Flat or Sharp …
Question: Does A# equal Bb?
Answer: Yes and no.
Actually, in reality, there is a really small difference in interval between sharps and flats. While sharps actually raise a tone, flats lower the tone. The interval by which they officially raise or lower the tone are not the same, i.e., they do not meet each other in the middle. So, from a scientific point of view, A# is not the same as Bb.
However, since Werckmeiser devised a tuning that allowed for all keys to be played on a single keyboard, and since Bach emphasized the tuning problems through “Das WohlTemperiertes Clavier” (commonly referred to as WTC), A# became the same as Bb. The writings of Mozart and Beethoven were actually influenced by WTC.
While A# became the same as Bb in the Werckmeister Tuning, there are actually “rules” as to whether you refer to a raised A or a lowered B as A# or Bb. From a practical point of view, if writing a scale down as raised intervals would contain more sharps than you would get writing a scale down as lowered intervals, you would use the lowered notation since that helps to lessen confusion amongst the performer.
