Michael Cooper writes:

>Think of it mathematically.  Time-values of notes and time signatures are
>all fractions.  1, or 1/1 or 4/4, etc., has the value of a whole note, 1/4
>has one quarter the value of a whole note, 1/8 has one-eighth the value
>of a whole note, and 1/3 has the value of...  Now at this, thousands of
>learned musicians cry foul!  "There is no such thing as a third note!"
>they sputter, with righteous indignation.  Well, don't call it a third note
>then, although that's what a triplet-half note (NOT a>dotted-quarter) _is_.

The distinction to be made is between meter and rhythm.  That there is in
effect a rhythmic X/3 is a fact.  A time signature however refers not to
rhythm, but to meter or measure.  As you point out, X/3 is measured as a
triplet half-note in a 4/4 (or equivalent) bar.  For that matter, you could
notate every waltz as triplets in duple meter, but very few people do.
It's probably eccentric.  X/3 is a meter and a notational issue, not a
rhythmic one.

As someone (I forget who, sorry) pointed out, the interesting thing is how
to move from, say, 4/4 to in rhythmic effect 5/3, where we get 5 beats of
a triplet half-note of the previous 4/4 measure.  Various composers have
done various things.  Some notate 5/4 and add that the new quarter note
the old triplet half.  Others actually compute the number of new quarter
notes per minute.  For example, if the old quarter note = 60, the new
quarter note 45 (? - is my arithmetic right?).  Still others have invented
idiolectic symbols (ie, they haven't caught on with most other composers).

>>Triplets do not come into the argument of time signatures at all,
>
>There is no musical reason they cannot, only a notational one, which is
>really no problem at all.  (See below).

But that's the point.  It *is* a notational issue or measuring issue, rather
than a rhythmic one.

Steve Schwartz