These upward and downward “leading tones” act like a pair of pincers, powerfully converging on the note between:

Now see how Bach harnesses this pincer power:

The operative key coming into the cascade we’re looking at is A major; the high A that begins the cascade is therefore that most stable of scale steps, the tonic. But this A is also the upper note of a diminished third, and in this highly unstable capacity it tugs strongly downward to G-sharp (abetted by the upward thrust to G-sharp of its pincer-partner, F-double-sharp). In a stunning move, Bach uses this tug from the note A to the note G-sharp to pull the key down from A major to G-sharp major. The strongest modulations occur between “remote” keys, those that have few notes in common. Keys don’t come much more remote from one another than A major and G-sharp major (they have only two notes in common):

When Bach modulates from the former to the latter, we feel the world fall away beneath us— only to reestablish itself a few feet down.