Albert Ayler Festival (Saturday) The cultish and revered saxophonist Albert Ayler — whose wild, ecstatic performances inspired generations of jazz experimentalists — would have celebrated his 74th birthday this month had he not died (somewhat mysteriously) in New York in the autumn of 1970. A gaggle of acolytes — including William Hooker, Flow Trio, Charles Gayle, Gunter Hampel, Giuseppi Logan and the New Atlantis Sextet with Marshall Allen — are celebrating his life and work at this daylong outdoor festival. In between sets, crate diggers can scour a record fair featuring new and used vinyl, hosted by ESP Records in Brooklyn. At 2 p.m., Riverwalk Commons, Roosevelt Island , albertaylerfestival.com; free. (Amanda Petrusich)
Evil Eye/Brewed by Noon (Wednesday) Each of these bands harnesses the propulsive energies of rock as well as the creative license of avant-garde jazz, coming up with something off to the side rather than in the middle. Evil Eye, jointly led by the drummer Mike Pride and the saxophonist Jonathan Moritz, further includes Ben Gerstein on trombone and Ken Filiano on bass; the band goes on at 8 p.m. Brewed by Noon, led by the drummer Sean Smith, follows at 9 with a lineup that includes the guitarist Aram Bajakian. L.I.C. Bar, 45-58 Vernon Boulevard, Long Island City, Queens , (718) 786-5400, licbar.com; no cover. (Chinen)
Mary Halvorson Trio (Friday) The guitarist Mary Halvorson, a calmly prickly and increasingly prominent presence on the avant-garde landscape, leads a working trio with John Hébert on bass and Ches Smith on drums; their repertory will include new music as well as pieces from “Dragon’s Head” (Firehouse 12), Ms. Halvorson’s superb 2008 debut. At 8 p.m., Roulette at Location One, 20 Greene Street, at Grand Street, SoHo , (212) 219-8242, roulette.org; $15; $10 for students and 60+; and free for members. (Chinen)
Tom Rainey Group (Thursday) Tom Rainey, a starkly suggestive drummer, digs in here with Ingrid Laubrock, a German-born, British-based tenor and soprano saxophonist, and a rhythm section that includes the bassist Chris Lightcap and the pianist Gary Versace. From 9 p.m. to midnight, Smalls, 183 West 10th Street, West Village , (212) 252-5091, smallsjazzclub.com; $20 cover. (Chinen)
Jenny Scheinman’s Mischief & Mayhem (Tuesday through Thursday) Ms. Scheinman often goes for rustic charm as a violinist, but she never ceases to be an adventurous improviser. Her partners in this band, formed a few years ago, are the engagingly prickly guitarist Nels Cline, the stalwart bassist Todd Sickafoose and the polymorphically propulsive drummer Jim Black. (Through July 18.) At 9 and 11 p.m., Village Vanguard, 178 Seventh Avenue South, at 11th Street, West Village , (212) 255-4037, villagevanguard.com; $25 cover, with a $10 minimum. (Chinen)