The anteroom probably was right outside the bigger room in which you received your musical instruction. Anterooms lead into larger rooms; they're often waiting areas that serve as gathering spaces before you're allowed entrance to a main room.
From Leo Tolstoy's War and Peace:
During his service, chiefly as an adjutant, Prince Andrew had seen the anterooms of many important men, and the different types of such rooms were well known to him. Count Arakcheev's anteroom had quite a special character.
From A Connecticut Yankee in King Arthur's Court, by Mark Twain:
Sir Uwaine, at a sign from his mother, went to the anteroom and called some servants, and meanwhile madame went rippling sweetly along with her talk.
And, from George Eliot's Adam Bede:
On entering the small anteroom which was called a dressing-room, but which Arthur really used only to lounge and write in, he just cast his eyes on the writing-table, and saw that there were several letters and packets lying there . . .