The question did (appropriately) extend the question to cultures beyond Europe and European colonial transmission of music. I think the easiest way to transmit music over the most years is to be a stable culture, or at least a durable population in one spot for a very very long time. The original populations on any given continent would then be in the running, as hilda fish pointed out. The continent that at this time is considered to have been inhabited the longest is Africa, so I would start my search there and extend outward to adjacent landmasses. Arabia, the Middle East, the sub-continent of India. China, as a homogeneous nation now, is composed of regions with recorded cultures going back a very long time. I would put forward the hypothesis that the culture with the oldest recorded information would be the best candidate, because writing probably outlasts the oral tradition, and as new findings of old materials turn up (and as they are interpreted) then there is a possibility that the words to songs will be part of that material. Only if someone can figure out if melody was transcribable can they figure out more.

So, pulling these thoughts together, I'd look to see who produced the earliest ethnographic materials about "Other" cultures. It would be nice if they produced ethnographic materials as well about their own cultures, but so often we don't record what is around us; like the fish in water, it often isn't aware of that water unless it is completely removed from it. Early chroniclers in many instances were colonizing religious folk. So, to get records from the New World, as an example, you need to find direct descendents of those in situ populations (difficult after colonization, but not impossible) or records of the colonizers. Since many colonizers destroyed the records of those they colonized, that might be as close as you can get to the old tunes.

At any rate, it is an interesting process, but bound to be quite difficult and the path is filled with cultural, social, and political landmines.

SRS