I originally wrote
A "transcription" and an "arrangement" are not at all the same thing.
But it seems that this was not quite correct.
Merriam-Webste definers the musical sense of "transcription" as:
2 a : an arrangement of a musical composition for some instrument or voice other than the original
2 b : a recording (as on magnetic tape) made especially for use in radio broadcasting
4 a the arrangement of a composition for a medium other than that for which it was originally written.
So it seems that a transcription, in this sense, is a kind of arangement
A transcription means writing down in some form of musical notation, or just words, all or some aspect of s musical work. In some cases it means just writing down the lyrics of a song. In some cases it means jut writing down the notes. But it seems that it often mean reworking the music for different instruments, which may mean changing the notes to accommodate the ranges of those instruments. It doesn't matter what tools or technology is used, it is the result that matters. In many cases a transcription does not contain any original content, and so is not able to be protected by copyright in US law.
Merriam-Webste defines the musical sense of "arrangement" as:
2 b : a piece of music that has been hanged so that it can be performed by particular types of voices or instruments
6 a the adaptation of a composition to voices or instruments, or to a new purpose.
An arrangement is often a different but closely related piece of music. It may be as simple as shifting to another key. It may involve adding or removing sections of the work, adding or removing voices, adding or removing repeats, or altering the lyrics. There are special provisions in US copyright law doe cover versions to be made without the permission of the copyright owner, but these do not always apply. An arrangement may be original enough to get its own copyright. In any case, an arrangement of a PD work does not need permission from anyone, because no one owns the copyright to the original work.
Statement from the Question
I created a new work that evolved from the public domain work. I applied a great deal of creativity in choosing which music to assign to which instrument.
That sounds like an arrangement (not a transcription) that is quite likely protected by its own copyright. Copyright protects an eligible work as soon a it is "fixed in a tangible form" and a computer file or printout counts.
Whether a work is creative enough to be protected is a very fact-intensive determination, and one that this site cannot make. One might, in such a case, want to consult a lawyer with music copyright expertise. If this happens in the US, one might want to register a copyright in the work.