I thank you for your superb pronunciation of "Tschaikovsky"'s name. Transliteration between languages with different orthographies is a vexing task, especially if one wishes to retain ability to actually recreate these sounds. We have certainly seen this since the end of the cold war, and the cultural upheavals in China.
What has linguistics to do with art? Ultimately, quite a bit; immediately, not much. It's fun, though. I think speakers of English would do well to spell the Guy's name "Pyot'r Ill'ich Chaikovskii". That corresponds closely to English phonemic structure.