When the Washington Post asserted last summer that Vice President Cheney keeps three “man-sized safes” in his office, it immediately became a great source of speculation and humor among comedians and pundits. The bassist and composer Ben Allison, however, has obviously figured it out: The vice president is clearly a jazz fan, and he stashes his private collection of great and rare albums in those safes. On his last album, 2006’s “Cowboy Justice,” the composer dedicated one original tune to Mr. Cheney, affectionately titled “Tricky Dick,” and now in the VP’s honor, Mr. Allison has christened his new band Man-Sized Safe.

To further exhibit his respect for the political process, Mr. Allison chose to introduce the band and his new album, “Little Things Run the World,” in a special mini-concert at the National Arts Club on Super Tuesday. Mr. Allison even launched the show with a tactic designed to distract everyone who had spent the day to that point kvelling over the victories of Senator McCain and, of course, the New York Giants. He began by having trumpeter Ron Horton blow into the wrong end of his horn. Hello! Well, that got my attention. This led into a brief exchange of seemingly random phrases between Mr. Horton and tenor saxophonist Michael Blake, and before we knew it, we were into the melody of the title track from the new record — although on the CD, it begins with Mr. Allison playing a bass intro. The composer explained that the title, “Little Things Run the World,” refers, rather literally, to insects and bacteria, as the basis of our ecosystem. Although the opening suggests Ornette Coleman, the body of the tune, like much of Mr. Allison’s work, has the feel of a ’60s pop hook, and the way he uses the horns in contrast to the rhythm (Steve Cardenas on guitar and Michael Sarin on drums) has as much in common with Blood, Sweat and Tears and Chicago as it does with the Jazz Messengers.

The evening’s second tune (and first on the album), “Respiration,” showcased Mr. Allison’s technique of building a composition out of many different levels of melody and rhythm, played on different combinations of instruments. Often, the guitar played one line in one tempo while the saxophone played another melody in another tempo. It was impossible at first to discern what the two lines had in common, but the more one listened, the more sense it made. Mr. Allison constantly sent the five members of his ensemble in their own directions and then brought them back together. This was in addition, mind you, to the solo improvisations, and the leader was smart enough not to have everyone solo on every number.

Mr. Cardenas (who was celebrating his birth on Super Tuesday) often uses his guitar more like an electronics kit with nylon strings, in roughly the same manner as Bill Frisell and Ben Monder, supplying a background soundscape rather than the familiar harmonic road map expected of pianos and guitars. However, for his own tune on Tuesday night, “The Language of Love,” Mr. Cardenas played notes and melodies. Here, Mr. Allison took his first extended solo of the evening, laying down a 3/4 pattern, over which the others played different melodies in divergent time signatures; the use of a brief catchy tune over uncountable meters recalled Burt Bacharach. As with most of Mr. Allison’s music — and, indeed, much of the best millennial jazz — so long as Man-Sized Safe kept the beat going, its musicians were free to try virtually anything.

Just as Mr. Allison keeps everything in the groove, I’ll wager that this is one album Mr. Cheney will want to keep in the vault.

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Following Man-Sized Safe’s show in Gramercy Park, I hopped in a cab, dodged the remaining Giants fans still celebrating in the streets, and sped up to the Algonquin. Like Mr. Allison, Lorraine Feather is expanding the jazz repertoire in her own idiosyncratic way and showcasing the power of composition as much as the power of performance. Ms. Feather is one of the few singer-songwriters in contemporary jazz worth listening to and, along with inspirations Jon Hendricks, Dave Frishberg, and Jay Leonhart (who happens to be playing bass for her this week), virtually the only one whom I want to hear sing nothing but her own lyrics.

Ms. Feather’s general strategy is to take classic instrumentals, most often by Fats Waller and Duke Ellington, and write new words for them. Unlike Mr. Hendricks, she does not put lyrics to horn solos; by sticking mostly to piano pieces, she adheres to the pre-written parts rather than improvisations. Also unlike Mr. Hendricks, Ms. Feather’s tactic is to discard the original titles and impose her own, a habit that becomes especially apparent when she takes Ellington’s metrocentric melodies and relocates them to the far-flung corners of the globe: “Harlem Air Shaft” is now “Calistoga Bay,” and “Dooji Wooji” has become “Sweet Honolulu.”

A further inspiration for Ms. Feather is Ira Gershwin, who famously took his younger brother’s syncopated lines and attached vibrant, slangy language to them. Ms. Feather does the same with melodies that are even more fragmented, staccato, and reliant on sheer rhythm and speed for much of their effectiveness; it’s a testament to her skills as a singer that she can keep up with and articulate her own lyrics. “Imaginary Guy” is a vividly re-imagined text to “Dancers in Love,” while “Trying To Get Over” and “Indiana Lana” fully capture the jazzy insouciance, sophistication, and hip energy of “Doin’ the Voom Voom” and “Jubilee Stomp” (also lyricized as “Music Is a Woman”).

Of her songs not based on famous instrumentals, the funniest is “Hit the Ground Runnin’,” a Frishbergian compendium of sports announcer clichés. Further illustrating Ms. Feather’s allegiance to the Gershwins, the song uses the chord changes to “I Got Rhythm” and swings in the general direction of Thelonious Monk’s “Rhythm-A-Ning.”

On Tuesday night, the only ESPN expression I noticed that she omitted was “they came to play,” which Ms. Feather, Mr. Leonhart, and the pianist Shelly Berg (a West Coaster best known for Oscar Peterson-style modernism who shows, in his work with Ms. Feather, a strong talent for stride) most assuredly did. Apart from Ms. Feather, Abbey Lincoln, and one or two others, original songs (or even semi-original) by contemporary jazz singers are generally to be avoided, almost as much as scatting. I only hope that her remarkable success in this area will not encourage others.

wfriedwald@nysun.com