“Rainbow” is a song written by Alfred Bryan and Percy Wenrich in 1908. It was a type of song called an “Indian intermezzo,” which was popular back then. Other songs similar to it were “Silver Bell” and “Red Wing.” The tune was catchy and found its way into various genres of recordings. There’s even a recording of “Rainbow” by Grandpa Jones. His recording is called “Come Be My Rainbow.”

Here is the sheet music:
Rainbow

Here is a recording from a piano MIDI file:

And here’s the 1908 hit on Victor 17233 by Billy Murray And The Haydn Quartet, recorded on September 2, 1908: