As a medical doctor, I suggest AINE`s like diclofnac, 50 mg, 12/12H, "per os", as first line therapeutic. If this does not result, then prednisolone, 1 f. IM 1 or 2 days only and then diclofenac. IF you have not gastric ulcer or gastritis. For gastric protection, during this treatment, you may take Omeprazole, 20 mg/day "per os". This is the best medical treatment for finger tendinitis.
As pianist, I think something is or may be wrong with your technic to cause tendinitis. Perhaps muscular tension or a bad position of your hand, I dont know...
Best wishes.