Looking at the score for Beethoven's 9th symphony, it surprises me how the second movement is written in a breakneck 3/4 at 116 dotted half notes per minute.
Why did Beethoven opt for this 3/4, rather than halving all notes values and going with 3/8 at 116 dotted quarter notes per minute?
Because it is a Scherzo. In Beethoven's time, a Scherzo was expected to be a "light-hearted" and fast-moving humourous composition in the rounded-binary form. It had some characteristics, but three chief ones were as follows:
And so he chose to go along with the convention and respect it. Beethoven has written some more Scherzi before this, starting with the 3rd movement of his Piano Sonata No. 2 in A Major, Op 2 No. 2 published in 1796. You can find the same characteristics in that. Or the 3rd movement of the Piano Sonata No. 3 in in C Major, Op 2 No. 3. Of course, variations to this form began to appear later, and the rules began to be broken once in a while, but the form endured.
So, by convention, 3/4 was the right choice. However, Beethoven wanted more. The difference between 3/4 and 3/8 is on the emphasis. In 3/4 you have three quarter notes, and each quarter note is a beat, and the emphasis is on every one of them. In 3/8, you have three 8th notes instead, and the emphasis is on the first beat only.
His desire to respect and defy the convention at the same time however can be observed in the choice to embellish the first beat of every bar by adding that flute note at the top which makes it sound like a 3/8 composition, with the emphasis on the first beat. This was very novel at the time and can still thrill today, nearly 200 years later.
You can read more about the form here: https://en.wikipedia.org/wiki/Scherzo
The conventional structure for a symphony or sonata in the Classical period was four movements, of which the inner two are a slow movement and a minuet & trio. The minuet & trio movement is written in 3/4. One way the fashion changed was for the minuet & trio to be played much faster than anyone could dance a minuet to.
An example is the third movement of Beethoven's Symphony 1 in C, op. 21. This is to be played very fast --- Breitkopf's score gives it the metronome mark of a bar to 108. However, Beethoven calls it a minuet (using the spelling Menuetto, which is neither the German Menuett nor the Italian minuetto), and accordingly notates it in 3/4.
Later, it became usual to call such a movement a Scherzo. It's just that it never became usual to notate in dotted crotchets (quarters). The convention to notate in dotted minims (halves) had already been established.
Convention. And because music notated with an 8th beat is not automatically twice as fast as music notated with a quarter beat. For instance marches were traditionally written in 2/4 time, nowadays they're quite likely to be notated in 'cut time' 2/2. This doesn't mean marches used to be twice as fast.
