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:

1. 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.
2. Retrograde (R): the original row in reverse order. The transposition of the retrograde is determined by the last pitch class of the row.
3. Inversion (I): the original row with the ordered pitch-class intervals reversed. The first pitch class determines transposition.
4. 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.

To follow up on pitch-class sets, previously talked about in Part 1:

First, a clarification that may help you to better understand inversions of pitch-class sets in general. The easiest way to visualize the inversion of a pitch-class set is to start with the top pitch-class member (right-most) of the pitch-class set, and working your way backwards, go up from that set member the same distance as you go down in the original set. For example:

To invert the set [5,6,9], start with pitch-class 9. Working backwards gives an ordered pitch-class interval of -3 (9 down to 6). In the inverted set, add +3 to 9 to arrive at the second inverted set member, 0. Continuing to work backward in the original set, 6 down to 5 is -1, so the third inverted set member is 0 + 1, or 1. In this way, [5,6,9] inverts to [9,0,1].

It is important to note, however, that this is not an inversion at T0I. This inversion relationship, or index number, is actually 6 (9 + 9; 6 + 0; 5 + 1).

The final two ideas that we need to discuss relating to pitch-class sets involve set classes and the prime form of a set.

A set class contains all the pitch-class sets (in normal order) that are related by transposition, and all the transpositions of the inversion of the set. For example, start with the randomly chosen set, [9,1,2]. The first twelve sets show all the transpositions of [9,1,2].

[9,1,2]
[10,2,3]
[11,3,4]
[0,4,5]
[1,5,6]
[2,6,7]
[3,7,8]
[4,8,9]
[5,9,10]
[6,10,11]
[7,11,0]
[8,0,1]

Now you add all the transpositions of the previous twelve sets.

[2,3,7]
[3,4,8]
[4,5,9]
[5,6,10]
[6,7,11]
[7,8,0]
[8,9,1]
[9,10,2]
[10,11,3]
[11,0,4]
[0,1,5]
[1,2,6]

The twelve transpositions and the twelve transpositions of the inversion form a single set class. We refer to a set class by its Prime Form. The prime form a set always starts on pitch class 0, and travels clockwise or counter-clockwise around the clock face by virtue of which direction starts with the smallest interval. In the above case, the inversion [0,1,5] starts with a smaller interval than [0,4,5], so [0,1,5] is the prime form of this set class.

Prime forms are usually written in parentheses, without commas. In this system, use “t” and “e” for 10 and 11. Using this system, the prime form of the above set class would be written (015). Using this system we can distinguish between a specific pitch-class set as it appears in a piece of music and the set class to which it belongs.

I started posting this as a comment to Basic Concepts for Atonal Theory, but decided it needed its own life.

Important additional comments/clarifications regarding interval terminology:

Unordered and ordered pitch intervals are the same except that ordered pitch intervals include  a + or – sign.

Pitch-class intervals can never be larger than 11, since a pitch-class assumes octave equivalence.

Ordered pitch-class intervals can be counted in either direction (up/down, clockwise or counter-clockwise). The ordered pitch-class interval from C to A is both +9 and -3. Usually the positive number is chosen (by convention).

Unordered pitch-class intervals drop the + and – minus signs. Therefore, the unordered pitch-class interval between C and A is (again) both 9 and 3. However, we choose the smaller number to name an unordered pitch-class interval, so we would only call it a 3.

By choosing the shortest distance to label an unordered pitch-class interval, we can say that an unordered pitch-class interval is the same thing as an interval class. An interval class includes all the pitch intervals that can be expressed as the same unordered pitch-class interval. For instance, unordered pitch intervals 15, 9, 3, 21, and 27 all belong to the interval class 3. (15 – 12 = 3; 9 is the inversion of 3; 21 = 21 – 12 = 9, which is the inversion of 3; 27 = 27 – 12 = 15 – 12 = 3) In this way, you can think of an interval class as being a collection of equivalent pitch intervals.