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.

Just to expand and what Dan said about acoustic spaces: if we project songs from several populations 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, if dialects are present, we can expect to find clusters, and the individuals within a cluster should belong to the same population. This can be used as a decision rule if we add a statistical test (like a Mantel test of acoustic vs spatial distances). This approach can be pretty flexible given that the acoustic space can be estimated using pretty much anything: measured features, pairwise similarity measures, repertoire composition, syntax, etc.

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 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.

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.