I would also advise using established modules for this, but here's another regex-only way to remove the first n directories from an absolute path. It looks like you're using a *nix path, and I haven't made the effort to define a proper directory pattern in $dir (I don't think that any character other than a / is allowed), so you will have to fix up the [^/] character class accordingly.
c:\@Work\Perl\monks>perl -wMstrict -le
"my $filespec = '/usr/share/directory/piano_book.mp4';
print qq{absolute: '$filespec'};
;;
my $dir = qr{ / [^/]+ }xms; # fix [^/]
;;
my $n = 2;
die qq{'$filespec' could not be made relative}
unless $filespec =~ s{ \A (?: $dir){$n} }{.}xms;
print qq{relative: '$filespec'};
"
absolute: '/usr/share/directory/piano_book.mp4'
relative: './directory/piano_book.mp4'
Please see perlre, perlretut, and perlrequick.
Update: Added error check to substitution in example code.
Give a man a fish: <%-{-{-{-<
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