Currently there are 2 major organizations that regulate live streaming music licensing: Ascap and BMI.
Typical license costs around $100 / Year. Please contact Ascap and/or BMI for further information on how to obtain proper licensing to play music that accompanies your live stream classes.
Ascap - http://www.ascap.com/licensing. With an annual ASCAP music license, you can play millions of songs from our 565,000 songwriter, composer, and music publisher members, however you want - including live music by a band or DJ, or recorded music played off a CD, iPod, online streaming service, or most other means. Your license fee goes straight back to our members, so they can keep making the music that your customers love.
BMI - http://www.bmi.com/licensing. BMI offers the ability to license most businesses and organizations online. Simply choose your business type and click “Apply for a License”, then enter your Username & Password, which are provided for you on your correspondence from BMI or by calling a BMI representative. Once licensed, you can also pay your fee and manage your account online. If your business type does not yet have BMI online licensing capability, you can download a license from the site and return it to BMI at 10 Music Square East, Nashville, TN 37203 or call a BMI representative at (888) 689-5264 to license.