You're using the REST routes. 'artists/1/edit' is the expected result
for this.
If you want the default routes, you could just erase
map.resources :artists
and then the routing of artists/edit/1 will fall back onto the default
route, if this is still at the bottom of your routes.rb:
map.connect ':controller/:action/:id'
this way 'artists/edit/1' will work.
That doesn't follow the rails way of doing things. You will find your life a lot easier if you "go with the flow" in regards to rails traditions. To get that url you could just stop using map.resources and specify <%= link_to 'Edit', { :controller > "artists", :action => "edit", :id => i.id } %> instead.