Copyright law can vary from jurisdiction to jurisdiction, but a creative work is typically considered to be copyrighted as soon as it is created. In order to copyright music under this principle, all you need to do is record it or write down the notation. Once you've done that, it should be automatically copyrighted, and thus protected by law. It is still possible to register a copyright in many jurisdictions, and while this isn't required, it is often considered as prima facie that your copyright is valid, or rather proof that your copyright existed prior to that of any other claimant.
There are a number of international agreements regarding copyright law. One of the primary agreements, known as the Berne Convention, is what may allow you to copyright music as soon as you've created it. So long as your creation is original, it doesn't matter whether you've written a story, recorded a song, made a movie, or created any other tangible creative work, your work will be protected. Many countries may require that the copyright symbol be included on all creative works in order for this protection to exist, though others — such as the U.S. — have made the inclusion of this mark entirely optional.
While the process to officially copyright music isn't strictly necessary, it can help if there are any questions as to ownership at a later date. The process isn't an incredibly complicated one, though your own personal knowledge, abilities, and time will dictate whether you choose to do it on your own or get help. If you opt to find assistance to copyright music, many lawyers specialize in this field. Going this way may not only help to ensure that the process goes smoothly, but you will also know that everything was done correctly and all the paperwork is in place, should any copyright questions ever arise.
If you want to copyright music on your own, the process can vary widely, depending on the jurisdiction in which you live and work. The process can be fairly straightforward in the U.S., with all the paperwork you need being available from the website of the national Copyright Office. Those without access to the Internet may also obtain all the necessary paperwork via the U.S. mail, by contacting the Copyright Office and requesting the correct forms. It may then be as simple as filling out the forms and mailing them back, along with a registration fee and a copy of the protected work.