Music is neither inherently Jewish nor gentile. There may be a couple of exceptions to this statement. First, ta’amei mikra (trop notes) are musical patterns specifically intended for Jewish musical use. Second, some knowledgable people hold that some of the scarbala nigunim (used on the yomim nora’im at various points) trace back to nigunim of the levi’im in the beis ha’mikdash. I’m not sure if any other music qualifies as innately Jewish. All other music is “stam” music, unless such music’s use in avoda zara has rendered it off-limits. Gregorian chants, for example, cannot pass muster for use in shul. Otherwise, music appears to carry no innate issur. Certain types of music may raise issues of taste that affect a hashkafic perspective, but that is a separate issue.
The more difficult question becomes, is the takana against listening to instrumental music outside of a chasana (assuming there is such a takana, as the kitzur shulchan aruch appears to hold) applicable, or has it become one which is too hard for the observant Jewish world to uphold (as evidenced by the observant Jewish world’s performing, recording, sale and listening to instrumental music outside of chasanas that is pervasive today)?