In the case of my own great-grandparents, whose surname was BRAUTMAN, they
just simply started using the name BRANTMAN instead over time. I suppose
they thought it was easier to pronounce. They did not take any formal action
to change their name, they just started using it, many years after having
been naturalized. There didn't seem to be any specific moment in time when
they made a definitive change, but more like a period of many years when
they were "in transition", BRAUTMAN in some situations and BRANTMAN others.
You could see the change over the years in the New York City Directory, and
there was even a few years when they were listed under both variants. By the
end of their lives, the change had pretty much taken hold, and it certainly
had for the next generation. Even my great-grandmother's official death
certificate is indexed both ways.
I suspect mine was not the only family who changed their name in this casual
manner. It sounds like there were a variety of ways and times and
circumstances that families changed their name, and not any one general
rule. Makes our jobs as genealogists a bit harder, but the challenge is part
of the fun, isn't it?
Shana tova l'kol,
Los Angeles, CA