[hmqcvppp] Musical time signatures

Musical time signature A/B:  The 1/B note gets the beat, and there are A of them in a measure.

If the dotted quarter note gets the beat, then B=4/(1.5)=8/3=2+2/3; that is, 2 and 2/3 dotted quarter notes equals 1 whole note.  There are two of them per measure, so A=2, resulting in the crazy looking time signature

2
-------
2 2/3

But this is usually written 6/8.  The eighth note does not get the beat in 6/8.  If we had wanted a 6-beat measure, the time signature would have been written 6/4.

I guess I've seen, somewhat more rationally, 2/(dotted quarter note symbol).

So an 8 as a denominator to a time signature means "group eighth notes in groups of at 1, 2, 3 (typically 3) and call that a beat".  All groups cannot be 2, or else it would have a denominator of 4.  All group cannot be 1, or else that would be a waste of ink on the eighth note flags.  For weird time signatures of 5/8, it needs additional annotation of whether the beat is e.g., 2+3 or 3+2 or 2+2+1.

16 as a denominator expands the possible groupings from 1 to 7.

In general, nothing with a flag ever (constantly) gets the beat.

There's still something wrong with this analysis if 8/8 is permitted as a half-time time signature, like 2/2 is a synonym for cut-time.  Then, the eighth note actually does get the beat.  But I'd argue 8/8 should be rewritten 8/4 with all the notes doubled in written length.  8/8 would be a fine time signature for 3+3+2.  I think I've also seen 4/8 (as 1+1+1+1), wasting ink on flags.

There's probably a tradition of 2/2, 4/8, 8/8 as it keeps the notes and measures looking "familiar", like 4/4, 2/4, and 4/4 respectively, allowing the musician to recognize it visually as the same rhythm from elsewhere but played twice or half as fast. Without annotation, it's impossible to tell if 6/8 is the (traditional) 3+3 or 3/4 played at half-time.

What happens if weirder notes get the beat, for example the triplet quarter note?  The denominator becomes 6.  A regular quarter note takes 1.5 beats.  One can imagine craziness like 5/6.

Even crazier a denominator of 7.  The 1/7th note gets the beat, technically written as a whole note with a 7 bracket underneath.  Probably better to write it with a denominator of 28, then it's a 7-bracketed quarter note, for which integer multiples are more reasonably written, e.g., 7-bracketed half notes.