You have two possibilities to keep the music playing while your users are browsing through your site, so that the music does not stop when changing the pages.

1. Using Pop-Up Player

Enable the Pop-up Player in the Audio Player options. This will open the audio player in a Pop-Up window.



2. Ajaxify your Theme

The other possibility is that you need an ajaxified theme, where only a part of the page reloads and not the whole page. 
I found some plugins, that makes your wordpress site ajaxified. I test it in twenty eleven and it was working like a charm for me: