Hi,
As I learn music theory, I know that Moonlight is in C# minor.
But as I test that, and play just C#, at some point during the 1st movement the C# suddenly sounds off key for a while. If you need the spot where it goes awry, I will find the measure. But I was hoping that all you experts would know the spot already.

Does the piece change key? It doesn't indicate it anywhere on the sheet music. What is going on exactly? Please feel free to use technical language.
Thank you!