At most jazz gigs, musicians improvise. When they finish a solo, they get a round of applause, like a circus elephant who’s just done a trick.

That doesn’t happen at Neil Cowley gigs. That’s because pianist Cowley doesn’t really play solos as such. He plays simple melodies which are stated and restated, played loudly and softly, slow and fast, until each starts to throb. Some songs, such as Vice Skating, feature just a single note, bashed at various rates of RSI-inducing intensity. Others, like We Are Here to Make Plastic, feature a complicated, Lisztian jangle of notes. Others resemble the neurotic minimalism of Michael Nyman, or the piano-thumping ebullience of Fats Domino.

Cowley can actually improvise as well as any jazz pianist. During the Charlie Brown-style funk of Couch Slouch, for instance, he moves from one end of the piano to the other, spraying whimsical, off-the-cuff chromatic riffs. But for the most part, his jazz chops are held in reserve. Often it’s his bassist, Rex Horan, who does the soloing. He’s an extravagantly bearded Australian who looks like a daguerreotype of an American civil war veteran, and he’ll often play slithering counter melodies while Cowley remains static or plays the simple but effective flourishes he provides on pop sessions.

This was a gig of two halves. The second was a greatest-hits set of direct themes from the trio’s first four albums: demented nursery rhymes, soundtracks in search of a film, instrumental versions of stadium-rock classics that have yet to be performed. But the first hour – a full recital of their latest album, Touch and Flee – was the more satisfying. These songs are richer and more harmonically complex than anything they’ve done before, yet retain that direct, melodic USP.

In a jazz world full of introverts, Cowley is a rare musical extrovert, someone who can actually communicate with a broader audience. It’s something to cherish.

• This article was amended on 7 October 2014, as it was published with the wrong star rating