Note: You should read Constructing Major Scales (if you are unaware of how to do so) before reading this.
Recall that a major tetrachord consists of a 4 notes in ascending order, separated by the following sequence of intervals: whole step, whole step, half step. Also, remeber that a major scale consists of two tetrachords separated by a whole step.
Since all major tetrachords have the same construction it is fairly obvious that each major tetrachord exists in two major scales.
For example, take the C major tetrachord: C-D-E-F. We find this sequence of notes in two scales. The sequence occurs as the lower tetrachord of the C major scale (C-D-E-F-G-A-B-C) and as the upper tetrachord of the F major scale (F-G-A-Bb-C-D-E-F).
In the same fashion, the G major tetrachord (G-A-B-C) is upper tetrachord of the C major scale and the lower tetrachord of the G major scale.
We can keep building these relationships for all twelve tones used in Western music, and eventually form a pattern called the circle of fifths, named so because traveling clockwise, each movement represents a five-to-one relationship. Going counter-clockwise, it really becomes the circle of fourths.
See Soujirou's writeup for a beautiful ASCII diagram of the circle of fifths.
⟨⟨ Constructing Major Scales | 
