Why is 3/4 called "simple triple" if we can divided the beats by more than 2?

Why is 3/4 called “simple triple” if we can divided the beats by more than 2?

It's called triple for the three beats per measure not because of the subdivisions.

In simple meters beat subdivision is by 2. Dividing by another number is either for compound meters with subdivisions of 3 or you need to use a tuplet to subdivide the beat by a number other than 2.

I think of it in terms of the default subdivision.

Simple meters subdivide by 2.

Compound subdivide by 3.

When you use rhythms with the little number of the beams...

...it's saying "the beat which normally subdivides by 2 is now subdivided by 3."

If it isn't already understood, those three "triplet" eighth notes are smaller fractions of the beat than two "default" eighth notes. If played a steady pattern of alternating the two...

...it should have a odd, lurching feel as the eighth notes speed up a bit for the triplets and slow down a bit for the "duplets."

That particular difference in rhythmic feel - the difference of a beat subdivided by 2 versus subdivided by 3 - is the basic difference between simple and compound meter.

For the sake of argument, if you wrote a piece of music that constantly used...

It might make more sense to change the meter to...

The first example is a triplets and the second example is septuplets. In both these examples, the number of beats in the measure would still be 3. So it would still be triple meter. But they wouldn't be "simple" because the beats aren't divided into 2.

It would still be simple 3/4.

Septuplets may not be simply (easy) to play, but the septuplet 7 is saying "not a normal subdivision by 2s". The expectation is the septuplet rhythm would return to the default subdivision by 2s pretty soon.

But I find this confusing because in 3/4, there are 3 quarter notes. And they haven't yet been divided into anything.

Yes, I suppose that if a piece literally never divided its beats, the simple/compound dichotomy would break down, but this very rarely happens in metred music. These terms, "simple" and "compound" do not exist in an abstract theoretical vacuum, they exist to describe music we actually play -- and the music we actually play tends to divide beats.

Some pieces don't stick to one particular division, but they tend to overwhelmingly favour one. These words describe how the listener hears the metre -- if a listener's grooving, are they going to be nodding up-and-down with quavers, or air-drumming along to the triplets? This is what defines "simple" vs. "compound", not the sheet music (and this is generally true of theory -- theory describes music, which is sound, not paper).

But what if I wanted to divide each beat by 3s?

Then, as a couple of answers have pointed out, you would have compound triple. Same as if you wrote a piece in 4/4 but wrote every single bar in nested minim/crotchet triplets -- you would have written in compound triple and anyone reading it would be entitled to punch you for being obtuse.

I could also divide each beat by 7.

I'm not sure there's a word for this, but there really should be; dividing beats into 5s has become relatively popular recently, and again, I'm not aware of a word for it. This is just a limitation of language, though; sometimes things just don't have names yet.

why would we say 3/4 is "simple triple" if the beats can be divided into other values besides 2?

I would personally avoid saying that "3/4 is simple triple", in favour of something like "3/4 is almost always used to notate simple triple". While your triplets example is "incorrect" in the sense that it would come out a lot nicer in 9/8, you might reasonably decide that 3/4 in septuplets looks nicer than 21/16.

Yes, you could take a bar of 3/4 (a time signature that implies Simple Triple meter) and divide each quarter note into triplets. If you did this consistently there would be a good argument for calling the meter 'Compound Triple' - and for re-writing it as 9/8.

If you're going to sub-divide in 2, choose 3/4. If you're going to sub-divide in 3, choose 9/8. If you're going to mix it up, choose one or the other and use tuplets where needed.

(Rhythmic styles have loosened up a lot since the textbooks were written, and even since Bernstein decided to notate 'America' with alternating bars of 3/4 and 6/8. We very often DO 'mix it up' now and rely on note grouping to show the meter rather than constantly changing time signature.)

If you're not going to sub-divide at all - just have notes with whole numbers of beats - you could theoretically use either 9/8 or 3/4. (But you'd use 3/4.)

The previous answer I wrote and you linked went over a lot of this, but I think there are some ideas that should be stressed to help clear up some confusion. From the previous answer:

A simple meter is a meter that has the beat assigned to a simple note duration represented by the lower number of a time signature. The subdivision of the beat of a simple meter are in two.
...

A compound meter is a meter that has the beat assigned to 3 of the note duration represented by the lower number of a time signature (so for 6/8 the beat is a dotted quarter note which is 3 eighth notes). The subdivision of the beat of a compound meter are in three.

The distinction here is the metrical hierarchy and how subdivisions are delegated. The time signature dictates metric hierarchy and what to rhythmically expect accent and stress wise. You can always drift between rhythmic feels by adding tuples and manual accents, but at some point you may be venturing out of what the time signature represents. If you are writing all eight note triplets in 3/4 you're really not writing in 3/4, but 9/8 as you have 3 eight notes to one beat.

Think of what a key signature represents harmonically and how you could write a piece in A major with the key signature of F major, but you'd need to add a lot of accidentals and you'd be writing against what was expected of a piece in F major. This is the same idea applied to time signatures but instead of harmonic expectations, there's metrical expectations.

Simple Time is the antonym of Compound Time. Simple Time means beats without dots and Compound Time means beats with dots, it is as simple as that.

You want a time signature to tell you three basics things. Does the beat have a dot or not, what is a beat and how many of them are there in a bar? Simple / Compound is the terminology that pertains to the question of whether a beat is dotted or not.

Simple is simply what it says. There is only one basic way the bar can be divided.3/4 is simple, as the basic count can only be 1-2-3, 1-2-3. (Your triplets 3/4 are 9/8, by the way). Compound is simply what it says - it's not simple! The bar can be divided in more than one way. Let's look at 6/8. It can be counted 1--2--, 1--2--, as in two time - duple time.Using the dotted quaver as the beat. Or, it can be counted !23, 456, 123,456.Using the quavers as the beat. So, it's a mix, with two different, but intertwined feels.

Compound usually has the 'basic' beat that can be sub-divided into thirds. So 6/8, 9/8, 12/8 are all regarded as compound. Your 3/4 into triplets is one of those compounds. 12/8 is sometimes written in 4/4, but with a legend at the top, turning it into a swing, beause it's easier to write and read that way. So simple (4/4) > compound (12/8).