I'm trying my best to understand musical meter, but it's just not making sense to me. In particular, I don't understand how one might deduce from the two numbers in the time signature (i.e. 3/4 or 4/4 etc.) the exact number of divisions and subdivisions of the beat. That information is necessary in order to determine whether a meter is simple/compound etc.
I'll quickly expound my (likely flawed) understanding. The number on top is the number of beats per bar, the number below tells you which type of beat. As there is a choice over how fast to play a piece, the lower number is arbitrary, or at least only important as regards ease of notation. As far as I can see, the top number is more important.
Simple meter divides each beat (or pulse) into two smaller beats. Compound meter divides each pulse into 3 smaller beats. My first question is: why are these are taken to be the only types of meter? Might we not group a meter with 10 beats per bar into two pulses of 5 sub-divisions, for example? This would classify as neither simple nor compound.
Secondly, suppose I'm given a time signature of 6/8. Numerically, 6 can be divided in four ways: either one group of six, six groups of one, three groups of two, or two groups of three. Unless I'm mistaken, the time signature 6/8 is always taken to represent the last of these possibilities: two pulses of 3 semiquavers. Now, I understand that 3/4 meter can represent either 3 groups of 1, or 3 groups of 2,4,8,16, ... etc. due to the fact that the lengths of notes in musical notation are related by powers of 2 (for example, a minim is twice the length of a crotchet), which accounts for the 3 groups of 2 possibility. This still leaves the question: Why shouldn't 6/8 represent one group of six beats (or six groups of one as they amount to the same thing)?
Is there some kind of system I've not yet come across, or is the only thing relating the groupings of beats to the number of beats per bar some pre-existing conventions which students of music theory simply have to learn by heart?