Everyone knows Jule Styne's songs--from ""Sunday"" in 1926 to Sinatra's hits to Merman's ""Coming Up Roses"" to Streisand's Funny Girl repertoire--but he has remained a rather low-profile public figure, especially when compared to his self-touting, onetime collaborator, Sammy Cahn. Yet Styne is a colorful, earthy, Runyonesque personality; and Taylor has caught much of that color in a better-than-average showbiz bio, always lively if never illuminating, that draws judiciously on Styne's own words. Born Julius Stein to Jewish immigrants in London, little Jule hit Chicago in time to drift from being a so-so child prodigy into a full-fledged lover of Chicago's post-WW I jazz. (""Whatever music I write. . . it was all made in Chicago."") Bandleader, songwriter, vocal coach--in Chicago, N.Y., and at last Hollywood, where he had to crank out songs for any situation (""I Love Watermelon""), go on the road with Constance Bennett, and keep up with Shirley Temple--""tough as nails, stamina of a steam engine."" Hits for Sinatra changed all that (a rocky relationship: ""one doesn't easily leave Frank Sinatra. . . the privilege of departure seems to be reserved for Mr. Sinatra""), leading finally to Broadway and other egos: neurotic Bert Lahr, insecure Barbra, sneaky Gower Champion, and imperious Merman (""She's tough and knows what she wants and she's usually right""). Plus other problems--compulsive gambling, revolving wives, and IRS investigations. There are fewer terrific anecdotes here than in some other backstage tales, but the tone throughout is down-to-earth and jaunty, flavored with ""Styne-ese""--the near-non-sequiturs that form the conversation of energetic, optimistic, and slightly foolish Jule. One big quibble, though: Taylor has no insight into what makes Styne the great underrated composer that Kenneth Tynan and others have called him. Otherwise--a warm, open shmooze of a biography, written with more loving care than usually goes into such projects.