Modulating without modulatory chord

[theholygideons]

I'm analyzing the prelude in c sharp major from the well tempered klavier. Can someone explain how bach modulates without using any chords with the leading tone?

https://i.imgur.com/kFiruFz.png (here's the picture larger)

[klavieronin]

There are, of course, different ways to analyse a piece of music but to me the modulations aren't happening where you have circled. The first one, into the dominant key (G#), strictly speaking doesn't happen until bars 14-15 with a V7 - I cadence (the F double sharp is the leading tone). The next, into d#, is at bar 22-23, again a V - i cadence (the C double sharp is the leading tone). Hope that helps.

[theholygideons]

How come, if he hasn't modulated to d sharp minor yet in bar 16, he already introduces f sharp and b natural in bar 16 and g sharp in bar 18, which are not in the previous key, g sharp major, but in the key d sharp minor? Are you saying that these are all chromatic notes?

[brogers70]

How come, if he hasn't modulated to d sharp minor yet in bar 16, he already introduces f sharp and b natural in bar 16 and g sharp in bar 18, which are not in the previous key, g sharp major, but in the key d sharp minor? Are you saying that these are all chromatic notes?

I would say, yes, those are just chromatic notes; they are indeed setting the stage for a modulation, but you don't know that yet. For a new key to be established, you need a clear cadence in the new key. So I would not call the modulation until the V-I or V-i cadences that klavieronin identified above.

[dw4rn]

I'm not sure I know exactly what you mean by "modulatory chord" or "chords with the leading tone". Are you referring to the use of a "pivot" chord to somehow smooth the change from one key to another? In that case, I would say that these are not always needed, and that this is far from the only example where a composer engages in a much more straightforward juxtaposition of keys. 

[theholygideons]

My understanding is that a modulation commonly employs a chord common to both keys followed by a modulatory chord with the leading tone. In the example I gave, I thought that the latter was missing. Now, it seems, owing to the input of others, that the modulation occurs later than I thought.