David Peterson wrote:
> There are not a small number of vowel systems that have just four >
> vowels,where the four vowels are /a/, /e/, /i/ and /o/.

My conlang Tlapˇa has the same vowels and has some affixes that work
with vowelharmony.
Tlapˇa discriminates between front vowels (/i/ and /e/) and
back vowels (/a/ and /o/). So, for example: the non-control suffix
-nLuh (where `L' stands for Low vowel) can appear as /neuh/ if the
preceding vowel is a /i/ or /e/, and as /nauh/ if the preceding vowel
is an /a/ or /o/.

nachi  `see'  --> nachi-neuh `happen to see'
pah    `hear' --> pah-nauh   `happen to hear'

<snip the interesting examples>

> Is such a system feasible?
Why not?

> Do any four-vowel languages like the above even have a vowel
> harmony system?

I don't know about any natlangs, but as to a conlang: mine has, as
you can see. And it works too, I really like it.
I like it too the way it works in your lang.

> Or how about three vowel languages?   For some
> reason it doesn't seem likely.

Three vowels looks more awkward to me too.