What might have been the source of the music that Jonas heard behind him in The Giver?

The music Jonas hears could be from a house in Elsewhere. 

The ending of the novel is ambiguous.  That means that, based on the evidence in the novel alone, it is not possible to tell what really happened at the end.  Jonas hears music.  It could mean one of two things.  Either the music is real, and he has reached Elsewhere, and a community that does not have Sameness, or the music is not real, and it is from a memory.  In that case Jonas is hallucinating and close to death.

There are three other books in the series, and Jonas appears in them. This means that, although they are not direct sequels, they take place after the events in this book and Jonas did indeed survive.  Based on that evidence, we can infer that the music was real. 

Jonas’s community has no music. Part of Sameness involves a quest to eliminate any kind of emotion.  It makes sense that they would not want music, because music is designed to make you feel.  Thus, the first time Jonas hears music is in a memory, where he experiences many of the things that used to be in the world before Sameness. 

Music is highly significant for Jonas.  He has what he learns is called the Capacity to See Beyond, but it turns out that when The Giver first experienced his special ability, it was not through seeing colors.  He heard something that did not exist, instead of seeing something that did not exist.  He called it hearing-beyond. 

Jonas frowned, trying to figure that out. "What did you hear?" he asked.

"Music," The Giver said, smiling. "I began to hear something truly remarkable, and it is called music. I'll give you some before I go." (Ch. 20) 

In this case, how can Jonas experience something that does not exist?  He takes Gabriel and flees, enduring a great deal of pain and suffering.  They almost starve to death, and almost freeze to death.  In fact, it is when they are at their lowest point that Jonas hears the music.  He is knee-deep in snow, and comes across a sled.  Is it a real sled, or the sled from the memories? 

Suddenly he was aware with certainty and joy that below, ahead, they were waiting for him; and that they were waiting, too, for the baby. For the first time, he heard something that he knew to be music. He heard people singing. (Ch. 23) 

We have to assume that Jonas has reached another type of world, where people still sing.  He hears music, but comments that it might just be “an echo.”  This is more of the ambiguous information.  Did he and Gabriel live?  We know they did only because there are other books in the series.  Based on this information alone, he could have been dying or dead.