Thinking about it, I realize I don't know much about the Eastern Roman Empire's record when it comes to their academia, gender/sexual norms, or religious homogeneity, i.e. degree of "freedom of conscience" within practicers of the Orthodox faith, as well as social policies toward non-Orthodox (Latins, Jews, etc). I have some vague understanding of their political system and political economy, though I didn't know they performed so well, relatively speaking, in the latter (though, AIUI, they did have far less per capita "demand" for slavery than their islamic neighbors, which is a plus for our purposes). I'm kind of curious what @Carp or @LSCatilina have to say on this... or I suppose any of this board's many byzantine enthusiasts, for that matter.
The Byzantine Empire was religiously oppressive and culturally not particularly dynamic or creative. In all its history, it produced little of value in terms of poetry, philosophy (apart from Gemistus Pletho), scientific advancement or original thought. For my money, I'd probably go with Persia, in the period from c750AD to 1750. The achievements of Persia in the arts, in poetry, in philosophy, in the Sufism, were spectacular. Some of the more interesting and esoteric Sufi/Shia/Ismaili sects led a pretty wild sexual life, too. And the poetry of the Sufi love poets is full of reference to love and wine. Moderns would probably find much to enjoy in that environment. Although of course there was a deep spiritual side too which would require quite a bit of adjustment for a modern secular westerner, but it's not an impossible jump by any means.