The compliment came from an unlikely source. ''You're so tasteful,'' gushed Bette Midler. Lena Horne's reply was something of a surprise: ''I'm tired of being tasteful.'' In this family history, Gail Lumet Buckley reveals the source of her mother's weariness and, en route, shows that fatigue can be contagious. Lena, it appears, was no sudden black star, up from ghetto poverty. Her ancestors, the ''old'' Hornes, had settled in New York City before the turn of the century. From the evidence of the book's many photographs, they were all attractive, intelligent people who paid a good deal of attention to clothes and carriage. Lena's grandmother Cora was a college graduate--uncommon even among white women of her time. Her grandfather Edwin was an alternate delegate to the 1884 Republican Convention, as well as a teacher, journalist and entrepreneur. He spelled out the code of the emerging black bourgeoisie: ''To be the full equal of the white man, there are two particular things we need--education and wealth.'' For most of the Hornes, Buckley says, ''racism seemed the only bad fairy at the family party. One could imagine that if it were not for race . . . Edwin might have been President of the United States, or at least a Republican senator.'' Lena's father was the first family member to break the code: Teddy Horne divorced his wife, who later became an actress. He was also a major-league gambler with underworld connections. One of them paid off: Dutch Schultz's mob guaranteed ''protection'' for Teddy's daughter when at 16 she began her career as a dancer at Harlem's Cotton Club. Yet Lena, though she followed in her parents' wayward footsteps, remained very much the proper granddaughter, combining ''ante-bellum manners and New England values.'' In later years she would go through a divorce and marry a white man, the orchestra leader and arranger Lennie Hayton. Between marriages she would even dare a brief fling with Joe Louis. But until she was 19, Lena had never had a boyfriend. Rather disastrously, she married one of the first men she dated: Louis Jones, a minister's son. Led by her mother, little Gail became part of a new and vibrant phase of black bourgeois life, in which respectability became the dancing partner of chic. As Lena moved up in the world, displaying her flawless cheekbones and elegant diction at bistros from New York's Cafe Society to Hollywood's Little Troc, she and her daughter crossed most of the boundaries set up by racial discrimination. Sailing on ''every ship of the French Line,'' they mingled offstage with the celebrities Lena had enchanted onstage in London and Paris --the Laurence Oliviers, Noel Coward, Yves Montand, Edith Piaf--until Lena herself became a name to drop. The pools of Hollywood opened up to Lena when she became one of the first black film stars to escape the costume of a maid's uniform, in Cabin in the Sky and Stormy Weather. John Barrymore kissed her hand. Orson Welles tried, unsuccessfully, to go further. Lena and Gail settled down across the street from the Humphrey Bogarts, and Gail went to school with Natalie Wood. If Lena's durable career seems to have been plucked from the book of an MGM musical of the '40s, Gail's life edges closer to something between The Cosby Show and Dynasty. After Radcliffe, she reports, she did everything an Ivy Leaguer should: worked as a volunteer for Jack Kennedy, made TV commercials, served a stint on LIFE magazine. ''The black bourgeoisie,'' she writes, ''took great pride in its separateness from ordinary black culture.'' It was Lena redivivus, including marriage to a white man--in this case Director Sidney Lumet, the ex-husband of Gloria Vanderbilt. Buckley admits that black history took place for her like news from another planet. The Supreme Court decision desegregating schools had less significance than a scene in Las Vegas featuring an infatuated Marlene Dietrich pursuing Frank Sinatra. The year Rosa Parks refused to give up her seat on a Montgomery bus, the teenage Gail was dancing at the Waldorf with Harry Belafonte. All along, Buckley chooses a policy of never apologize, never explain. She refuses to elaborate on a mysterious religious conversion that took place % before her divorce from Lumet and led, presumably, to this book. And there are many occasions when the family chronicle becomes more like a family scrapbook in its sketchy and elliptical nature. But the omissions occur less because of carelessness than because emotional confession falls beyond the range of the Horne style. It may be that inside every black princess an empress of the blues is struggling to come out, shouting her salty grief like Bessie Smith. But like her mother, the author is too polite to shout, and too honest to fake it. The story of Lena and Gail can be measured in privilege and recognition; what remains incalculable is the withholding tax that both women are still paying for their lives. BOX: Excerpt ''Lena was mad about her husband. As a high school dropout addicted to the movies, she saw him as the rather unlikely combination of her three favorite stars: Noel Coward, Leslie Howard, and George Raft. Unfortunately, Louis was somewhat less sophisticated than Lena's idols. He had, for example, old- fashioned ideas about wifely duties. He believed in perfectly ironed shirts, piping hot biscuits, and demon needlework. Lena . . . could sing, dance, and make intelligent conversation, but she could barely boil an egg.''