If we project songs from several populations and several individuals per population/song neighborhood into a low dimensional space (let's say 2 dimensions) representing variation in acoustic structure and in which each point is an individual (so points close to each other have similar songs), then a couple of things can be expected if dialects are present:
 1. There should be clusters 
 2. Individuals within a cluster should belong to the same population/song neighborhood
 
Ideally the acoustic space should represent the most varying/relevant song features, but it could be estimated using measured features, pairwise similarity measures, repertoire composition, syntax, etc.