Please read the QUOTING section in bash (1) ( man bash ).
In the first example
echo "C:\Users\Ted\Documents\Unix\Scripts" | xargs echo
the shell interprets (and then removes) the backslash in double quotes, so the first echo command only sees the line C:UsersTedDocumentsUnixScripts .
In your second example
echo 'C:\Users\Ted\Documents\Unix\Scripts' | xargs echo
single quotes correctly protect backslashes, as you can see by running only the first echo command without | xargs echo | xargs echo .
In all other examples
cyg0 C:\Users\Ted\Music\Enigma cyg1 C:\Users\Ted\Music\Enigma ...
since the argument ( C:\Users\Ted\Music\Enigma ) is incorrect, the shell is interpreted again and removes the backslash before the command ( cyg0 , cyg1 , ...) ever sees them. AND...
I want to be able to type C: \ Users \ Ted \ Music \ Enigma without quotes.
Sorry, this is not possible. If you do not put the backslash in single quotes, the shell will interpret each of them as a quotation mark for the subsequent character, and then remove the backslash.
You have several options:
Use single quotes on the command line to protect backslashes:
cyg4 'C:\Users\Ted\Music\Enigma'
Specify each backslash character separately, doubling it, for example.
cyg4 C:\\Users\\Ted\\Music\\Enigma
Use slashes instead:
cyg4 C:/Users/Ted/Music/Enigma
Option 3 may indeed be your best bet. Unfortunately, the backslash character is a very special character for all Unix shells, and this causes a lot of link problems when using DOS / Windows paths. But it is not known that DOS / Windows also enjoys using a slash ( / ) as a path separator. Give it a try.
Note. Of your few cyg * functions, only cyg1 and cyg4 are correct. Others use incorrect quoting. cyg1 is useful with only one argument, since it combines all the arguments together on one line, and cyg4 can accept and pass multiple arguments to cygpath. However, since cyg4 only passes its quoted arguments, it does not add any value; it really is no different from, for example,
cygpath C:/Users/Ted/Music/Enigma
Good luck.