The difference between 3/2 and 3/4

Question

So, today, the tides washed up this question from years ago: In 3/2 time why is a whole-note rest used as a bar's rest when a whole-note doesn't fill the measure?

This reminded me of a noob question I've been wondering about for years: Is there actually any difference between, say, a 3/2 at ♩ = 60 and a 3/4 at ♩ = 120?

I assume that a composer and/or engraver would choose between those two depending on what would be easier to read for a certain piece of music. But aside from that, is there any difference in how you'd count and emphasize those? Or could I take any generic waltz in 3/4, halve the tempo, halve all the note durations, replace the 3/4 with a 3/2, and get the same piece of music?

Sorry if this sounds like a stupid question, but I'm mostly educated in maths and physics, and as I begin to understand music theory it seems to me that, while music is rooted in maths and physics, and often borrows mathematical concepts, they are not always applied in a way that is consistent with their uses outside of maths.

Answer 1

At various historical periods there have been different conventions. Old British military marches were usually written in 2/4. Now 2/2 (or 'cut time' - that C with a vertical line one) is much more likely. (Unless it's a 6/8 march, of course.)

Waltzes have pretty generally been written in 3/4. Maybe 3/8. 3/2 would be unlikely, though I'm sure someone will find an example! Incidentally, writing a piece in 3/8 rather than 3/4 was by no means always an indication that the music was fast.

But yes,apart from this sort of convention, 3/2 at ♩ = 120 and a 3/4 at ♩ = 60 are exactly the same. (You got it the wrong way round in the question!) Except that we'd express the 3/2 tempo as half=120 the beat is the half-note not the quarter.

The 'whole bar' rest looks very like a whole note rest except that it is centered in the bar. It is used for a complete bar's rest in (almost) any time signature (there are exceptions for bars of 3/16 or shorter, 4/2 or longer).