Towards the grand challenge of achieving human-level manipulation in robots, playing piano is a compelling testbed that requires strategic, precise, and flowing movements. Over the years, several works demonstrated hand-designed controllers on real world piano playing, while other works evaluated robot learning approaches on simulated piano playing. In this work, we develop the first piano playing robotic system that makes use of learning approaches while also being deployed on a real world dexterous robot. Specifically, we use a Sim2Real2Sim approach where we iteratively alternate between training policies in simulation, deploying the policies in the real world, and use the collected real world data to update the parameters of the simulator. Using this approach we demonstrate that the robot can learn to play several piano pieces (including Are You Sleeping, Happy Birthday, Ode To Joy, and Twinkle Twinkle Little Star) in the real world accurately, reaching an average F1-score of 0.881. By providing this proof-of-concept, we want to encourage the community to adopt piano playing as a compelling benchmark towards human-level manipulation in the real world. We open-source our code and show additional videos at www.lasr.org/research/learning-to-play-piano .