Hi... this is just a quick reply because I am in a direct line with Emma Louise Vaeth through her father Johann Vath in Syracuse NY late 1880s. This line includes Ludwig Vath. The "e" seems to have been added after naturalization. US census lists Baden.
In recent days several specific towns have come to light so I shall be busy.
Fett spelling may be found on IGI via familysearch. org
Judy