THE HORNES: AN AMERICAN FAMILY by Gail Lumet Buckley (Knopf: $18.95; 320 pp.; illustrated)

We expect indecent exposure from books by children of celebrities. Not so from Gail Lumet Buckley, one of Lena Horne’s two children from her marriage to Louis Jones of Pittsburgh. Rather, Buckley has written a family portrait with her mother and herself at the center, which reaches out to include her mother’s people--the Hornes of Tennessee and the Calhouns of Georgia. Her object is not to diminish the celebrity of the mother but to show the family and tradition on which Lena Horne stands, which has sustained her in a career and industry notoriously cruel and destructive to talent, especially when black and female.

Her object, also, is her own quest and self-discovery. She is, after all, at that certain age--in her second marriage and mother to her own children--when the urge is most keen to retrace traveled paths. A childhood which, casually considered, one might envy: growing up amid the glamour and glitz of Hollywood and show biz, along the corridors and lobbies of nice-to-fancy hotels with whims catered to by liveried Room Service, part of an international set regularly crossing the Atlantic aboard luxury liners, at home so to speak in all the European capitals and watering places, yet with time and talent for an excellent education in Quaker private schools and Radcliffe College; such a childhood has its costs. Now is time for a gathering in, for finding ground on which to stand, especially for a black woman like the author whose life has been removed from what might be called a typical, middle-class, black experience, and whose children’s lives are likely to be even more removed.

This book is Buckley’s effort to rediscover the close, insular, middle-class black world from which her mother came, which the young Gail could only see at a distance, and which no longer really exists.

The Calhouns and the Hornes, where her story starts, were relatively privileged Southern blacks, comfortable, well-educated and aspiring. Typical of their age and class, however, Cora Calhoun and Edwin Horne, Gail’s great-grandparents, understood their privilege to demand responsibilities of racial improvement through protest, reform and uplift.

They established their family home in Brooklyn, where Edwin switched from teaching to politics, becoming a cog in the Tammany machine and supporting his family quite well, thereby. Cora was a club woman and social activist throughout her life. Of their children, only Lena’s uncle, Frank Horne, followed in that path. Frank was considered a promising black poet in the 1920s, but he went into government service, becoming part of the so-called “black cabinet” to the New Deal.

Teddy Horne, Lena’s father, was handsome, a playboy of sorts. After the failure of his marriage to Edna Scottron, Lena’s mother, Teddy lived the life of a “sport,” mainly in Pittsburgh where they owned a hotel managed by his second wife. Teddy was never, it seems, without money, fine cars and clothes. But Lena bounced between her neurotic mother and the Brooklyn family. It was Lena’s mother who pushed her into show business by getting her in the chorus at Harlem’s Cotton Club.

One is struck by how thin was this class in which Lena Horne was nurtured, it was cramped and confined by American race and caste patterns. Yet, it was a tough and resilient group, perhaps explaining Lena Horne’s grit and force of character which has held her steady, preserving her integrity and sense of social responsibility in the face of a corrupt and spineless industry that has destroyed the likes of Billie Holiday and Dorothy Dandridge. This family portrait may give hints as to the sources of that strength of character.

Hints are as far as this short book goes on many things. One would like to know more about the author’s father, about Lena’s second husband, Lennie Hayden, about Gail’s marriage to Sidney Lumet. Who is Mr. Buckley? She holds back most when the subject comes close, pulling back the curtain but never quite revealing what is there.

The reader will want, mainly, to know about Lena Horne. Her story, now familiar, is here, placed within the context of family as seen by her daughter. It is a fresh perspective, and worthy in that alone.

Be warned, however. Buckley is no flasher, exposing scandal. Rather, she is discreet, decent and respectful of a past she may now, only for the first time, be coming to terms with.