Ahhh, that's not /aasd then, is it? Seriously, just anchor the match - ^ for the beginning, $ for the end:

$var =~ s{/[^/]+$}{};
[download]

jeffa

L-LL-L--L-LL-L--L-LL-L--
-R--R-RR-R--R-RR-R--R-RR
B--B--B--B--B--B--B--B--
H---H---H---H---H---H---
(the triplet paradiddle with high-hat)