Young Man With A Horn by Dorothy Baker; Gary Giddins

Rick Martin loved music and the music loved him. He could pick up a tune so quickly that it didn't matter to the Cotton Club boss that he was underage, or to the guys in the band that he was just a white kid. He started out in the slums of LA with nothing, and he ended up on top of the game in the speakeasies and nightclubs of New York. But while talent and drive are all you need to make it in music, they aren't enough to make it through a life.

Very minor lesbian incident in a jazz novel. Look for multiple reprints with different covers.