This article is from the Final-Fantasy and other Square Soft Games FAQ, by firstname.lastname@example.org (Nick Zitzmann) with numerous contributions by others.
First, you need to be able to speak the language of the Lefeinish in
order to deal with them.
If you can't speak the language of the Lefeinish, you can by first
finding the Stone Slab, and taking it to Dr. Unne in Melmond. The Slab
is on the top floor of the Sea Shrine, and it's rather hard to find, but
you should be able to turn it up.
Once you do this, you can deal with the Lefeinish. Talk to all of them,
and one of them will give you a chime. Once you have this chime, you
will be able to enter the Mirage Tower.