You must contact the publisher/producer and ask for permission. They will ask you questions about what you plan on using the music for, and whether you are profiting from its use. They will charge you accordingly.
The venue needs a license from the performing rights organizations, such as ASCAP and BMI.
In music, the lyrics and music can be registered separately or together, and sound recordings have their own rights.
It is not legal. If you do not own rights to the music it is copyright infringement.
Copyright, patents, rights in music, and performance rights are all aspects of intellectual property
In the US, copyright is addressed in US Code Title 17. Music downloads are still called "phonorecords" in Title 17, and the rights pertaining to them are called mechanical rights. Downloading music is duplicating a phonorecord. Uploading music is distributing a phonorecord.
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Performance rights are generally handled by the performing rights organizations, ASCAP and BMI.
The Mechanical Copyright Protection Society entered into an agreement with the Performing Rights Society, and now is part of PRS for Music.
The national anthem is public domain, nobody has copyright on the words or music.
The book has a copyright. The music has a copyright. The musical production has copyright. The movie has a copyright. The sound track has a copyright. Music not used in the movie has copyright. The play was first produced on Broadway in 1957, meaning it is copyrighted until 95 years later. The movie was produced in 1962 and has 95 years of copyright. The sound recordings of the music produced prior to 1973 have no federal copyright but are protected by state laws until 2067, not including those works that were also part of the 1962 dramatic audiovisual work, which are covered by federal copyright for 95 years.
Your music would be protected by copyright as soon as it was written down or recorded, and a publisher is not immediately necessary. You may wish to become a member of a performing rights society, however.
While the copyright in the sound recordings (master rights) are owned by Universal Music Group, the copyright in the compositions (publishing rights) are owned by Fifty Six Hope Road Music/Odnil Music and administered worldwide by Blue Mountain Music. See http://www.bluemountainmusic.tv for further information.