The title of this song has no question mark, and its punchline is that it is anything but a simple melody. When Barry Manilow presented a programme about Irving Berlin for BBC Radio in 2011 - part of a series on American songwriters - he said this piece, which Berlin wrote in 1914, was the first of his famous double songs, ie it is in counterpoint like "You're Just In Love" which he wrote three and a half decades later.
Also known as "Play A Simple Melody," it is written in 4/4 time, was published by Berlin himself, and appeared in Watch Your Step, Berlin's first musical.
Suggestion credit: Alexander Baron - London, England, for above 2
Walter Van Brunt and Mary Carson recorded a popular version in 1915, followed by Elsie Baker (aka Edna Brown) and Billy Murray the following year.
This was resurrected in 1950 with a string of hit covers. Gary Crosby peaked at #2 when he recorded a version with his dad, Bing. Jo Stafford landed at #18, followed by Georgia Gibbs & Bob Crosby at #25, and Phil Harris at #30.
Ethel Merman and Dan Dailey sang this in the 1954 tribute film to Irving Berlin, There's No Business Like Show Business.