Are there any examples of non-Western music which have the equivalent of predominant and dominant passages as well as tonic passages, or even just the notion of dominant vs tonic? If not, are there any close analogies? To be specific, are there other cultures where there are ways of grouping scale degrees or classifying chords with a much smaller number of labels)? Are there other cultures which have particular scale degrees associated with strong tendencies towards another scale degree? I have heard many times that Western music is the only source of functional harmony, but since "functional" is so ill-defined I didn't know if that actually means no concept of dominant-tonic relationship.
I would assume that the concept of scale degrees is present in all cultures. Notes sound good together when they have higher harmonics in common (e.g. the third harmonic of C coincides with the second harmonic of G) -- a concept universal to all humans.
"Harmony" however is the occurrence of these notes at the same time, as opposed to in succession, which is the case in some other melody-based music cultures. Western music is harmony-heavy in this sense.
An example from Indian Classical music : most Ragas (scales of 5-7 notes on which the entire piece is based) have "Vaadi" (most important/frequent) and "Samvaadi" (second-most important/frequent) notes. They must have a dominant-tonic relationship with each other. Harmonic relationships also play a role for determining which microtones are supposed to be present in the Raga
The meaning of "functional harmony" is pretty clear.. https://en.wikipedia.org/wiki/Function_(music) ...but that particular label should be distinguished from a general use of the word "function."
Sure, other musical cultures will have tonal systems, and you could talk about how elements function, but it won't be "function" in the Riemann sense.
No one should misconstrue that to mean one musical culture is superior to another.