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Oh now I see.

6/8, in typical musical theory fashion, is an awful name for something like (3/4)/2

It is technically precise and in practice misses the point completely.

(And don't get me started on mode names. Just don't)




More like 6/4 = 2×(3/8) but 3/8 is not a scalar because it also carries division and subdivision information, and × is a new operator that is barely similar to multiplication.

________

A time signature N/B is actually a tuple...:

    (division-of-N, subdivision-of-N, B)
...with units:

    (beats/measure, base_notes/beat, base_notes/whole_note)
...where division-of-N and subdivision-of-N are inferred from N (see explanation at [0] and table at [1]).

E.g. 3/4 is actually a tuple:

    (3 beats/measure, 2 base_notes/beat, 4 base_notes/whole_note).
________

The × operator works like...:

      A × (N/B) =
    = A × (D, S, B) =
    = (A * 1 division, D * 1 subdivision/division, B / 2)
...where X is a unitless scalar.

E.g.

      2×(3/8) =
    = 2×([division-of-N-3, subdivision-of-N-3, 8 base_notes/whole_note]) =
    = 2×([3 beats/measure, 2 base_notes/beat, 8 base_notes/whole_note]) =
    = [2 beats/measure, 3 base_notes/beat, 8 base_notes/whole_note / 2] =
    = [2 beats/measure, 3 base_notes/beat, 4 base_notes/whole_note]
    = [division-of-N-6, subdivision-of-N-6, 4 base_notes/whole_note]
    = 6/4
________

_UNINTENTIONALLY BURIED THE LEDE, IT'S HERE_

But honestly this just works only with traditional time signatures which follow this neat table-simple-compound-duple-triple-quadruple nonsense; for other compound N like 5/4, 7/4 etc. it won't work.

E.g. 5/4 is either (2/4 + 3/4) or (3/4 + 2/4) depending on the song (can't think any of the former off the top of my head, but Mission:Impossible's theme and Take Five are representatives of the latter). It can even be (1/4 + 3/4 + 1/4).

Or, like Gorillaz's 5/4 which sounds like (2.5/4 + 2.5/4) (which I'd argue is actually not 5/4 but 10/8 and the guitar definitely sounds like it). Also the lyrics+drums are actually in 4/4 so it's just a polyrhythm and both rhythms synchronize every LCM(5, 4) = 20 beats (4 guitar bars, 5 lyrics+drums bars) and in fact the macro song structure changes every 20 beats instead of 16 like in traditional 4/4 so maybe it is 5/4... or 10/8. Or 4/4 with 5 measures per hypermeasure? Or both? Or neither?

It's all just implicit. It's on your ears. Listen for the subdivisions.

This IMHO is the right way to look at time signatures.

[0] https://www.musictheory.net/lessons/15

[1]

    - N=2 simple duple        - 2 beats/measure * 2 base_notes/beat = 4  base_notes/measure
    - N=3 simple triple       - 3 beats/measure * 2 base_notes/beat = 6  base_notes/measure
    - N=4 simple quadruple    - 4 beats/measure * 2 base_notes/beat = 8  base_notes/measure
    
    - N=6 compound duple      - 2 beats/measure * 3 base_notes/beat = 6  base_notes/measure
    - N=9 compound triple     - 3 beats/measure * 3 base_notes/beat = 9  base_notes/measure
    - N=12 compound quadruple - 4 beats/measure * 3 base_notes/beat = 12 base_notes/measure
Notice how N=3 and N=6 share the same base_notes/measure hence the confusion.




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