In order for the main idea presented in this article to be the most useful and make the fullest sense, the reader should already have a firm grasp over all seven modes of the Major scale for all possible tonal centers.
In a nut-shell, Pitch Axis Theory acts as a guide during improvisatory soloing to inform the musician as to what scales will sound the most awesome over a series of changes (aka chord progressions). In order to obtain this guidance from Pitch Axis theory, the improvisee need only make two quick mental calculations in order to figure out what scale to invoke in his improvising.
1.) What is this chord progression's Pitch Axis?
-The answer to this question is most frequently found in either the Chord progression's initial key, or in whatever tone is most common to the chords of that progression. Once you determine pitch Axis, you know what tonal center you will be using in your improvising.
Example: Suppose that I start comping the following progression for you to solo over
G Dominant 7th---> A Minor 7th---> D Minor 7th--->G Dominant 7th
Right of the bat, it appears for a number of reasons that this progression's pitch axis is "G". It is the Starting tone or, "tonic" of this chord progression, and it is also a common tone to the first,second, and final chord in this progression.
2.) Now that I know my tonal center, what accidentals (sharps/flats) will be implemented in the scale that I am going to use?
The answer to this question is to look to the sharps/flats that are being used in the chord progression that you are to improvise over. So for our improvising for the above chord progression, we will use a G-Mixolydian scale because 1.) G is the progression's Pitch-Axis, and therefore our tonal center, and 2.) the chords in this progression have no sharps or flats, and G-Mixolydian happens to be the only modal scale built on G that possesses no sharps or flats.
So the next time you are playing over a series of changes, you will know what scale best fits over them, and you will be able to tailor your soloing around the chords that are being played through. (-: