The denominator is not as important as the numerator. (The Wikipedia page on time signatures makes that clear)One thing about 6/8 that is not literally communicated by the notation is the rhythm: the first and fourth beats of the bar are stressed. But not equally, because that would be indistinguishable from 3/8.

 The denominator is not important at all, though there seems to be a convention of keeping the rhythm fraction around 1, so 6/8 instead of 6/4, but say, 5/4 instead of 5/8.And 6/8 says nothing about the beat structure within the 6 beats. You're stating the most common occurrence, but that's not the definition of 6/8.
 You're not the only one in the comments on this article who has asserted that some pieces in 6/8 have a different pattern of stresses. So I'm totally prepared to hear about it, even if basic music theory doesn't cover those cases.The point stands that time signatures are not rational numbers (as you acknowledge, the unit of the denumerator can be scaled) so 3/4 and 6/8 aren't part of some equivalence class. They have different meanings which are only sustained by the presence of a cyclical pattern of stressed notes. So if there's some music theory you can link to which elaborates on all the possible different rhythmic patterns of 6/8 vs 3/4, that would provide examples of the distinction we both agree exists.

Search: