetic to emotional, ranging from whimsical to wistful.

It helped that the full house contained more than a few friends of the quartet’s members. “All my friends are here! I’m so excited. I should play in Hamden more often,” said Moran, who lives and works in Hamden, before the show started.

“Do you want to sit in?” Cleary asked a student of his who was sitting in the audience. “Read some really hard charts?” Said charts had all been written by Vasi, who applied his compositional skill to arrange pieces across a variety of genres for the quartet. “Good music is great no matter what genre,” Vasi said. His own compositional voice in his arrangements lent the genre-hopping a grounding unity, while bringing out what was unique to each of the pieces that he arranged.

“This reminds me of the places we used to haunt in the the Village in the ‘60s,” Vasi said, surveying the crowd. It was appropriate, then, that the quartet began with an arrangement of Thelonious Monk’s “Blue Monk.” Vasi’s arrangement put Monk’s famous melody into a round, passed from instrument to instrument, before creating a texture that didn’t lose the swing of the original and allowed room for each of the players to improvise.

The next piece was an arrangement of a piece by early jazz pianist Jelly Roll Morton called “King Porter Stomp.” Vasi explained that during his career in New Orleans, Morton used to engage in piano battles with other players. “King Porter was Jelly Roll’s greatest rival and he wrote this in his honor.”

The quartet then switched to Bach. Vasi’s arrangement of “Jesu Joy of Man’s Desiring” gave the piece’s meandering line to the soprano and the slower melody to the alto, while tenor and baritone made a harmonic bed for them both. Later in the piece, however, Vasi let the melody move from instrument to instrument again, showcasing how good the musicians were at passing musical ideas around and listening hard.

“I think Bach would have dug that,” Vasi said. The audience laughed, but Vasi was serious. “Bach was a cool guy,” he continued, explaining that the composer was constantly rearranging his own compositions and the works of others on different instruments. “He was way hip to that stuff,” Vasi said.

The quartet marked the season with “The Holly and the Ivy” and selections from the Charlie Brown Christmas special that have become classics in their own right thanks to the talents of Vince Guaraldi. “Vince was a great composer, great jazz pianist, died way too young,” Vasi said. They visited Miles Davis and his piece “So What.” “Some people think Kind of Blue is the greatest album ever made,” Vasi ventured, suggesting that he might just agree.

One of the discoveries of the night, for this reporter, was a Vasi arrangement of a lullaby by George Gershwin. “He wrote this when he was a teenager and nobody knew about it until after he died,” Vasi said. “He was a genius, and his genius showed at a young age.”

A medley of Ellington songs that Vasi had woven together found the quartet arranging paper on their stands to accommodate it.

“Too many pages!” Vasi said.

“Blame the arranger,” Hagner said.

The quartet also played one of Vasi’s own compositions, which he called “Random Thoughts.” “The only thing I can say about it is that it’s totally random,” Vasi said.

“Thought so,” Moran said. But “Random Thoughts,” with its shifts in meter and texture, also showed Vasi’s hand as a composer who appreciated thought and emotion in equal measure, who valued conversation and playfulness. Listening to it, it was easy to understand what drew him to the form of a saxophone quartet in the first place, and how he got it to succeed both on the page and in performance.

The medley of Cole Porter was supposed to be the quartet’s final number, but as the applause died down, nobody in the audience actually moved from their seats.

“You all look like you’re waiting for an encore,” Vasi said. The crowd agreed. So the quartet returned to another holiday tune, “Sleigh Ride.” “If you don’t know it, you’ve been living on the moon for the past 100 years,” Vasi said. The quartet, and Vasi’s arrangement, made the tune instantly recognizable, but also connected it to the range of music they had played, and to the joy of playing together, whatever music they set their minds to.