The only usage of "major circle" that I'm familiar with is the "major circle of fifths", a sequence of musical notes of which each is the relative fifth of the previous.
Starting at C (you can start on any note), the major circle goes C - G - D - A - E - B (Cb) - Gb (F#) - Db (C#) - Ab - Eb - Bb - F - C.
In terms of keys, each successive key on the chart has one more sharp (or one less flat) than the previous one. The three "doubled" ones can be written with either sharps or flats (for example, the keys Db and C# have the same actual tones, but are written with 5 flats or 7 sharps respectively).
You can also use the circle to tell you the order in which sharps (or flats) should be added. For flats, you start with Bb and work backward: Bb, Eb, Ab, Db, Gb, Cb, Fb (which is equivalent to E). For sharps, you start with F (sharp) and work forward: F#, C#, G#, D#, A#, E#, B#.
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