In a time when jazz is barely a smudge on the cultural radar, the marriage of improvisation and popular music continues nowhere more apparently than with the Vermont rock band Phish. Other artists may be touring and improvising—and they are—but they don’t sell out Madison Square Garden for three nights in a row and continue to host a series of annual, one-band festivals that draw upward of seventy thousand people, all for the adventure of musical improvisation.

A highly divisive band, known best for their obsessive, vagabond following, Phish remain a baffling success in the music industry. Since they began, their musical style has continued to be a fluid spate of genres, most of which have very little in common with contemporary music, and some of which are laughably silly. A typical live show will include streaks of calypso, ’70s hard rock, jazz fusion, salsa, labyrinthine prog-rock, old-timey music, new wave and barbershop quartet, and, at any moment, one of these genres might be stretched out to forty-five minutes of wordless improvisation. Unlike many equally successful rock bands, Phish are relatively ignored (or dismissed) by the media, and do little to engage with the publicity cycles that dictate the peaks and valleys of most bands’ careers.

Trey Anastasio, the singer, guitarist, and sometimes composer for the band, is one of the revered, old-fashioned rock guitarists from the last quarter decade, playing the type of soaring, lyrical guitar melodies that have been all but banished from pop music since the early ’90s. Phish—and Anastasio in particular—are often cited as the musical heirs to the Grateful Dead and leaders of the new generation of so-called jam bands, but while the audiences may overlap, the band’s music bears little similarity to the bluegrass rock of the Dead or the electro-psychedelic “livetronica” style that dominates the current jam-band culture.

Over the last decade, Phish has gone on hiatus, disbanded, dispersed into solo projects, and reunited. During breaks, Anastasio performed solo acoustic sets, founded a new band with a full horn section, and composed a lengthy composition, “Time Turns Elastic” which has been performed by the New York Philharmonic at Carnegie Hall. Currently, Phish is preparing for a summer tour and for Super Ball IX, their next three-day summer festival, to be held in upstate New York.

I met with Anastasio in New York at Soho House for a conversation over soup. He talked about his long-standing interest in the art of improvisation, playing seven-hour-long shows, set lists, and the problems of conjoining art with daily life. After we spoke, I attended a Phish show at Madison Square Garden at which the audience was at full, standing applause for forty-five minutes before the band...