>the chord turns to d when the tune note is A

I read this as being some form of D chord...? (My C# suggestion above referred to a possible secondary dominant.) Hope Flora will come back with further explanation!

My professors used to constantly intone that Bach (& Co) were allowed to break the rules, citing the same reasoning; but one of them seemed to take a particular sadistic delight in reminding us of our inferiority. As if we needed to be told.