Saxophonist Virginia Mayhew knows that the essence of jazz is group interplay, which is why she’s been so successful at assembling a series of thrilling bands over the past decade.

That’s not to imply that Mayhew is any kind of a slouch on the horn. She’s attained a beautiful, brawny sound on the tenor, and can swing ferociously in even the oddest meters. But it’s her work as a bandleader that really stands out.

Mayhew brings her latest quartet to the Jazz & Blues Company for a Blackhawk Live concert, broadcast live on KRML (1410-AM), on Saturday. The band features bass virtuoso Harvie S (formerly known as Harvie Swartz), guitarist Ken Wessel, whose credits include a long stint with Ornette Coleman’s Prime Time, and veteran drummer Victor Jones, who has worked extensively with Mingus Dynasty and the Mingus  Big Band.

Mayhew’s ability to consistently attract top flight players flows directly from her gift for writing music that inspires her collaborators. “Her music is always challenging,” says Mr. S, who has recorded with numerous jazz giants and is a noted bandleader in his own right. “She’s not content to just play some standards. She uses different time signatures, Afro-Cuban and Brazilian rhythms mixed with jazz. When I play with Virginia, I know I’ve got to play my best. She demands it and her music demands it.”

Mayhew credits her band director at Redwood City’s Sequoia High, Edward C. Harris, with playing a key role in her decision to become a musician. She started on clarinet and then started doubling on alto. By the time she moved to San Francisco in 1979 she showed enough promise that veteran trumpeter John Coppola became a mentor, expanding her jazz education by turning her on to the ineffable beauty of tenor saxophonist Lester Young.

“Some of the first professional gigs I ever did were in San Francisco with Johnny Coppola,” Mayhew says. “We’d work out some background riffs, and learn the melodies. He’d continue to up the ante. He’d work with me on phrasing and breathing, and listening.”

She played widely around the Bay Area in the ’80s, until she made the move to New York City in 1987. While studying at the New School for Social Research—she won the first annual Zoot Sims Memorial Scholarship—Mayhew worked frequently with trumpeter Rebecca Coupe Franks, and was eventually hired for a recording with former Basie trombonist Al Grey. It turned out to be a breakthrough gig that brought her to the attention of players like the legendary trumpeter Clark Terry and vibraphonist Terry Gibbs.

“It was amazing, playing with Clark Terry and all these really great musicians.” Mayhew says. “About a year after that I got a phone call from Al, and he wanted me to do some gigs with the band, because the album was out. I got a lot of exposure playing with him.”

At Grey’s urging and with the opportunity to join the all-female big band DIVA, she switched from alto to tenor, finally taking up the horn of her heroes Dexter Gordon, George Coleman and Archie Shepp. It’s the instrument with which she’s gained a firm foothold in the highly competitive world of jazz, a position she’s maintained with her acute musicianship and astute improvisational sensibility.

“She’s a strong player,” Swartz says. “But she’s not a flashy player. That’s not her thing, and I think that’s to her credit. She’s not trying to show how fast she can play, or how high she can play. She’s really just trying to show how musically she can play, and that’s what she does.”

Virginia Mayhew plays Saturday, June 4 at 7:30pm at the Jazz & Blues Co., San Carlos and 5th, Carmel (above The Hog’s Breath Inn). $45. 624-6432. The show will be broadcast live on KRML radio (1410-AM).