It will depend on the school. For many schools it is part of their accreditation criteria that they cannot accept MS students who have not completed a four year bachelor program. This was true for instance when I went to Georgia Tech—often to the frustration of international students who had a three year degree that their home country considered equivalent to a four year program. So it will depend on the criteria for each school, but it is probably most common in the US that you must have a regionally accredited undergraduate degree before starting a graduate program at a major university.

I did a MSc with a BMus in piano performance. I went on and got a PhD in CS, afterwards.

Normally, when doing a Master's degree, you are required to do "levelling" courses to ensure that you know the basics. These courses are usually the core competencies from the BSc in that university, and don't count towards the Master's itself. It just so happens that, if you have the BSc from that university, you will also have the levelling courses automatically.

You can do the MSc without a BSc if you have enough professional experience.

I however already have a BSc in an unrelated field.