Pitch-Class Sets

When we talk about a group of pitch classes as a unit, we call that group a pitch-class set, often abbreviated “pc set.” Any group of pitch classes can be a pitch-class set.

Normal Order

Normal order is the most compressed way to write a given collection of pitch classes, in ascending order. Normal order has a lot in common with the concept of root position. Root position is a standard way to order the pitch classes of triads and seventh chords so that we can classify and compare them easily. Normal order does the same, but in a more generalized way so as to apply to chords containing a variety of notes and intervals.

Following are a mathematical and a visual method for determining normal order.

Mathematical method

Occasionally you’ll have a a tie for largest interval in step 3. In these cases, the ordering that is most closely packed to one side or the other is the normal form. If there is still a tie, choose the set most closely packed to the bottom.

Visual method (clock face method)

If you don’t like the process described above, the video in Example 1 clearly explains how to use the clock face to quickly find normal order.

Example 1. Using the clock face to find normal form.

Transposition

In post-tonal music, transposition is often associated with motion: take a chord, motive, melody, and when it is transposed, the aural effect is of moving that chord, motive, or melody in some direction. In two disconnected passages from Claude Debussy’s La cathédrale engloutie, the opening motive <D, E, B> or <2, 4, 11> is transposed four semitones higher to <F♯, G♯, D♯> or <6, 8, 3> in m. 18 (both in blue in Example 2), representing the cathedral’s slow ascent above the water. When we hear the passage at m. 18, we recognize its relationship to the passage in m. 1 because the same set of ordered intervals return, but starting on a different pitch. Transposition preserves intervallic content, and not only that, it preserves the specific arrangement of the intervals.

Example 2. Transposition of a motive, ordered and unordered.

The same motive is preserved in a more obscured fashion going from m. 18 into m. 19 (in red in Example 2). Here, the unordered pitch class set is transposed, but not the ordered set from before. This may not look as much like a transposition as the first two sets of motives, but if the pitches are put into normal order, the transposed intervallic relationship becomes clear.

Transposition is often abbreviated T_n, where n, the index number of a transformation, represents the ordered pitch-class interval between the two sets. Transposition is an operation—something that is done to a pitch, pitch class, or collection of these things. Alternatively, transposition can also be a measurement—representing the distance between things.

Transposing a set

To transpose a set by T_n, add n to every integer in that set (mod 12, meaning numbers wrap around after reaching 12, like on a clock face).

Given the collection of pitch classes in m. 1 above and transposition by T₄:

[latex]\begin{alignat*}{2}&& [11, 2, 4]  \\ {}+ && 4\ 4\ 4\ \\ \hline && [3, 6, 8] \\ \end{alignat*}[/latex]

The result is the pitch classes in m. 18.
T₄ [11, 2, 4] = [3, 6, 8]

Identifying transpositions and calculating the index number

To determine the transpositional relationship between two sets, subtract the first set from the second. If the numbers that result are all the same, the two sets are related by that T_n. For example, to label the arrows in Example 1, an analyst would “subtract” the pitch class integers of m. 1 from the pitch-class integers in m. 18. Note that both sets should be in normal order.

[latex]\begin{alignat*}{2} && [3, 6, 8] \\ {}- && [11, 2, 4] \\ \hline && 4\ 4\ 4\ \\ \end{alignat*}[/latex]

[3, 6, 8] and [11, 2, 4] are related by T₄.

Inversion

Inversion, like transposition, is often associated with motion that connects similar objects. The passage in Example 3 from Chen Yi’s Duo Ye (2000) is an example: just as in the transpositionally related passages, these two gestures have the same intervallic content, so our ears recognize them as very similar. Unlike transposition, however, the interval content of these two gestures is not arranged in the same way: both have the same intervals, but the [1, 4, 6] set has the interval 3 on the bottom instead of on the top (Example 4).

Example 3. [2, 4, 7] is inverted to become [1, 4, 6].

General inversion

If you are asked to invert a set and are not given an index number, assume you are inverting the set mod 12. This means taking the complement of each number mod 12. The complement of each integer x mod 12 is the number y that is the difference between x and 12. For example, the complement of 4 is 8: 4+8=12. The complement of 6 is 6: 6+6=12. The complement of 0 is 0: 0+0=0, which is 12 mod 12.

Inverting [2, 4, 7] in this way would yield [5, 8, 10].

I_n: Invert-then-transpose method

Sometimes, sets are inverted and then transposed, as in Example 3. The abbreviation for this is I_n.

In Example 3, the first set [2, 4, 7] is inverted by I₈. To invert a set by I₈, follow this process, in this order:

1. Invert: [2, 4, 7] becomes [5, 8, 10].
2. Transpose: adding 8 to every number in [5, 8, 10] yields [1, 4, 6].

I_n: Subtraction method

You can calculate the new set created by I_n by subtracting all the pitch classes of your first set from n.

What is I₈ of [2, 4, 7]?

[latex]\begin{alignat*}{2} && 8\ 8\ 8\ \\ {}- && [2, 4, 7] \\ \hline && [6, 4, 1] \\ \end{alignat*}[/latex]

I₈ [2, 4, 7] = [1, 4, 6].

Identifying inversions and their index numbers

Any two pitches related by inversion can be added together to form the index number. This makes sense as a logical extension of the subtraction method above: if the inverted pitch y is the result of n–x, then it is also true that n = x + y.

Another way to visualize this is on the clock face. If you have two sets that are 1) both in normal order and 2) related by inversion, the notes within each set will map onto one another in reverse order, as shown in Example 5 below. Write the two sets in normal form on top of one another, then add the opposing integers of each set together as illustrated in Example 6 to yield the index number of the I relation. If the sum of each number pair is 12 or more, subtract 12 so that your n is in mod 12.

Using the Clock Face to Transpose and Invert

If you prefer a more visual method for transposing and inverting, watch the video lesson in Example 7.

Example 7. Using the clock face for transposition and inversion of pitch class sets.

• Blank clock faces (integer notation)
• Blank clock faces (letter names)
• Set Theory Quick Reference Sheet: summarizes the definitions of pitch vs. pitch class, intervals vs. interval classes, and sets vs. set classes.

1. Worksheet on normal form and transformations (.pdf, .docx). Asks students to calculate normal form of various sets, and to calculate T_n/I_n relationships in “Nacht” by Arnold Schoenberg.
2. Composition prep worksheet (.pdf, .docx). Prepares students for the set class composition by asking them to find sets and transformations.