One of the most significant developments in music theory during the Twentieth Century was the "Second Viennese School," spearheaded by German composer Arnold Schönberg. He developed a technique called serialism, which kept music devoid of tonality by using all twelve tones of the chromatic scale equally.
This unnatural equality was produced using a tone matrix.
A tone matrix is a 12 x 12 grid. Each row or column of the grid is called a series, hence the name of the system. There are four steps to filling out the matrix.
One begins a tone matrix by randomly (or not) lining up all twelve tones of the chromatic scale and putting them on the top line of a grid. This is called the prime series, or P0.
Then, down the left column, the tone row is written using the same intervals, but upside-down. This is called the inverse series, or I0.
Next, the tone row is transposed, so that the intervals remain the same, onto the second line. Since this is a transposition of the prime series, it is notated by a "P" and the number of semitones in the transposition: in this case, P6.
This process continues to fill out the entire matrix.
When composing in a serialist manner, a composer must use a full series before he or she is allowed to repeat a note. There are forty-eight possible series:
The prime series, or any of its transpositions (i.e. any row going left-to-right).
The inverse series, or any of its transpositions (i.e. any column going top-to-bottom).
The retrograde series, or any of its transpositions (i.e. any row going right-to-left... the first row would be R0).
The retrograde-inverse series, or any of its transpositions (i.e. any column going bottom-to-top... the first column would be RI0)