There is no specific law in California which establishes the decibel level for noise.
However, the landlord is probably correct that a tenant cannot conduct business within a residential apartment, and having a music studio would constitute having a business.
Moreover, the landlord has a duty to ensure quiet enjoyment for all tenants within the apartment building.
There are questions as to whether this music studio is authorized by local zoning/use regulations. If it is not, then it may not be appropriate to have a studio there. Regardless of the law, you should try to find a way to work this out without the law. It would be hard to believe that this studio would be worth the hundreds or thousands to prove your rights in court. If it is, then find another place to live.
This answer does not create an attorney client relationship. Consult an attorney before you decide on a course of action. A review of all documents and a consultation at a minimum is necessary to render effective legal advice.