Why different half note time signatures at different tempo extremes?

I've seen the half note take the beat in 2 tempo extremes, the extremely slow and extremely fast. And I see different time signatures more often in these, 2/2 often in extremely fast and 3/2 often in extremely slow. I can see why it occurs in both extremes. In extremely fast tempi, the momentum is very fast, so much so that it often feels like eighth notes but maybe the notes aren't quite furious enough to be designated Presto by the composer even though that's what it feels like so you get an Allegro in 2/2 instead of a Presto in 4/4. ex. Pathetique Sonata First Movement by Beethoven

In extremely slow tempi the momentum is very slow, so much so that the quarter note actually feels too fast to be the beat. The quarter note feels like your normal eighth note. So instead of an Adagio in 3/4, you get a Largo in 3/2. ex. Egmont Overture, Beethoven(albeit, only the introduction of this feels slow, probably because it's in French Overture form(which often forgos the second slow section))

Both of these situations have to do with the quarter note feeling like the eighth note. But why is it more common for me to see 2/2 in pieces with the speed of a Presto such as the Pathetique Sonata and even in Prestos themselves such as the Presto of Chopin's Piano Sonata in C minor(I mean the presence in Prestos could simply have to do with the fact that duple meter is more common than quadruple meter in Prestos(most Prestos I've heard are in 2/4, though some are in 3/4 or 4/4) and 3/2 in pieces that are super slow when in theory, any half note time signature could be used for both tempo extremes?

Fast 2/2 examples:

First is Allegro and the second is Presto

Slow 3/2 examples:

I virtually never see 2/2 in slow pieces or 3/2 in fast pieces. I wonder why that is though? I mean theoretically any half note time signature has 2 possible interpretations, a slow one and a fast one. Is it just centuries old convention that 2/2 is a cut 4/4, ergo it's faster and 3/2 is an expanded 3/4, ergo it's slower? And the overall frequency differs as well. From Renaissance to Romantic, I see about the same frequency for 2/2, maybe just a bit more common in Beethoven and beyond. I've even seen differences in the time signature of editions of Impromptu in Gb Major op. 90. no. 3 by Schubert, where the German editions have this dual 2/2 time signature and other editions have either a single 2/2 or a single 4/4 time signature instead. But the frequency change of 3/2 is very steep, from common in the Renaissance and Baroque to almost non-existent by the times of Chopin and Schubert, with Beethoven being one of the last composers I know to write anything in 3/2.

Are you sure that this doesn't just happen because composers don't want to write the time signature as 6/4 because it might be misconstrued as being in 2 rather than in 3.  Writing it in 3/2 makes it clear that the bar is divided in 3.  Anyway - that's my reasoning behind it.

3 minutes ago, PaperComposer said:

Are you sure that this doesn't just happen because composers don't want to write the time signature as 6/4 because it might be misconstrued as being in 2 rather than in 3.  Writing it in 3/2 makes it clear that the bar is divided in 3.  Anyway - that's my reasoning behind it.

 

That might explain why 3/2 is used for very slow tempi, but it doesn't explain why even though in theory 2/2 could be interpreted as being slow like 3/2 is, in practice 2/2 is virtually always used for the very fast tempi.

14 hours ago, caters said:

But why is it more common for me to see 2/2 in pieces with the speed of a Presto such as the Pathetique Sonata and even in Prestos themselves such as the Presto of Chopin's Piano Sonata in C minor(I mean the presence in Prestos could simply have to do with the fact that duple meter is more common than quadruple meter in Prestos(most Prestos I've heard are in 2/4, though some are in 3/4 or 4/4) and 3/2 in pieces that are super slow when in theory, any half note time signature could be used for both tempo extremes?

I don't think there's a good answer to this question, because we would have to survey composers long dead as to their reasons for using 2/2 vs. 4/4 vs. 3/2, etc. It probably has something to do with the time signatures and tempi of popular dance music of the early Baroque period. For example the courante (at least the French version) was in a slow and stately 3/2 time. The bourée and gavotte were in cut or double time, and were faster, more lively dances. These time signatures and tempi likely became associated with each other when early Baroque composers were in their heyday, and the practice continued over time.

There is something to be said of the different beats, however. For example, a song in 2/2 and the same song in 2/4 would have different beats emphasized, and my guess is that the 2/2 would "sound" faster, even if it really isn't—something of an optical illusion for the ears. But that's all very subjective.