The Secret Sense by Isaac Asimov.
The sense is the ability to see electric fields.
In the story the Earthman Lincoln Fields is talking to a Martian Garth Jan. Fields is a music afficionado and the story starts with Lincoln pitying his friend because Martians are mostly tone deaf and cannot appreciate music. Jan reveals that Martians have other sense and tells Fields:
Fields opened his mouth to speak but the Martian motioned him testily into silence. “I know. You’re going to babble foolishness about sunsets or something. But what do you know of beauty? Have you ever known what it was to witness the beauty of the naked copper wires when an AC current is turned on? Have you sensed the delicate loveliness of induced currents set up in a solenoid when a magnet is passed through it? Have you ever attended a Martian portwem?”
Jan reveals it is possible to grant Earthmen this sense, but only for five minutes and then the sense is lost forever. Fields forces Jan to do this for him and for five minutes is able to enjoy the experience of hearing a Martian portwem play:
There beat upon his brain something that consisted of pure waves of enjoyment-something that lifted him out of himself and pitched him headlong into a universe unknown to him earlier. He was falling through an endless eternity of- something. It wasn’t sound or sight but it was-something. Something that enfolded him and hid his surroundings from him-that’s what it was. It was endless and infinite in its variety, and with each crashing wave, he glimpsed a farther horizon, and the wonderful cloak of sensation became thicker-and softer-and more beautiful.
Then the five minutes is over and Fields is left broken and bitter that he can never experience it again:
And within, that tiny, bitter voice, repeating over and over again, “You entered a normal man! You leave blind-blind-BLIND.”