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It’s a spectacular week for jazz, folks.

Thurday, June 23
It’s been a long time—-over a year and a half, in fact—-since D.C. has seen a gig by the Cricket Fusion Quartet. The collective is the brainchild of Joe Brotherton, a trumpeter (and a hip one) with a taste for two things: spontaneous group improvisation and groove. Brotherton thus organized a small ensemble that was as conversant in funk, fusion, and hip-hop as in jazz… and one that could, with equal ease, play standards, write originals, or take off on in-the-moment, free-form runs. Indeed, the written material tends to launch powerfully into the free-form direction, too. It was a hell of a band, tight and endlessly creative; its members, however, had busy gig schedules, as did Brotherton, who also wanted to take some time to write music the band could record. Fast forward to tonight: Brotherton has re-formed the Cricket Fusion Quartet with bassist Blake Meister, drummer Terence Arnett, and saxophonist Elijah Balbed, and landed them a new weekly gig at the WXYZ bar at National Harbor. Tonight is the inauguration (although Balbed will be replaced by Herbert Scott for the evening), featuring a special guest, hip-hop artist Yusef. They hit at 6 p.m. at the WXYZ Bar in the Aloft hotel, 156 Waterfront St. at National Harbor. Free.

Saturday, June 25

Dizzy Gillespie is dead and gone, but his legacy is enormous—-as a trumpeter, as a composer, as a bandleader, and as a thinker. In D.C., much of his legacy is embodied in the DC Jazz Festival, in many respects created in Dizzy’s image; in the world at large, his legacy remains enormous but finds a particular focal point in Jon Faddis. Faddis was a Gillespie protégé, remains his standard-bearer, and is the undisputed heir to his title as the most virtuosic jazz trumpet player alive (who in many aspects has surpassed his onetime guru). He was head of Dizzy’s United Nation Orchestra, the Dizzy Gillespie Alumni All-Stars, and the Dizzy Gillespie Alumni All-Stars Big Band. Here’s a guy who takes his “keeper of the flame” status very, very seriously. But Faddis isn’t exactly free of his own accomplishments; he leads an orchestra under his own name, has released multiple Grammy-nominated recordings, and is even the composer of a jazz opera (Lulu Noire). He also passes the torch on himself, as a teacher of music at SUNY Purchase. And, in this case, he’s a performer in Washington, where he’ll play at 8 and 10 p.m. at Blues Alley, 1073 Wisconsin Ave. NW. $37.75.

Sunday, June 26
Fans of the Bohemian Caverns Jazz Orchestra will know its trombonist Shannon Gunn. What they may not know is that she leads her own big band—-an all-female big band. Shannon Gunn and the Bullettes was formed in the summer of 2010, when Gunn (then a graduate student in George Mason University’s Jazz Studies department) did the math and realized there were enough women in enough instrumental positions for such a thing to be possible. She began organizing rehearsals and raising money for a recording session. As a result, the Bullettes had their first record (the Introducing the Bullettes EP, with five sharp, elegantly crafted arrangements) before they had given their first public performance. And this is their first performance. At least 15 women will assemble as the Bullettes at 4 p.m. at Northern Virginia Community College’s Ernst Cultural Center Theatre, 8333 Little River Turnpike, Annandale. $15.

Wednesday, June 29
The work of bassist/composer Anne Mette Iversen (as documented on her new album The Milo Songs) is seriously tricky stuff. It’s tuneful, but can lose you going around its abrupt rhythmic corners; it seems to hide mysteries that can’t be unlocked; and it forces the musicians playing it to engage in some serious instrumental cartwheels—-sometimes just to find their own ways around it. Nevertheless, it’s incredibly compelling music, full of gorgeous musical ideas and proof that intensity has nothing to do with the amount of volume or force applied to the instrument. Iversen, Danish by birth, is a member of the Brooklyn Jazz Underground, a creative coalition dedicated to the cause of progressive jazz. But she has a special sense of what “progressive” means; in the case of her quartet (pianist Danny Grissett, tenor saxophonist Jerome Sabbagh, drummer Otis Brown III), it’s getting challenging artistic concepts across in intriguing ways, even if the paths can be a bit bumpy and jarring. The Anne Mette Iversen Quartet performs at 8 and 10 p.m. at Twins Jazz, 1344 U St. NW. $10.