During the first part of episode 9 of Boogiepop Phantom, we meet Saki Yoshizawa, a high school student who plays piano. When her cell phone rings while they are on their way to a piano lesson, Saki's kouhai points out that Saki is using "Bach's étude" as her ringtone, and Saki, who is a second year student, replies that it is her "test piece" for "next year."
This appears to also be the piece that plays in the background during the students' piano lesson. (The music starts playing while Saki's kouhai is having her lesson, but given that it is Saki who has to prepare the piece, it is uncertain whether this is what the kouhai is actually playing; the kouhai's responses in the conversation about the ringtone also do not strike me to be those that someone playing the same piece would have.)
I do not remember J.S. Bach to be a composer who wrote "études" for keyboard: Bach did write pedagogical material, but I cannot recall a single piece of his that is labelled as an étude. I also tried searching for
bach "étude" site:imslp.org to see if the IMSLP archive had something relevant, but nothing came up.
With this in mind, what is the actual title of the Bach piece that Saki is learning and using for her ringtone? If the piece is part of a multi-movement work, I would also like to know the specific movement that is quoted in Boogiepop Phantom,