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Key changes for modulations

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@JSSchreiber wrote:

Hello !

I am currently having some students transcribe the Moonlight Sonata.
Measure 10 there is a modulation from E major (III) to E min (or E min as iii of C major).
I don’t find this chord or any equivalent in the key of C#min.
This modulation would require a change of key. Is it possible?
It seems it is possible because the transcription of Moonlight sonata that serves as an example in TABS shows a change of key between the parts (C#min to E min).
Anyone can tell me how I can do this change of key without changing the rest of the chords? Or how to do this modulation?

Thanks!

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