They don’t have Q ratings in jazz, and even if they did, New York pianist and composer David Berkman wouldn’t score very high, despite the critical acclaim that met his first two albums (including several year-end best-of lists). Why? His compositions are cerebral, certainly: their splendidly crafted harmonies and form reveal subtly original thinking on his part and invite the same from his soloists. But Berkman goes way beyond theoretical constructs, turning his ideas into lyrical melodies and eminently likable tunes that should please even those who don’t know a diminished chord from a turnaround. And, like such spiritual forebears as Elmo Hope and Horace Silver, Berkman knows as a pianist how to best exploit his own compositions: he uses his rich solos sometimes to explore a song’s theme, sometimes to limn the harmonic potential, and sometimes to link the work of the other soloists into a unified performance. While his lauded 1998 recording debut, Handmade, featured a conventional jazz quintet, Berkman seems to have settled on a preferred instrumentation: the new Leaving Home, like his 2000 disc, Communication Theory (both, like the debut, on Palmetto), employs a sextet with soprano, alto, and tenor saxophones. The format affords Berkman a wonderful latitude in his writing; the reeds allow him to fully illustrate his harmonies and counterthemes, while the absence of brass instruments keeps the tonal palette contained and even intimate. Unfortunately the expense of traveling with a sextet prevents Berkman from bringing the whole band to Chicago; we get just one of the saxists, the versatile Dick Oatts, plus bassist Ugonna Okegwo and drummer Mike Sarin. But Berkman has no trouble adapting his music to a given set of circumstances, and his two previous Chicago dates, both with a quartet, remain among the more memorable shows of the new century. Friday, April 12, 9 PM, and Saturday, April 13, 8 PM, Green Mill, 4802 N.Broadway; 773-878-5552.