[glitcher]

May or may not be helpful, but the way I learned it:

numerator = beats per measure

denominator = which note "gets the beat"

3/4 time has 3 beats per measure, where quarter (1/4) notes get the beat.

6/8 time has 6 beats per measure, where eighth notes get the beat.

etc...

[abruzzi]

Not quite. It is correct to say that the numerator is the count and the denominator is what is counted. So 3/4 is 3 quarter notes, and 6/8 is 6 eighth notes. However which notes get the beat is a bit more fuzzy. 6/8 is usually but not always two beats per measure, with three eighths per beat (counted 1-tee-ta 2-tee-ta)

That gets to the concept of simple and compound meter. In simple meter the the beat gets divided in two and in compound the beat gets divided in three. (Complex is a mix, like 7/8 may be divided 1-and 2-and 3-tee-ta).

Ultimately, the beat can be more interpretive. The 2nd movement of Beethoven’s 9th symphony is in (mostly) 3/4 but it is so fast, no conductor beats it in three, they beat in in one.

[delinka]

"... two beats per measure, with three eighths per beat (counted 1-tee-ta 2-tee-ta)"

I've never heard this. I've only ever heard it as "1 2 3 4 5 6"

[yesenadam]

or 1 and a 2 and a..

[lenzm]

Additionally, when you are listening to music the denominator has no meaning. You can hear how many beats to the bar (the numerator) but the way that it is notated (quarter note gets the beat or eighth note) is just a matter of how it is written.

[theoh]

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.

[pishpash]

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.

[theoh]

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.