In bustling Galata step off the street into a tranquil garden with a cemetery. Inside, there's a beautiful old building with a wooden chamber in which dervishes hold ceremonies at times. Mevlevi ceremonies used to be performed for visitors on the last Sunday of each month (but check if this is still the case).
Galata Mevlevihane, Galip Dede Caddesi.
Updates
The ceremony is wonderful, but PLEASE respect it and don't be one of the many other tourists who ignored the multiple requests not to take flash photos and spent the whole time doing just that and moving around disturbing everyone else in their selfish desperation to get the perfect photo. You can buy some lovely postcards etc afterwards to remember it by. You can find all the times at www.emav.org. tickets cost 25YTL. I thoroughly recommend it.
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