“Clement Greenberg: A Critic’s Collection,” which has been touring the country and currently has settled at the Katonah Museum of Art, is something of an accidental assortment of roughly 40 paintings and sculptures from five decades. These were sometimes chosen by Greenberg but more often were given to him or his family by the artists he championed.

Greenberg lived with most of his collection, some parts of which have since been deaccessioned. But this exhibit still feels piecemeal. Though it has cohesion in terms of period (the paintings, mostly color-field, are housed in the gold-edged, dark wood, modern frames popular during the 1960s), the collection betrays its hodgepodge raison d’etre. Some of the best work by the best artists Greenberg championed (Hans Hofmann, David Smith, and 1940s Pollock and de Kooning) are hardly represented at all.

This gives the show a very skewed tilt toward the artists most willing to give up the goods – Kenneth Noland, Friedel Dzubas, Jules Olitsky, Larry Poons, Jack Bush, and Daryl Hughto. Still other artists who were generous toward the critic, such as Anthony Caro (who could not be well represented in the show because of the sheer size of his sculptures), are not represented in a manner at all reflective of their importance for, and impact upon, the critic.

The Katonah Museum is a small, symmetrical, Modernist building comprising two modest galleries, and this show is overhung. Most of the work, cramped together, suffers as a kind of decorative wallpaper. The exhibition, to its credit, recreates the kind of close quartered hanging said to have existed in Greenberg’s apartment on the Upper West Side. I do not believe that this is justifiable for any reason, though, especially when one of the gems of the collection, Ann Truitt’s blue-and-white, sculpted column “Bonne” (1963), is pushed against a wall next to an emergency exit door. In a sense, however, as it betrays the work, the hand is true to Greenberg’s push toward art as expansive decoration.

Some of the artists in the show were not well known, or they have not maintained the reputations Greenberg bestowed upon them 40 years ago. And the artists in the exhibition who have gained stardom because of the critic are better represented in most museums’ 20th-century American wings than they are here.

The question arises, then: What is the real purpose of the exhibition, beyond satisfying a mild curiosity about what, exactly, Helen Frankenthaler, Jackson Pollock, or Adolph Gottlieb gave to Greenberg? It represents his taste, his “eye,” yet it doesn’t. It represents his stable of artists, yet it doesn’t. When all is said and done, the show errs on the side of misrepresentation and irresponsibility. But I hope at least it will generate interest as much in the artists as in the man.

If viewers want to get the real story behind Greenberg, they will have to go much deeper than a smattering of artworks given to him. They will have to hit the books and read his criticism – then hit the museums and judge for themselves. The paintings and sculptures in this show will not by themselves bring us closer to the man who owned them. This is one of those rare instances when, contrary to popular belief, a picture is not worth a thousand words.

Until April 17 (Route 22 at Jay Street, Katonah, N.Y., 914-232-9555).