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I know that in 3/4 you group eights by two and in 6/8 you group them by three.

My question is simply, is this simply a convention or could I derive this from the time signature itself somehow? Or is there maybe a bigger picture of conventions I don't know about?

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As mentioned in the other answer, it is convention, but there is a bit of logic behind the convention. A time signature of 3/4 is taken to mean that there are three quarter notes in each measure, so the fact that there are six eighth notes in a measure of 3/4 follows from subdividing each of the three quarter notes into two eighth notes.

Logically, a time signature of 6/8 tells you nothing about how six eighth notes should be organized into groups of two and/or three, but if you assume that three groups of two is out because that organization is properly expressed as 3/4, the only other option is two groups of three. A time signature that would make this explicit isn't possible using the traditional two-figure system. Mathematically, you could say that it ought to be 2/(8/3), but that is rather complicated to figure out and somewhat counterintuitive (the modern approach to teaching time signatures overlooks their original function as fractions). In recent decades, you might see 2/(dotted quarter note), which is a bit more intuitive.

For compound meters with a larger number of beats per measure, another approach that has arisen in recent decades is to make the groupings explicit in the numerator, so for example 7/8 might be expressed as (3+2+2)/8 or (2+2+3)/8, but this can also be expressed with other notational conventions such as beaming, so the practice is far from universal. Besides, in some pieces with compound meter the grouping changes from one measure to the next. In any event, I've never seen 6/8 expressed as (3+3)/8.

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    Right. And lots of ⁵⁄₄ pieces should arguably better be written in ³⁺³⁺⁴⁄₈, but aren't. ⁶⁄₈ is at least pretty unambiguous by convention. – leftaroundabout Jan 19 at 23:33
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    @leftaroundabout I would have expected (3+3+4)/8, if not written literally that way, to be written 10/8, not 5/4. Now I'm curious. Can you give an example of a (3+3+4)/8 whose signature is written 5/4? – David K Jan 20 at 18:21
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    @DavidK - 5/4 manifests itself most commonly as (3+2)/4. Think Mars, think Take Five as starters. No need for (x+y+z)/8. – Tim Jan 21 at 16:38
  • @Tim I once had an argument with someone about a piece that I thought was in 5/8 (One two three four five, one two three four five) and he thought was in syncopated 5/4 (One and two and three and four and five and, or (3+2+3+2)/8). But he wasn't counting the "ands," just counting five quarter notes over what I would have called two measures of 5/8. His cross rhythm was quite catchy, but I didn't see much evidence for it in the piece we were listening to. It's very subjective :-) – phoog Jan 21 at 16:41
  • @phoog - quite often I listen to new pieces and are torn between say 4/4 or 4/8. Depends somewhat on how fast I count! And it is more of academic interest than anything. Without beats and bpm notated, could be either! And listening to 5/4 and syncopated 5/8 - maybe only the composer really knows?! You can still syncopate 5/4 to sound like... well, you know. – Tim Jan 21 at 16:47
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It is convention. The two are different, even though mathematically they contain the same amount of notes. In 3/4, the count is 1-2-3,1-2-3.Simple time. Thus, as you state, there would be 3 beats, each containing two quavers.

Since 6/8 is compound time, it will be written out differently. It's basically two 'beats', each comprised of three quavers.So it's written as two separate halves in each bar. The count is 1--2--,1--2--. Imagine marching to the beat, L--R--,L--R--. That works in 6/8, but you'd probably trip up trying it in 3/4!

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