*Classical serialism* typically refers to the 12-tone composition technique developed by Schoenberg and his followers.

The basic premise of the 12-tone system is the *row*, which is an ordered arrangement, or set, of pitch classes. Each pitch class occurs once, and only once. The row has four basic forms:

*Prime*(P): the original ordered set (row). The transposition of the prime form is determined by the first pitch class of the row. The prime form of the row is usually the first occurrence of the row.*Retrograde*(R): the original row in reverse order. The transposition of the retrograde is determined by the last pitch class of the row.*Inversion*(I): the original row with the ordered pitch-class intervals reversed. The first pitch class determines transposition.*Retrograde**Inversion*(RI): the inversion in reverse order. Like the retrograde form, the last pitch class determines the transposition level.

The *matrix* contains all 48 versions of the row – the four basic forms, each with 12 transpositions. Although theorists disagree on how to construct and label rows, we will adopt the technique that begins with the prime form of the row transposed to pitch-class 0.

As an example, start with the initial row: 2 11 10 5 6 8 4 3 0 1 7 9.

Transposed to pitch class zero: 0 9 8 3 4 6 2 1 10 11 5 7.

Start by filling in the top row of a 12×12 matrix with the P-0 form of the row.

Then fill in the inversion of P-0 going down the left-most column. Remember that you can invert either by reversing the ordered pitch-class intervals, or by subtracting each pitch class from 0.

Finally, fill in the remaining prime transpositions of the rows, beginning with the pitch class in the first column, and using the interval between it and pitch class 0 as your interval of transposition.

Notice how pitch class 0 moves diagonally through the matrix. You can use this property to check yourself. You shhould also check yourself by making sure that you haven’t repeated any pitch classes in any row or column (Sudoku-like).

The row across the bottom, starting with 5 and moving left-to-right, is P-5. Moving across the bottom right-to-left is R-5.

The row starting from the top with pitch class 4 is I-4. The reverse of that row (starting at the bottom with 9) is RI-4.

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