Firstly, let me admit that I have never been to a museum that curates history of Indian music. When I stumbled upon this Bangalore-based museum on the web, I didn't have to think twice to add it to my list of must-visit places.
Center for Indian music experience (IME) is an attempt to trace the history and evolution of Indian Music to make Indian music, including classical music, relevant in this modern age. The exhibits are curated based on themes that can provide perspective on the timelines of various events happened in India.
If you want to be a patron and support this initiative, check the donations page for different ways you can contribute.
Also, do check out this project's director's TED talk here.

Manasi Prasad connects us to our roots by infusing the beauty of music from different times to present day creating relevance for us to understand it nuances better. Her musical talk takes us through the harmonious tunes of tradition woven into a modern context.