I have read other questions, but this is a different situation.
I've been working as a web developer in this company for more than a year. I've found that listening to music via headphones while I work helps me to concentrate, helps prevent distraction from outside noises, make my day much more enjoyable, and improves the quality of my work due to my greater energy and motivation.
However, a couple of months ago, the administrator (she handles company finances and is my boss's wife) told me not to use headphones anymore. I asked why, but she declined to give a reason other than "because headphones here are not good," and told me I could close my door if I like.
The headphones I use are small ones designed for running (where safety requires you to hear outside noises). Some of my co-workers do listen to the radio via speakers, but since the music I listen to is not enjoyed by most of my co-workers, I feel uncomfortable playing it over the speakers.
How can I convince my employer to let me listen to headphones while I code?
EDIT/UPDATE: I tried in these days to close the door, and to listen to only to one ear, so I can avoid safety issues, being concentrated (because actually lots of noise comes out the door), giving the impression I am not necessarily listening to music and do my job. But my administrator came in the room... to do nothing but leaving the door open, on purpose, because she didn't came back to do other stuff. This is getting kind of ridiculous. At least I am doing my job. But I still can't figure out how to talk to her, because I don't want to appear unprofessional (even if thanks to you I am understanding that this is not really a employee-friendly environment.