"Just one more waltz with you
That melody so entrancing
Seems to be made for us two"
Context:
This song by Paul Whiteman is heard at Gatsby's party when Nick, Tom, and Daisy are conversing among themselves, discussing how Gatsby grew to acquire his wealth. During this conversation, Nick thinks about the possibility of another girl coming to the party and distracting Gatsby from his underlying love for Daisy.
Rationale:
This song being placed here makes Gatsby seem to be sort of a meddler, a controller of fate. He moved up to the Egg just to be near her and threw these parties in hopes of finding his love. His choice of playing the song at the party with those guests shows Jay's commitment to making that dream of him and Daisy come true. The lyrics of the song seem to parallel this, with one lyric being "That melody so entrancing Seems to be made for us two," (Whiteman). This song influences the mood by giving the party an atmosphere a feel of nostalgia from what Daisy said was the waltz of that year.
Video:

"Ev'ry morning, ev'ry evening
Ain't we got fun?
Not much money, Oh, but honey
Ain't we got fun?
The rent's unpaid dear
We haven't a bus
But smiles were made dear
For people like us"
Context:
In my opinion, this is the most interesting of the three songs discussed. It is played when Daisy meets with Jay at his house for the first time. This is interesting because of the historical background to this particular song. The rationale section will better explain the historic context of the song.
Rationale:
This song is utilized here by Fitzgerald for multiple reasons. First, the song addresses several critical social and economic changes that happened during America's Roaring Twenties. Secondly, the song has an interesting historical use besides being in Fitzgerald's novel. It was utilized by Warner Brothers Looney Tune cartoon characters Bugs Bunny and Daffy Duck. They would use this song when the two were hatching mischievous plans. This correlates with Fitzgerald showing to the reader the mischief Gatsby's plan had been all along. Gatsby is the puppet-master of his own future, having a relationship with Daisy. This song influences the mood of the novel by actually having Gatsby's plans become a reality. The reader gets the sense that there is a mischievous undertone to what Gatsby is doing.
Video:

Backbeat, the word was on the street
That the fire in your heart is out"

Rationale:
This song from the 90's has the same historic feel as "Three O'Clock in the Morning". As they describe Whiteman's song as the waltz of that year, everyone from the modern generation has the lyrics to "Wonderwall" seared into their minds forever. It has been blasted over the radio for years with no intention of stopping. It would carry the same general mood because the lyrics regarding lust for a specific girl, much like the song Fitzgerald chose for the novel.