Oral traditions are not necessarily about the transmission of songs or tunes. In many cultures what is transmitted orally is the musical framework that allow you to compose your own. In many peoples of North and South America is was expected that nobody ever re-used somebody else's flute tune, and much the same happened in eastern Europe with the "doinas" (slow unaccompanied fantasias) of Romania and nearby places (and later, Askenazi Jewish klezmer musicians both sides of the Atlantic) - a doina was composed rather than improvised, and each musician had their own which was played with little if any variation. Learning to play somebody else's, as is usually done today, misses the point.

I can't thing of anything quite like that from an Anglophone culture but it probably happens somewhere.